Numbers are the lengths of lists which are the flattenings of trees which are the spannings of graphs. Unlike the first three, graphs have two underlying types of interest –the vertices and the edges– and it is getting to grips with this complexity that we attempt to tackle by considering their ‘algebraic’ counterpart: Categories.
In our exploration of what graphs could possibly be and their relationships to lists are, we shall mechanise, or implement, our claims since there will be many details and it is easy to make mistakes –moreover as a self-learning project, I'd feel more confident to make bold claims when I have a proof assistant checking my work ;-)
Assuming slight familiarity with the Agda programming language, we motivate the need for
basic concepts of category theory with the aim of discussing adjunctions with
a running example of a detailed construction and proof of a free functor.
Moreover, the article contains a host of exercises
whose solutions can be found in the
literate source file. Raw Agda code can be found here.
Since the read time for this article is more than two hours, excluding the interspersed exercises, it may help to occasionally consult a some reference sheets:
Coming from a background in order theory, I love Galois Connections and so our categorical hammer will not be terminal objects nor limits, but rather adjunctions. As such, everything is an adjunction is an apt slogan for us :-)
-- This file has been extracted from https://alhassy.github.io/PathCat/ -- Type checks with Agda version 2.6.0.
( Photo by Mikhail Vasilyev on Unsplash )
Lists give free monoids \(ℒ\, A = (\List\, A, +\!+, [])\) —a monoid \(𝒮 = (S, ⊕, 0_⊕)\) is a triple consisting of a set \(S\) with a binary operation \(⊕\) on it that is associative and has a unit, \(0_⊕\). That it is ‘free’ means that to define a structure-preserving map between monoids \((\List\, A, +\!+, []) \,⟶\, (S, ⊕, 0_⊕)\) it suffices to only provide a map between their carriers \(\List\, A → S\) —freedom means that plain old maps between types freely, at no cost or effort, give rise to maps that preserve monoid structure. Moreover, the converse also holds and in-fact we have a bijection: \[ (ℒ\, A ⟶ 𝒮) \qquad≅\qquad (A ⟶ 𝒰\, 𝒮) \] Where we write \(𝒰\, (S, ⊕, 0_⊕) = S\) for the operation that gives us the 𝒰nderlying carrier of a monoid.
Loosely put, one says we have an ‘adjunction’, written \(ℒ ⊣ 𝒰\).
Observe that natural numbers ℕ ≅ List Unit
are a monoid whose operation is commutative.
By using different kinds of elements A
–and, importantly, still not imposing any equations–
we lose commutativity with List A
.
Then by generalising further to binary trees BinTree A
, we lose associtivity and identity
are are only left with a set and an operation on it —a structure called a ‘magma’.
This is the order that one usually learns about these inductively built structures. One might be curious as to what the next step up is in this hierarchy of generalisations. It is a non-inductive type called a ‘graph’ and in this note we investigate them by comparison to lists. Just as we shifted structures in the hierarchy, we will move to a setting called a ‘category’ —such are more structured than magmas but less restrictive than monoids.
For those who know category theory, this article essentially formalises the often seen phrase “consider the category generated by this diagram, or graph”. Indeed every category is essentially a free category over a graph but with additional equations that ‘confuse’ two paths thereby declaring, e.g., that one edge is the composition of two other edges.
In our exploration of what graphs could possibly be and their relationships to lists are, we shall mechanise or implement our claims since there will be many details and it is easy to make mistakes –moreover as a self-learning project, I'd feel more confident to make bold claims when I have a proof assistant checking my work ;-)
Before reading any further please ingrain into your mind that the Agda keyword
Set
is read “type”! This disparity is a historical accident.
Since the Agda prelude is so simple, the core language doesn’t even come with Booleans or numbers by default —they must be imported from the standard library. This is a pleasant feature. As a result, Agda code tends to begin with a host of imports.
module PathCat where open import Level using (Level) renaming (zero to ℓ₀ ; suc to ℓsuc ; _⊔_ to _⊍_) -- Numbers open import Data.Fin using (Fin ; toℕ ; fromℕ ; fromℕ≤ ; reduce≥ ; inject≤) renaming (_<_ to _f<_ ; zero to fzero ; suc to fsuc) open import Data.Nat open import Relation.Binary using (module DecTotalOrder) open import Data.Nat.Properties using(≤-decTotalOrder ; ≤-refl) open DecTotalOrder Data.Nat.Properties.≤-decTotalOrder -- Z-notation for sums open import Data.Product using (Σ ; proj₁ ; proj₂ ; _×_ ; _,_) Σ∶• : {a b : Level} (A : Set a) (B : A → Set b) → Set (a ⊍ b) Σ∶• = Σ infix -666 Σ∶• syntax Σ∶• A (λ x → B) = Σ x ∶ A • B -- Equalities open import Relation.Binary.PropositionalEquality using (_≗_ ; _≡_) renaming (sym to ≡-sym ; refl to ≡-refl ; trans to _⟨≡≡⟩_ ; cong to ≡-cong ; cong₂ to ≡-cong₂ ; subst to ≡-subst ; subst₂ to ≡-subst₂ ; setoid to ≡-setoid)
Notice that we renamed transitivity to be an infix combinator.
Let us make equational-style proofs available for any type.
module _ {i} {S : Set i} where open import Relation.Binary.EqReasoning (≡-setoid S) public
We intend our proofs to be sequences of formulae interleaved with
justifications for how the formulae are related. At times, the justifications
are by definition and so may be omitted, but we may want to mention them
for presentational –pedagogical?– purposes. Hence, we introduce the
combinator notation lhs ≡⟨" by definition of something "⟩′ rhs
.
open import Agda.Builtin.String defn-chasing : ∀ {i} {A : Set i} (x : A) → String → A → A defn-chasing x reason supposedly-x-again = supposedly-x-again syntax defn-chasing x reason xish = x ≡⟨ reason ⟩′ xish infixl 3 defn-chasing
While we’re making synonyms for readability, let’s make another:
_even-under_ : ∀ {a b} {A : Set a} {B : Set b} {x y} → x ≡ y → (f : A → B) → f x ≡ f y _even-under_ = λ eq f → ≡-cong f eq
Example usage would be to prove
mor G (mor F Id) ≡ mor G Id
directly by ≡-cong (mor G) (id F)
which can be written more clearly as
functor F preserves-identities even-under (mor G)
,
while longer it is also much clearer and easier to read and understand
–self-documenting in some sense.
Interestingly, our first calculational proof is in section 5 when we introduced an large 𝒞𝒶𝓉egory.
What's a graph? Let's motivate categories!
A ‘graph’ is just a parallel-pair of maps,
record Graph₀ : Set₁ where field V : Set E : Set src : E → V tgt : E → V
This of-course captures the usual notion of a set of nodes V
and a set of directed and labelled
edges E
where an edge e
begins at src e
and concludes at tgt e
.
What is good about this definition is that it can be phrased in any category: V
and E
are
any two objects and src, tgt
are a parallel pair of morphisms between them.
How wonderful! We can study the notion of graphs in arbitrary categories!
—This idea will be made clearer when categories and functors are formally introduced.
However, the notion of structure-preserving map between graphs, or ‘graph-maps’ for short, then becomes:
record _𝒢⟶₀_ (G H : Graph₀) : Set₁ where open Graph₀ field vertex : V(G) → V(H) edge : E(G) → E(H) src-preservation : ∀ e → src(H) (edge e) ≡ vertex (src(G) e) tgt-preservation : ∀ e → tgt(H) (edge e) ≡ vertex (tgt(G) e)
( The fancy 𝒢 and ⟶ are obtained in Agda input mode by \McG
and \-->
, respectively. )
This is a bit problematic in that we have two proof obligations and at a first glance it is not at all clear their motivation besides ‘‘structure-preserving’’.
However, our aim is in graphs in usual type theory, and as such we can use a definition that is
equivalent in this domain: A graph is a
type V
of vertices and a ‘type’ v ⟶ v’
of edges for each pair of vertices v
and v’
.
-- ‘small graphs’ , since we are not using levels record Graph : Set₁ where field V : Set _⟶_ : V → V → Set -- i.e., Graph ≈ Σ V ∶ Set • (V → V) -- Graphs are a dependent type!
Now the notion of graph-map, and the meaning of structure-preserving, come to the forefront:
record GraphMap (G H : Graph) : Set₁ where private open Graph using (V) _⟶g_ = Graph._⟶_ G _⟶h_ = Graph._⟶_ H field ver : V(G) → V(H) -- vertex morphism edge : {x y : V(G)} → (x ⟶g y) → (ver x ⟶h ver y) -- arrow preservation open GraphMap
Note edge
essentially says that mor
‘shifts’, or ‘translates’, types
x ⟶g y
into types ver x ⟶h ver y
.
While equivalent, this two-piece definition is preferable over the four-piece one given earlier since it means less proof-obligations and less constructions in general, but the same expressiblity. Yay!
Before we move on, let us give an example of a simple chain-graph. For clarity, we present it in both variations.
-- embedding: j < n ⇒ j < suc n `_ : ∀{n} → Fin n → Fin (suc n) ` j = inject≤ j (≤-step ≤-refl) where open import Data.Nat.Properties using (≤-step)
This' an example of a ‘forgetful functor’, keep reading!
[_]₀ : ℕ → Graph₀ [ n ]₀ = record { V = Fin (suc n) -- ≈ {0, 1, ..., n - 1, n} ; E = Fin n -- ≈ {0, 1, ..., n - 1} ; src = λ j → ` j ; tgt = λ j → fsuc j }
That is, we have n+1
vertices named 0, 1, …, n
and n
edges named 0, 1, …, n-1
with one typing axiom being j : j ⟶ (j+1)
. Alternatively,
[_] : ℕ → Graph [ n ] = record {V = Fin (suc n) ; _⟶_ = λ x y → fsuc x ≡ ` y }
However, we must admit that a slight downside of the typed approach –the two-piece definition– is now we may need to use the following ‘shifting’ combinators: They shift, or slide, the edge types.
-- casting _⟫_ : ∀{x y y’} → (x ⟶ y) → (y ≡ y’) → (x ⟶ y’) e ⟫ ≡-refl = e -- Casting leaves the edge the same, only type information changes ≅-⟫ : ∀{x y y’} {e : x ⟶ y} (y≈y’ : y ≡ y’) → e ≅ (e ⟫ y≈y’) ≅-⟫ ≡-refl = ≅-refl
Such is the cost of using a typed-approach.
Even worse, if we use homogeneous equality then we’d have the ghastly operator
≡-⟫ : ∀{x y y’} {e : x ⟶ y} (y≈y’ : y ≡ y’) → e ⟫ y≈y’ ≡ (≡-subst (λ ω → x ⟶ ω) y≈y’ e)
However, it seems that our development does not even make use of these. Lucky us! However, it is something to be aware of.
A signature consists of ‘sort symbols’ and ‘function symbols’ each of which is associated source-sorts and a target-sort –essentially it is an interface in the programming sense of the word thereby providing a lexicon for a language. A model or algebra of a language is an interpretation of the sort symbols as sets and an interpretation of the function symbols as functions between those sets; i.e., it is an implementation in programming terms. Later you may note that instead of sets and functions we may use the objects and morphisms of a fixed category instead, and so get a model in that category.
Math | ≅ | Coding |
Signature | Interface, record type, class | |
Algebra | Implementation, instance, object | |
Language Term | Syntax | |
Interpretation | Semantics, i.e., an implementation |
Formally, one-sorted signatures are defined:
open import Data.Vec using (Vec) renaming (_∷_ to _,,_ ; [] to nil) -- , already in use for products :/ -- one sorted record Signature : Set where field 𝒩 : ℕ -- How many function symbols there are ar : Vec ℕ 𝒩 -- Their arities: lookup i ar == arity of i-th function symbol open Signature ⦃...⦄ -- 𝒩 now refers to the number of function symbols in a signature
For example, the signature of monoids consists of a single sort symbol C
–which can be
interpreted as the carrier of the monoid– and two function symbols m , u
–which can be interpreted as the monoid multiplication and unit– with source-target
sort lists ((),C) , ((C,C), C)
—some would notate this by u :→ C , m : C × C → C
.
MonSig : Signature MonSig = record { 𝒩 = 2 ; ar = 0 ,, 2 ,, nil } -- unit u : X⁰ → X and multiplication m : X² → X
Generalising on monoids by typing the multiplication we obtain
the signature of categories which consists of three sort symbols O, A, C
–which can be
interepreted as objects, arrows, and composable pairs of arrows– and four function symbols
⨾ , src, tgt, id
with source-target sort lists (C,A) , (A,O) , (A,O) , (O,A)
—notice that only a language of symbols
has been declared without any properties besides those of typing. If we discard C, ⨾, id
we
then obtain the signature of graphs. Without knowing what categories are, we have seen that their
signatures are similar to both the graph and monoid signatures and so expect their logics to
also be similar. Moreover we now have a few slogans,
\[\color{green}{\text{Categories are precisely typed monoids!}}\]
\[\color{green}{\text{Categories are precisely graphs with a monoidal structure!}}\]
\[\color{green}{\text{Categories are precisely coherently constructive lattices!}}\]
( The last one is completely unmotivated from our discussion, but it's a good place for the slogan and will be touched on when we look at examples of categories. )
A signature can be visualised in the plane by associating a dot for each sort symbol and an arrow for each function symbol such that the arrow has a tail from each sort in the associated function symbols source sorts list and the end of the arrow is the target sort of the sort symbol. That is, a signature can be visualised as a hyper-graph.
𝒢
is an interpreation/realisation of the graph’s vertices
as sets and the graph’s edges as functions between said sets.𝒢
is nothing more than a graph morphism
𝒢 ⟶ 𝒮e𝓉
, where 𝒮e𝓉
is the graph with vertices being sets and edges being functions.
Notice that a Graph₀
is precisely a model of the graph • ⇉ •
, which consists of
two vertices and two edges from the first vertex to the second vertex.
We will return to this point ;-)
Before we move on, it is important to note that a signature does not capture any constraints required on the symbols –e.g., a monoid is the monoid signature as well as the constraint that the 2-arity operation is associative and the 0-arity operation is its unit. More generally, a specification consists of a signature declaring symbols of interest, along with a set of sentences over it that denote axioms or constraints. In programming parlance, this is an interface consisting of types and functions that need to be provided and implemented, along with constraints that are usually documented in comments. Unsurprisingly, models of specifications also form categories.
In this section we introduce the notion of a ‘‘poor-man’s category’’ along with the notion of structure preserving transformations and structure preserving transformations between such transformations. The former are called functors and the latter are known as natural transformations and are considered one of the most important pieces of the fundamentals of category theory; as such, we discuss them at length. Afterwards, we relate this section back to our motivating discussion of graphs.
A category, like a monoid, is a a few types and operations for which some equations hold.
However, to discuss equations a notion of equality is needed and rather than enforce one
outright it is best to let it be given. This is a ‘set’ in constructive mathematics:
A type with an E
-quivalence relation on it —also called a setoid or an E
-set.
However, then the structure must have a few added axioms: The operations must be congruences,
i.e., preserve the equivalence relation, and structure-preserving maps must also be congruences.
For our purposes we will use propositional equality and point-wise propositional equality, and as such most of the proofs fall out of the fact that propositional equality is an equivalence. However, this setoid structure becomes a bit of a noise, without providing any real insight for our uses, and the issues of equivalences will be a distraction from the prime focus. Instead, for our two cases where we use point-wise propositional, we will postulate two forms of extensionality. Without question this is not a general approach —then again, our aim is not to develop a library for category theory, which has already been done so elegantly by Kahl who calls it the RATH-Agda project.
module _ where -- An anyonomous module for categorial definitions record Category {i j : Level} : Set (ℓsuc (i ⊍ j)) where infixr 10 _⨾_ field Obj : Set i _⟶_ : Obj → Obj → Set j _⨾_ : ∀ {A B C : Obj} → A ⟶ B → B ⟶ C → A ⟶ C assoc : ∀ {A B C D} {f : A ⟶ B}{g : B ⟶ C} {h : C ⟶ D} → (f ⨾ g) ⨾ h ≡ f ⨾ (g ⨾ h) Id : ∀ {A : Obj} → A ⟶ A leftId : ∀ {A B : Obj} {f : A ⟶ B} → Id ⨾ f ≡ f rightId : ∀ {A B : Obj} {f : A ⟶ B} → f ⨾ Id ≡ f open Category using (Obj) open Category ⦃...⦄ hiding (Obj) -- Some sugar for times when we must specify the category -- “colons associate to the right” ;-) arr = Category._⟶_ syntax arr 𝒞 x y = x ⟶ y ∶ 𝒞 -- “ghost colon” cmp = Category._⨾_ syntax cmp 𝒞 f g = f ⨾ g ∶ 𝒞 -- “ghost colon”
However, similar to nearly everything else in this document, we can leave the setoid approach as an exercise for the reader, which of course has solutions being in the literate source.
Moreover, lest you’re not convinced that my usage of extensionality is at all acceptable, then note that others have used it to simplify their presentations; e.g., Relative monads formalised. Such ‘appeal to authority’ is for the lazy reader who dares not think for him- or her-self, otherwise one ought to read up on the evils of using equality instead of equivalence relations so as to understand when one thing is really another.
The diligent reader may be interested to know that Maarten Fokkinga has written a very gentle introduction to category theory using the calculational approach; I highly recommend it!
Anyhow, in place of strict equality, one uses categorical isomorphism instead.
open Category using (Obj) public record Iso {i} {j} (𝒞 : Category {i} {j}) (A B : Obj 𝒞) : Set j where private instance 𝒞′ : Category ; 𝒞′ = 𝒞 field to : A ⟶ B from : B ⟶ A lid : to ⨾ from ≡ Id rid : from ⨾ to ≡ Id syntax Iso 𝒞 A B = A ≅ B within 𝒞
Interestingly, we shall later encounter a rather large category named 𝒞𝒶𝓉 possessing the special property of being a “2-Category”: It has morphisms between objects, as expected, which are now called “1-morphisms”, and it has morphisms between 1-morphisms, also called “2-morphisms”.
That is, it has morphisms between morphisms.
Above we argued that equality should be deferred in favour of isomorphism at the morphism level. Hence, in a 2-Category, it is only reasonable to defer an equation involving objects to be up to isomorphism of 2-morphisms —this is then called an “equivalence”.
ℒHS ≃ ℛHS ⇔ Σ F ∶ ℒHS ⟶ ℛHS • Σ G ∶ ℛHS ⟶ ℒHS • F ⨾ G ≅ G ⨾ F ≅ Id
Hence when it comes to categories themselves, one usually speaks in terms of equivalences rather than isomorphisms.
For example, let 𝒫𝒶𝓇 be the supercategory of 𝒮e𝓉 with morphisms being ‘partial
functions’ (A ⟶ B) = (A → B + 𝟙)
where the extra element of 𝟙 = { * }
represents
‘undefined’ —also known as the Partial
, Option
, or Maybe
monads. Moreover,
let 𝒫𝒮ℯ𝓉 be the category of sets with an elected point. Then, 𝒫𝒶𝓇 ≃ 𝒫𝒮e𝓉
is
witnessed by (A ⟶ B) ↦ ( (A + 𝟙, *) ⟶ (B + 𝟙, *) )
and conversely
( (A , a) ⟶ (B , b) ) ↦ ( A - a ⟶ B - b)
where
X - x ≡ Σ y ∶ X • ¬(x ≡
y)
. Exercise: Work out the remaining details for the equivalence.
𝒮ℯ𝓉
-tingsLet us give some elementary examples of the notion of a category to exhibit its ubiquity.
The collection of small, say level i
, types and functions between them and usual function composition
with usual identity form a category and this is not at all difficult to see:
instance 𝒮e𝓉 : ∀ {i} → Category {ℓsuc i} {i} -- this is a ‘big’ category 𝒮e𝓉 {i} = record { Obj = Set i ; _⟶_ = λ A B → (A → B) ; _⨾_ = λ f g → (λ x → g (f x)) ; assoc = ≡-refl ; Id = λ x → x ; leftId = ≡-refl ; rightId = ≡-refl }
Sadly, this category is traditionally used to motivate constructions in arbitrary categories and as such people usually think of objects in an arbitrary category as nothing more than sets with extra datum —which is completely false.
For example, the category Div
consists of objects and arrows both being numbers ℕ
and there is an arrow \(k : m → n\) precisely when k × m = n
, so that an arrow is a
constructive witness that \(m\) divides \(n\). Notice that besides ℕ, no sets nor functions
were involved in the making of this useful number-theoretic category.
Recall that a type, or set, is nothing more than a specified collection of values.
Every set is also a category: There is a formal syntactic object associated with each element, the only morphisms are (formal)
identities, and composition is constantly a syntactic identity.
Some define a set to be a category with only identity morphisms; also called a
‘discrete category’ when one wants to distance themself from set theory ;)
—less loosely, a discrete category over a type S
has Obj = S
and (x ⟶ y) = (x ≡ y)
,
where the equivalence is classical, having two possible members true or false.
Discrete categories are quite an important space for hott people … that’s right, attractive people are interested in these things.
Observe that all arrows are invertible! —due to the symmetry of equality. Categories with this property are known as groupoids.
Recall that a monoid (M, ⊕, e)
is a type M
with an associative operation ⊕ : M × M → M
that has a unit e
.
Every monoid is also a category: There is one object, call it ★
, the morphisms are the monoid
elements, and composition is the monoid operation.
—less loosely, for a monoid (M, ⊕, e)
we take Obj = {★} , _⟶_ = M
.
In fact, some would define a monoid to be a one-object category! –I'm looking at you Freyd & Scedrov =)
Recall that a preordered set, or preset, is a type P
with a relation ≤
on
it that satisfies indirect inequality from above:
\[
∀ x , y •\qquad x ≤ y \quad⇔\quad (∀ z •\; y ≤ z ⇒ x ≤ z)
\]
Equivalently, if it satisfies indirect equality from below:
\[ ∀ x , y •\qquad x ≤ y \quad⇔\quad (∀ z •\; z ≤ x ⇒ z ≤ y) \]
If we also have \(∀ x , y •\; x ≤ y \,∧\, y ≤ x \;⇒\; x = y\),
then we say (P, ≤)
is a ‘poset’ or an ‘ordered set’.
Every (pre)ordered set is also a category:
The objects are the elements,
the morphisms are the order-relations,
identities are the relfexitivity of ≤
,
and composition is transitivity of ≤
.
To see this more clearly, suppose you have a group
\((M, ⊕, \_{}⁻¹, e)\) and you define \(x ≤ y \;⇔\; (∃ m : M •\; m ⊕ x = y)\)
then the this is a preorder whose induced category has a morphism
\(m : x → y\) precicely when \(m ⊕ x = y\)
–now sit down and define the remaining categorical structure of this
‘constructive’ preorder associated to the group.
Traditionally, classically, the relation ≤
is precicely a function P × P ⟶ 𝔹 = {true, flase}
and thus there is at-most one morphism between any two objects
–consequently, categories with this property are called poset categories.
In the constructive setting, the relation ≤
is typed P × P → Set
and then
for a preset (P, ≤)
we take Obj = P, _⟶_ = a ≤ b
and insist
on ‘proof-irrelevance’ ∀ {a b} (p q : a ≤ b) → p ≡ q
so that there is at most one morphism
between any two objects.
The restriction is not needed if we were using actual categories-with-setoids since then we would
define morphism equality to be
((a, b, p) ≈ (a’, b’, q) ) = (a ≡ a’ × b ≡ b’)
.
Observe that in the case we have a poset, every isomorphism is an equality: \[ ∀ x, y •\qquad x ≅ y ⇔ x ≡ y \] Categories with this property are called skeletal. Again, hott people like this; so much so, that they want it, more-or-less, to be a foundational axiom!
Poset categories are a wonderful and natural motivator for many constructions and definitions in category theory. This idea is so broad-reaching that it would not be an exaggeration to think of categories as coherently constructive lattices!
Equivalence relations are relations that are symmetric, reflexive, and transitive. Alternatively, they are preorder categories where every morphism is invertible —this is the symmetry property. But categories whose morphisms are invertible are groupoids!
Hence, groupoids can be thought of as generalized equivalence relations. Better yet, as “constructive” equivalence relations: there might be more than one morphism/construction witnessing the equivalence of two items.
Some insist that a true ‘set’ is a type endowed with an equivalence relation, that is a setoid. However, since groupoids generalise equivalence relations, others might insist on a true set to be a "groupoid". However, in the constructive setting of dependent-type theory, these notions coincide!
It’s been said that the aforementioned categories should be consulted whenever one learns a new concept of category theory. Indeed, these examples show that a category is a generalisation of a system of processes, a system of compositionality, and an ordered system.
Now the notion of structure-preserving maps, for categories, is just that of graphs but with attention to the algebraic portions as well.
record Functor {i j k l} (𝒞 : Category {i} {j}) (𝒟 : Category {k} {l}) : Set (ℓsuc (i ⊍ j ⊍ k ⊍ l)) where private instance 𝒞′ : Category ; 𝒞′ = 𝒞 𝒟′ : Category ; 𝒟′ = 𝒟 field -- Usual graph homomorphism structure: An object map, with morphism preservation obj : Obj 𝒞 → Obj 𝒟 mor : ∀{x y : Obj 𝒞} → x ⟶ y → obj x ⟶ obj y -- Interaction with new algebraic structure: Preservation of identities & composition id : ∀{x : Obj 𝒞} → mor (Id {A = x}) ≡ Id -- identities preservation comp : ∀{x y z} {f : x ⟶ y} {g : y ⟶ z} → mor (f ⨾ g) ≡ mor f ⨾ mor g -- Aliases for readability functor_preserves-composition = comp functor_preserves-identities = id open Functor public hiding (id ; comp)
For a functor F
, it is common practice to denote both obj F
and mor F
by F
and this is usually
not an issue since we can use type inference to deduce which is meant. However, in the Agda formalization
we will continue to use the names mor , obj
. Incidentally in the Haskell community, mor
is known as fmap
but we shall avoid that name or risk asymmetry in the definition of
a functor, as is the case in Haskell which turns out to be pragmatically useful.
A functor can be thought of as endowing an object with some form of structure
—since categories are intrinsically structureless in category theory—
and so the morphism component of a functor can be thought of as preserving relations:
f : a ⟶ b ⇒ F f : F a ⟶ F b
can be read as, ‘‘if a
is related to b
(as witnessed by f
)
then their structured images are also related (as witness by F f
)’’.
Recall the category Div
for constructive divisibility relationships ;-)
A functor among monoids –construed as categories– is just a monoid homomorphism; i.e., an identity and multiplication preserving function of the carriers.
(M, ⊕, e) ⟶ (N, ⊗, d) | |
= | Σ h ∶ M → N • ∀ x,y • h(x ⊕ y) = h x ⊗ h y ∧ h e = d |
By induction, h
preserves all finite ⊕-multiplication and, more generally,
functors preserve finite compositions:
\[ F (f₀ ⨾ f₁ ⨾ ⋯ ⨾ fₙ) \;\;=\;\; F\,f₀ ⨾ F\,f₁ ⨾ ⋯ ⨾ F\,fₙ \]
Cool beans :-)
In the same way, sit down and check your understanding!
Two examples of functors from a poset (category) to a monoid (category):
monus : (ℕ, ≤) ⟶ (ℕ,+, 0)
is a functor defined on morphisms by
\[ i ≤ j \quad⇒\quad \mathsf{monus}(i,j) ≔ j - i\]
Then the functor laws become i - i = 0
and (k - j) + (j - i) = k - i
.div : (ℕ⁺, ≤) → (ℚ, ×, 1)
is defined on morphisms by
\[i ≤ j \quad⇒\quad \mathsf{div}(i,j) ≔ j / i\]
The functor laws become i / i = 1
and (k / j) × (j / i) = k / i
.Hey, these two seem alarmingly similar! What gives! Well, they’re both functors from posets to monoids ;) Also, they are instances of ‘residuated po-monoids’. Non-commutative monoids may have not have a general inverse operation, but instead might have left- and right- inverse operations known as residuals —we’ll mention this word again when discussing adjunctions and are categorically seen as kan extensions. Alternatively, they’re are instances of ‘(Kopka) Difference-posets’.
For more examples of categories, we will need to reason by extensionality –two things are ‘equal’ when they have equivalent properties … recall Leibniz and his law of indiscernibles ;p
Categories have objects and morphisms between them, functors are morphisms between categories, and then we can go up another level and consider morphisms between functors. These ‘level 2 morphisms’ are pretty cool, so let’s touch on them briefly.
Recall that a poset ordering is extended to (possibly non-monotone) functions \(f , g\) pointwise \[f \overset{.}{≤} g \quad\iff\quad (∀ x •\; f\, x \,≤\, g\, x)\] As such, with posets as our guide, we extend the notion of ‘morphism between functors’ to be a ‘witness’ of these orderings \(η : ∀ {X} → F\, X ⟶ G\, X\) –this is a dependent type, note that the second arrow notates category morphisms, whereas the first acts as a separator from the implicit parameter \(X\); sometimes one sees \(η_X\) for each component/instance of such an operation.
\(\require{AMScd}\)
\begin{CD} \color{navy}{F\, A} @>\color{fuchsia}{η_A}>> \color{teal}{G\, A} \\ @V\color{navy}{F\, f}VV \!= @VV\color{teal}{G\, f}V \\ \color{navy}{F\, B} @>>\color{fuchsia}{η_B}> \color{teal}{G\, B} \end{CD}
However, then for any morphism \(f : A ⟶ B\) we have two ways to get from \(F\, A\) to \(G\, B\) via
F f ⨾ η {B}
and η {A} ⨾ G f
and rather than choose one or the other, we request that they
are identical —similar to the case of associativity.
NatTrans : ∀ {i j i’ j’} ⦃ 𝒞 : Category {i} {j} ⦄ ⦃ 𝒟 : Category {i’} {j’} ⦄ (F G : Functor 𝒞 𝒟) → Set (j’ ⊍ i ⊍ j) NatTrans ⦃ 𝒞 = 𝒞 ⦄ ⦃ 𝒟 ⦄ F G = Σ η ∶ (∀ {X : Obj 𝒞} → (obj F X) ⟶ (obj G X)) • (∀ {A B} {f : A ⟶ B} → mor F f ⨾ η {B} ≡ η {A} ⨾ mor G f)
The naturality condition is remembered by placing the target component η {B}
after
lifting f
using the source functor F
;
likewise placing the source component before applying the target functor.
Another way to remember it:
η : F ⟶̇ G
starts at F
and ends at G
, so the naturality also starts with F
and ends
with G
; i.e., F f ⨾ η {B} = η {A} ⨾ G f
:-)
It is at this junction that aforementioned problem with our definition of category comes to light: Function equality is extensional by definition and as such we cannot prove it. Right now we have two function-like structures for which we will postulate a form of extensionality,
-- function extensionality postulate extensionality : ∀ {i j} {A : Set i} {B : A → Set j} {f g : (a : A) → B a} → (∀ {a} → f a ≡ g a) → f ≡ g -- functor extensionality postulate funcext : ∀ {i j k l} ⦃ 𝒞 : Category {i} {j} ⦄ ⦃ 𝒟 : Category {k} {l} ⦄ {F G : Functor 𝒞 𝒟} → (oeq : ∀ {o} → obj F o ≡ obj G o) → (∀ {X Y} {f : X ⟶ Y} → mor G f ≡ ≡-subst₂ (Category._⟶_ 𝒟) oeq oeq (mor F f)) → F ≡ G -- graph map extensionality postulate graphmapext : {G H : Graph } {f g : GraphMap G H} → (veq : ∀ {v} → ver f v ≡ ver g v) → (∀ {x y} {e : Graph._⟶_ G x y} → edge g e ≡ ≡-subst₂ (Graph._⟶_ H) veq veq (edge f e)) → f ≡ g -- natural transformation extensionality postulate nattransext : ∀ {i j i’ j’} {𝒞 : Category {i} {j} } {𝒟 : Category {i’} {j’} } {F G : Functor 𝒞 𝒟} (η γ : NatTrans F G) → (∀ {X} → proj₁ η {X} ≡ proj₁ γ {X}) → η ≡ γ
Natural transformations are too cool to end discussing so briefly and so we go on to discuss their usage is mathematics later on.
𝒞𝒶𝓉
With the notions of categories, functors, and extensionality in-hand we can now discus the notion of the category of small categories and the category of small graphs. Afterwards we give another example of a functor that says how every category can be construed as a graph.
First the category of smaller categories,
𝒞𝒶𝓉
is a category of kind(ℓsuc m, ℓsuc m)
, wherem = i ⊍ j
, and its objects are categories of kind(i , j)
and so it is not an object of itself.Thank-you Russel and friends!
( You may proceed to snicker at the paradoxical and size issues encountered by those who use set theory. —Then again, I’ve never actually learned, nor even attempted to learn, any ‘‘formal set theory’’; what I do know of set theory is usually couched in the language of type theory; I heart LADM! )
instance 𝒞𝒶𝓉 : ∀ {i j} → Category {ℓsuc (i ⊍ j)} {ℓsuc (i ⊍ j)} 𝒞𝒶𝓉 {i} {j} = record { Obj = Category {i} {j} ; _⟶_ = Functor ; _⨾_ = λ {𝒞} {𝒟} {ℰ} F G → let instance 𝒞′ : Category ; 𝒞′ = 𝒞 𝒟′ : Category ; 𝒟′ = 𝒟 ℰ′ : Category ; ℰ′ = ℰ in record { obj = obj F ⨾ obj G -- this compositon lives in 𝒮e𝓉 ; mor = mor F ⨾ mor G ; id = λ {x} → begin (mor F ⨾ mor G) (Id ⦃ 𝒞 ⦄ {A = x}) ≡⟨ "definition of function composition" ⟩′ mor G (mor F Id) ≡⟨ functor F preserves-identities even-under (mor G) ⟩ mor G Id ≡⟨ functor G preserves-identities ⟩ Id ∎ ; comp = λ {x y z f g} → begin (mor F ⨾ mor G) (f ⨾ g) ≡⟨ "definition of function composition" ⟩′ mor G (mor F (f ⨾ g)) ≡⟨ functor F preserves-composition even-under mor G ⟩ mor G (mor F f ⨾ mor F g) ≡⟨ functor G preserves-composition ⟩ (mor F ⨾ mor G) f ⨾ (mor F ⨾ mor G) g ∎ } ; assoc = λ {a b c d f g h} → funcext ≡-refl ≡-refl ; Id = record { obj = Id ; mor = Id ; id = ≡-refl ; comp = ≡-refl } ; leftId = funcext ≡-refl ≡-refl ; rightId = funcext ≡-refl ≡-refl }
Some things to note,
functor F preserves-composition even-under mor G
is a real line of code!
It consists of actual function calls and is merely an alias for
≡-cong (mor G) (mor F)
but it sure is far more readable than this form!
We could have written id = ≡-cong (mor G) (id F) ⟨≡≡⟩ id G
,
but this is not terribly clear what is going on.
Especially since we introduced categories not too long ago,
we choose to elaborate the detail.
Likewise, comp = (≡-cong (mor G) (comp F)) ⟨≡≡⟩ (comp G)
.
assoc
is trivial since function composition is, by definition, associative.
Likewise leftId, rightId
hold since functional identity is, by definition, unit of function composition.𝒢𝓇𝒶𝓅𝒽
In a nearly identical way, just ignoring the algebraic datum, we can show that
Graph
's with GraphMap
's form a graph
𝒢𝓇𝒶𝓅𝒽 : Category 𝒢𝓇𝒶𝓅𝒽 = {! exercise !}
𝒞𝒶𝓉
's are 𝒢𝓇𝒶𝓅𝒽
'sForgive and forget: The 𝒰nderlying functor.
Let’s formalise what we meant earlier when we said graphs are categories but ignoring the algebraic data.
Given a category, we ignore the algebraic structure to obtain a graph,
𝒰₀ : Category → Graph 𝒰₀ 𝒞 = record { V = Obj 𝒞 ; _⟶_ = Category._⟶_ 𝒞 }
Likewise, given a functor we ‘forget’ the property that the map of morphisms needs to preserve all finite compositions to obtain a graph map:
𝒰₁ : {𝒞 𝒟 : Category} → 𝒞 ⟶ 𝒟 → 𝒰₀ 𝒞 ⟶ 𝒰₀ 𝒟 𝒰₁ F = record { ver = obj F ; edge = mor F }
This says that 𝒰₁
turns ver, edge
into obj , mor
--𝒰₁ ⨾ ver ≡ obj
and 𝒰₁ ⨾ edge ≡ mor
– reassuring us that 𝒰₁
acts
as a bridge between the graph structures: ver , edge
of graphs and
obj , mor
of categories.
Putting this together, we obtain a functor.
-- Underlying/forgetful functor: Every category is a graph 𝒰 : Functor 𝒞𝒶𝓉 𝒢𝓇𝒶𝓅𝒽 𝒰 = record { obj = 𝒰₀ ; mor = 𝒰₁ ; id = ≡-refl ; comp = ≡-refl }
We forget about the extra algebraic structure of a category and of a functor to
arrive at a graph and graph-map, clearly --≡-refl
– such ‘forgetfullness’ preserves identities
and composition since it does not affect them at all!
Those familiar with category theory may exclaim that just as I have mentioned the names ‘underlying functor’ and ‘forgetful functor’ I ought to mention ‘stripping functor’ as it is just as valid since it brings about connotations of ‘stripping away’ extra structure. I’m assuming the latter is less popular due to its usage for poor mathematical jokes and puns.
Before we move on, the curious might wonder if ‘‘categories are graphs’’ then what is the analgoue to ‘‘\(X\) are hypergraphs’’, it is multicategories.
Now the remainder of these notes is to build-up the material needed to realise the notion of ‘free’ which is, in some sense, the best-possible approximate inverse to ‘forgetful’ –however, forgetting is clearly not invertible since it can easily confuse two categories as the same graph!
Recall, that a natural transformation \(η : F \natTo G\) is a family \(∀ \{X \,:\, \Obj 𝒞\} \,→\, F\, X ⟶ G\, X\) that satisfies the naturality condition: \(∀ \{A \; B\} \{f \,:\, A ⟶ B\} \,→\, F f ⨾ η {B} \;≡\; η {A} ⨾ G f\).
∀{X : Obj 𝒞} → F X → G → X
and invocation η {X}
.\(\require{AMScd}\)
\begin{CD} \color{navy}{F\, A} @>\color{fuchsia}{η_A}>> \color{teal}{G\, A} \\ @V\color{navy}{F\, f}VV \!= @VV\color{teal}{G\, f}V \\ \color{navy}{F\, B} @>>\color{fuchsia}{η_B}> \color{teal}{G\, B} \end{CD}Let us look at this from a few different angles; in particular, what does the adjective ‘natural’ actually mean? It’s been discussed on many forums and we collect a few of the key points here.
Given two functors \(F , G\), for any object \(~x\) we obtain two objects \(F\, x\, , \, G\, x\) and so a morphism
from \(F\) to \(G\) ought to map such \(F\,x\) to \(G\, x\). That is, a morphsim of functors is a family
\(η \,:\, ∀ \{x : \Obj 𝒞\} \,→\, F \,x ⟶ G \,x\). Now for any \(f : a → b\) there are two ways to form a morphism
\(F\, a → G\, b\): \(F f ⨾ η \{b\}\) and \(η \{a\} ⨾ G\, f\). Rather than make a choice each time we want such
a morphism, we eliminate the choice all together by insisting that they are identical.
This is the naturality condition.
This is similar to when we are given three morphisms \(f : a → b , g : b → c , h : c → d\), then there are two ways to form a morphism \(a → d\); namely \((f ⨾ g) ⨾ h\) and \(f ⨾ (g ⨾ h)\). Rather than make a choice each time we want such a morphism, we eliminate the choice all together by insisting that they are identical. This is the associativity condition for categories.
Notice that if there’s no morphism \(F\, x ⟶ G\, x\) for some \(x\), they by definition there’s no possible natural transformation \(F \natTo G\).
That is,
\begin{align*} & \quad \text{it is a natural construction/choice} \\ = & \quad \text{distinct people would arrive at the same construction;} \\ & \quad \text{ (no arbitrary choice or cleverness needed) } \\ = & \quad \text{ there is actually no choice, i.e., only one possiility, } \\ & \quad \text{ and so two people are expected to arrive at the same ‘choice’} \end{align*}Thus, if a construction every involves having to decide between distinct routes, then chances are the result is not formally natural. Sometimes this ‘inution’ is developed from working in a field for some time; sometimes it just “feel”" natural.
Some would even say: Natural = God-given.
A natural transformation can be thought of as a polymorphic function
∀ {X} → F X ⟶ G X
where we restrict ourselves to avoid inspecting any X
.
mono
-morphic operation makes no use of type variables in its signature,
whereas a poly
-morphic operation uses type variables in its signature.is
a subclass of another thereby
obtaining specific information, whereas there is no such mechanism in Haskell.Inspecting type parameters or not leads to the distinction of ad hoc plymorphism vs. parametric polymorphism —the later is the kind of polymorphism employed in functional language like Haskell and friends and so such functions are natural transformations by default! Theorems for free!
For example,
-- Let 𝒦 x y ≔ Id {x} for morphisms, and 𝒦 x y ≔ x for objects. size : ∀ {X} → List X → 𝒦 ℕ X size [x₁, …, xₙ] = n
is a polymorphic function and so naturality follows and is easily shown –show it dear reader!
So we have always have
\[List\; f \;⨾\; size \quad=\quad size\]
Since 𝒦 ℕ f = Id
, then by extensionality: size : List ⟶̇ 𝒦
.
On the other hand, the polymorphic function
whyme : ∀ {X} → List X → 𝒦 Int X whyme {X} [x₁,…,xₙ] = If X = ℕ then 1729 else n
is not natural: The needed equation F f ⨾ η {B} = η {A} ⨾ G f
for any f : A → B
breaks as witnessed by
f = (λ x → 0) : ℝ → ℕ
and any list with length n ≠ 1729
,
and this is easily shown –do so!
One might exclaim, hey! this only works ’cuz you’re using Ramanujan’s taxi-cab number! 1729 is the smallest number expressible as a sum of 2 cubes in 2 ways: \(1729 = 12³ + 1³ = 10³ + 9 ³\). I assure you that this is not the reason that naturality breaks, and I commend you on your keen observation.
Notice that it is natural if we exclude the type inspected, ℕ. That is, if we only consider \(f : A → B\) with \(A ≠ ℕ ≠ B\). In general, is it the case that a transformation can be made natural by excluding the types that were inspected?
Before we move on, observe that a solution in \(h\) to the absorptive-equation \(F f ⨾ h = h\) is precisely a natural transformation from \(F\) to the aforementioned ‘diagonal functor’: \[F f ⨾ h \;=\; h \qquad⇔\qquad ∃ X : Obj \;•\; h ∈ F \overset{.}{⟶} 𝒦 X ~\]
In particular, due to the constant-fusion property \(g \,⨾\, 𝒦\, e \;=\; 𝒦\, e\), we have that \[∀ \{F\} \{X\} \{e \,:\, X\} \;→\; (𝒦\, e) \,∈\, F \overset{.}{⟶} 𝒦\, X \] Is the converse also true? If \(h ∈ F ⟶̇ 𝒦 X\) then \(h \,=\, 𝒦\, e\) for some \(e\)?
The idea that a natural transformation cannot make reference to the type variable at all can be seen by yet another example.
data 𝟙 : Set where ★ : 𝟙 -- Choice function: For any type X, it yields an argument of that type. postulate ε : (X : Set) → X nay : ∀ {X} → X → X nay {X} _ = ε X
Now naturality \(\Id \, f ⨾ nay_B \;=\; nay_A ⨾ \Id \, f\) breaks as witnessed by \(f \;=\; (λ _ → εℕ + 1) \;:\; 𝟙 → ℕ\) –and provided \(εℕ ≠ 0\), otherwise we could use an \(f\) with no fix points.
From this we may hazard the following: If we have natural transformations \(ηᵢ \,:\, ∀ {X : Objᵢ} →\, F X \overset{.}{⟶} G X\) where the \(Objᵢ\) partition the objects available — i.e., \(Obj \;=\; Σ i \,•\, Objᵢ\) — then the transformation \(η_{(i, X)} \;=\; ηᵢ\) is generally unnatural since it clearly makes choices, for each partition.
A family of morphisms is natural in x precisely when it is defined simultaneously for all x —there is no inspection of some particular x here and there, no, it is uniform! With this view, the naturality condition is thought of as a ‘simultaneity’ condition. Rephrasing GToNE.
The idea of naturality as uniformly-definable is pursued by Hodges and Shelah.
Recall that a functor can be thought of as endowing an object with structure. Then a transformation can be thought of as a restructuring operation and naturality means that it doesn’t matter whether we restructure or modify first, as long as we do both.
It may help to think of there’s a natural transformation from F to G to mean there’s an obvious/standard/canconical way to transform F structure into G structure.
Likewise, F is naturally isomorphic to G may be read F is obviously isomorphic to G. For example, TODO seq-pair or pair-seq ;-)
Sometimes we can show ‘‘F X is isomorphic to G X, if we make a choice dependent on X’’ and so the isomorphism is not obvious, since a choice must be made.
F f ⨾ η {B} = η {A} ⨾ G f
from left to right:
Mapping \(f\) over a complicated structure then handling the result
is the same as
handling the complex datum then mapping \(f\) over the result.
Lists give many examples of natural transformations by considering a categorical approach to the theory of lists.
The naturality condition can be seen as a rewrite rule that let’s us replace a complicated or inefficient side with a simplier or more efficient yet equivalent expression. I think I first learned this view of equations at the insistence of Richard Bird and Oege de Moor –whose text can be found here, albeit the legitimacy of the link may be suspect.
For example, recall the 𝒦onstant functor now construed only as a polymorphic binary operation:
_⟪_ : ∀{A B : Set} → A → B → A a ⟪ b = a
The above is a constant time operation, whereas the next two are linear time operations; i.e.,
they take n
steps to compute, where n
is the length of the given list.
-- This' map for List's; i.e., the mor of the List functor map : ∀ {A B : Set} (f : A → B) → List A → List B map f [] = [] map f (x ∷ xs) = f x ∷ map f xs -- Interpret syntax `x₀∷⋯∷xₙ₋₁` semantically as `x₀ ⊕ ⋯ ⊕ xₙ₋₁`, where ⊕ = cons. fold : ∀ {A B : Set} (cons : A → B → B) (nil : B) → List A → B fold cons nil [] = nil fold cons nil (x ∷ xs) = cons x (fold cons nil xs)
By Theorems for Free, or a simple proof, we have that fold
is a natural
transformation \(List \overset{.}{→} \Id\):
\[ List\; f \;⨾\; fold \; cons_B \; nil_B \qquad=\qquad fold \; cons_A \; nil_A \;⨾\; f \]
Note that here we are ranging over objects \(X\) equipped with \(nil_X : X, \; cons_X : X → X → X\);
as such the equation is not valid when this is not the case.
Now we map compute,
postulate A B : Set postulate nil-B : B postulate f : A → B -- possibly expensive operation head : List B → B head = fold _⟪_ nil-B compute : List A → B compute = map f ⨾ head
That is,
compute [x₀, …, xₙ₋₁] = head (map f [x₀, …, xₙ₋₁]) = head [f x₀, …, f xₙ₋₁] = f x₀ ⟪ f x₁ ⟪ ⋯ ⟪ ⋯ f xₙ₋₁ ⟪ nil-B = f x₀
However this approach performs the potentially expensive operation \(f\) a total of
\(n = \text{“length of input”}\) times! In spite of that, it only needs the first element of
the list and performs the operation only once! Indeed, by the naturality of fold
we have
an equivalent, and more efficient, formulation:
compute = head ⨾ f
This operation only performs the potentially costly f
once!
A more concrete and realistic example is to produce an efficient version of the function
that produces the average xs = div (sum xs, length xs)
of a list of numbers: This operation
traverses the input list twice, yet we can keep track of the length as we sum-along the list
to obtain an implementation that traverses the list only once!
div : ℕ × ℕ → ℕ div (0, 0) = 0 div (m, n) = m ÷ n average : List ℕ → ℕ average xs = div (fold _⊕_ 𝟘 xs) where 𝟘 = (0 , 0) _⊕_ : ℕ → (ℕ × ℕ) → ℕ a ⊕ (b , n) = (a + b , n + 1)
Given two functors \(F , G : 𝒞 ⟶ 𝒟\) let us construe them as only graph homomorphisms.
Then each is a model of the graph \(𝒰₀ \; 𝒞\) —each intereprets the nodes and edges of 𝒰₀ 𝒞
as
actual objects and morphisms of 𝒟— and a natrual transformation is then nothing
more than a morphism of models.
In the setting of types and functions, η : F ⟶̇ G
means we have η (F f x) = G f (η x)
which when read left-to-right says that η
is defined by pattern-matching on its argument
to obtain something of the form F f x
then it is defined recursively by examining x
and then
applying G f
to the result —of course there’s some base case F
definitions as well.
Alternatively, the input to η
is of the form F …
and its
output is of the form G …
–mind blown!
For example, I want to define a transformation \(\mathsf{List} ⟶̇ \mathsf{List}\).
[f x₀, f x₁, …, f xₙ₋₁]
–for arbitrary \(f : A → B\).[f y₀, f y₁, …, f yₘ₋₁]
where \(y \,=\, η\,x\).So my only choices are \(y : \List A\) and \(m : ℕ\)
Here are some possibilities and the resulting η:
[]
functionA functor among monoids –construed as categories– is just a monoid homomorphism:
\begin{align*} & (M, ⊕, e) ⟶ (N, ⊗, d) {{{newline}}} ≅ \quad & Σ h ∶ M → N • ∀ \{x \, y \} •\; h(x ⊕ y) = h x ⊗ h y \lands h e = d \end{align*}
A natural transformation (f, prf) ⟶ (g, prf’)
is a point \(n : N\) with
\(∀ x ∶ M \;•\; f x ⊗ n \,=\, n ⊗ g x\), a so-called ‘conjugation’ by \(n\) that takes \(f\) to \(g\).
Recall from the introduction \(𝒰(S, ⊕, e) \;=\; S\) was the underlying functor from monoids to sets.
Let \(𝒰 × 𝒰\) be the functor that for objects \(M \;↦\; 𝒰\, M \,×\, 𝒰\, M\) and for morphisms \(h \;↦\; λ (x,y) → (h\, x, h\, y)\). Then the monoid multiplication (of each monoid) is a natural transformation \(𝒰 × 𝒰 \natTo 𝒰\), where naturality says that for any monoid homomorphism \(h\), the application of \(𝒰\, h\) to the (monoid) multiplication of two elements is the same as the (monoid) multiplication of the \(𝒰\, h\) images of the two elements, and this is evident from the homomorphism condition.
Extending to finite products, \(ℒ \;≔\; (Σ n ∶ ℕ • ∏ i ∶ 1..n • 𝒰)\), the natural transformation
\(ℒ \natTo 𝒰\) is usually called fold, reduce, or cata and ℒ
is known as the
free monoid functor with notations \(A* \;=\; \List A \;=\; ℒ\, A\).
Loosely put,
ℒ₀ : Monoid → Set ℒ₀ M = Σ n ∶ ℕ • ∏ i : 1..n • 𝒰 M -- finite sequences of elements from M ℒ₁ : ∀ {M N : Monoid} → (M ⟶ N) → ℒ₀ M → ℒ₀ N ℒ₁ (h , prf) = λ (n , x₁, …, xₙ) → (n , h x₁ , … , h xₙ) fold : ∀ {M : Monoid} → ℒ₀ M → 𝒰₀ M fold {(M, ⊕, e)} = λ (n , x₁, …, xₙ) → x₁ ⊕ ⋯ ⊕ xₙ
–The reader would pause to consider implementing this formally using Agda's Data.Fin
and Data.Vec
;-)–
Now for any monoid homomorphism h
, applying induction, yields
h₀(x₁ ⊕ ⋯ ⊕ xₙ) = h₀ x₁ ⊕ ⋯ ⊕ h₀ xₙ where h₀ = 𝒰 (h₀, prf) = 𝒰 h
Which is easily seen to be just naturality – if we use backwards composition \(f ⨾ g \;=\; g ∘ f\) –
𝒰 h ∘ fold {M} = fold {N} ∘ ℒ h
Woah!
This is mentioned in the Barr and Wells' Category Theory for Computing Science text, citing Linton, 1969a-b.
For example, src, tgt
—from the graph signature— give natural transformations
\(V \natTo E\) from the vertex functor to the edge functor … keep reading ;)
Recall that \(V(G)\) is essentially \(ℙ₀ ⟶ G\) where
\(ℙₙ\) is the graph of \(n\) edges on \(n+1\) vertices named \(0..n\) with typing \(i \,:\, i-1 ⟶ i\),
which I like to call the path graph of length n; and in particular \(ℙ₀\) is the graph of
just one dot, called 0, and no edges. —Earlier I used the notation [n]
, but I’m using \(ℙ\) since
I like the view point of ℙaths.
What does it mean to say that V(G) is essentially ℙ₀ ⟶ G?
It means that the vertices functor
– \(𝒱 \;:\; 𝒢𝓇𝒶𝓅𝒽 ⟶ 𝒮ℯ𝓉\) that takes objects \(G ↦ V(G)\) and morphisms \(h ↦ \mathsf{ver}\, h\) –
can be ‘represented’ as the Hom functor \((ℙ₀ ⟶ \_{})\), that is to say
\[𝒱 \quad≅\quad (ℙ₀ ⟶ \_{}) \;\mathsf{within \; Func} \; 𝒢𝓇𝒶𝓅𝒽 \; 𝒮ℯ𝓉\]
--Func
-tor categories will be defined in the next section!–
Notice that we arrived at this expression by ‘eta-reducing’ the phrase V(G) is essentially ℙ₀ ⟶ G! ;)
More generally, we have the functor \(ℙₙ ⟶ \_{}\) which yields all paths of length \(n\) for a given graph.
Observe –i.e., show– that we also have an edges functor.
With a notion of morphisms between functors, one is led inexorably to ask whether functors as objects and natural transformations as morphisms constitute a category? They do! However, we leave their definition to the reader —as usual, if the reader is ever so desperate for solutions, they can be found as comments in the unruliness that is the source file.
instance Func : ∀ {i j i’ j’} (𝒞 : Category {i} {j}) (𝒟 : Category {i’} {j’}) → Category _ Func 𝒞 𝒟 = {! exercise !}
This is a good exercise as it will show you that there is an identity functor and that composition of functors is again a functor. Consequently, functors are in abundance: Given any two, we can form [possibly] new ones by composition.
It is a common construction that when a type \(Y\) is endowed with some structure, then we can endow the function space \(X → Y\), where \(X\) is any type, with the same structure and we do so ‘pointwise’. This idea is formalised by functor categories. Alternatively, one can say we have ‘categorified’ the idea; where categorification is the process of replacing types and functions with categories and functors and possibly adding some coherence laws.
There are people who like to make a show about how ‘big’ 𝒞𝒶𝓉 or Func 𝒞 𝓓
are;
these people adhere to something called ‘set theory’ which is essentialy type theory but
ignoring types, loosely put they work only with the datatype
data SET : Set where Elem : ∀ {A : Set} → A → SET
Such heathens delegate types-of-types into ‘classes’ of ‘small’ and ‘big’ sets and it’s not uniform enough for me. Anyhow, such people would say that functor categories ‘‘cannot be constructed (as sets)’’ unless one of the categories involved is ‘‘small’’. Such shenanigans is ignored due to the hierarchy of types we are using :-)
We must admit that at times the usage of a single type, a ‘uni-typed theory’ if you will can be
used when one wants to relise types in an extrinsic fashion rather than think of data as
intrinsically typed –E.g., graphs with src, tgt
then deriving a notion of ‘type’ with _⟶_
.
Everything has its place … nonetheless, I prefer (multi)typed settings!
Let 𝟙 ≔ [ • ]
be the discrete category of one object (and only the identity arrow on it).
Then 𝒞 ≅ Func 𝟙 𝒞
.
Let 𝟚₀ ≔ [• •]
be the discrete category of two objects.
Then the 𝒞-squared category can be defined 𝒞 ⊗ 𝒞 ∶≅ Func 𝟚₀ 𝒞
:
This category essentially consists of pairs of 𝒞-objects with pairs of 𝒞-arrows
between them.
The subscript 0 is commonly used for matters associated with objects and
the name 𝟚₀
is suggestive of the category of 2 objects only.
More generally, if 𝒩 is the discrete category of \(n\) objects, then
the n-fold product category is defined by
(∏ i ∶ 1..n • 𝒞) ∶≅ Func 𝒩 𝒞
.
These are also commonly denoted \(𝒞^2\) and \(𝒞^𝒩\) since they are essentially
products, and more generally Func 𝒳 𝒴
is also denoted 𝒴^{𝒳} and referred.
We can add an arrow to 𝟚₀
to obtain another category…
Let 𝟚 ≔ • ⟶ •
be the category of two objects, call them 0 and 1, with one arrow between them.
Then a functor 𝟚 ⟶ 𝒞
is precisely a morphism of 𝒞 and a natural transformation
f ⟶ g
boils down to just a pair of morphisms (h,k)
with h ⨾ g = f ⨾ k
.
Hence, the arrow category of 𝒞 is \(𝒞^𝟚 \;≅\; 𝒞^→ \;≅\; \mathsf{Func}\, 𝟚 𝒞\); which is essentially the category with objects being 𝒞-morphisms and morphisms being commutative squares.
Notice that a functor can be used to
Likewise, a natural transformation can be used to select a commutative diagram.
It is a common heuristic that when one suspects the possibility of a category 𝒞
, then one
can make probes to discover its structure. The objects are just functors 𝟙 ⟶ 𝒞
and the
morphisms are just functors 𝟚 ⟶ 𝒞
.
The category of presheaves of 𝒞 is the category PSh 𝒞 ≔ Func (𝒞 ᵒᵖ) 𝒮e𝓉
.
This is a pretty awesome category since it allows nearly all constructions in 𝒮ℯ𝓉 to be realised! Such as subsets, truth values, and even powersets! All these extra goodies make it a ‘topos’ aka ‘power allegory’ —the first is a category that has all finite limits and a notion of powerset while the second, besides the power part, looks like a totally different beast; the exhilaration!
The slice category of 𝒞 over B : Obj 𝒞 is the category 𝒞 / B ≔ Σ F ∶ Func 𝟚 𝒞 • (F 1 = B)
.
Essentially it is the category of objects being 𝒞-morphisms with target \(B\) and morphisms \(f ⟶ g\) being \((h,k)\) with \(h ⨾ g = f ⨾ k\) but a natural choice for \(k : B ⟶ B\) is \(\Id_B\) and so we can use morphism type \((f ⟶’ g) \;≔\; Σ h : \src f ⟶ \src g \;•\; h ⨾ g = f\).
This is seen by the observation \[(h, k) \;∈\; f ⟶ g \qquad⇔\qquad h \;∈\; (f ⨾ k) ⟶’ g\] Of course a formal justification is obtained by showing \[\_{}⟶\_{} \quad≅\quad \_{}⟶’\_{} \quad \mathsf{within \; Func }\; (𝒞 ᵒᵖ ⊗ 𝒞) 𝒮e𝓉 \] …which I have not done and so may be spouting gibberish!
Just as the type Σ x ∶ X • P x
can be included in the type X
, by forgetting the second
component, so too the category Σ F ∶ 𝟚 ⟶ 𝒞 • F 1 ≈ B
can be included into the category
𝒞 and we say it is a subcategory of 𝒞.
The notation Σ o ∶ Obj 𝒞 • P o
defines the subcategory of 𝒞 obtained by deleting
all objects not satisfying predicate P
and deleting all morphisms incident to such objects; i.e.,
it is the category 𝒟 with
\[ \Obj 𝒟 \quad≡\quad Σ o ∶ \Obj 𝒞 \,•\, P o
\qquad\text{ and }\qquad
(o , prf) ⟶_𝒟 (o' , prf') \quad≡\quad o ⟶_𝒞 o'
\]
This is the largest/best/universal subcategory of 𝒞 whose objects satisfy \(P\).
Formalise this via a universal property ;)
𝒮e𝓉
are Functor Categories\[ \Func \; S \; 𝒮e𝓉 \qquad≅\qquad 𝒮e𝓉 / S \] Where S in the left is construed as a discrete category and in the right is construed as an object of 𝒮e𝓉.
This is because a functor from a discrete category need only be a function of objects since there are no non-identity morphisms. That is, a functor \(f : S ⟶ 𝒮ℯ𝓉\) is determined by giving a set \(f\,s\) for each element \(s ∈ S\) —since there are no non-identity morphisms. Indeed a functor \(f : S ⟶ Set\) yields an S-targeted function \[ (Σ s ∶ S \,•\, f\, s) → S \quad:\quad λ (s , fs) → s \] Conversely a function \(g : X → S\) yields a functor by sending elements to their pre-image sets: \[ S ⟶ Set \quad:\quad λ s → (Σ x ∶ X \,•\, g\, x ≡ s)\]
Because of this example, 𝒞 / B
can be thought of as ‘𝒞-objects indexed by B’
–extending this idea further leads to fibred categories.
In a similar spirit, we can identify natural transformations as functors! \[\Func \, 𝒞 \, (𝒟 ^ 𝟚) \quad≅\quad (Σ F , G ∶ 𝒞 ⟶ 𝒟 \;•\; \mathsf{NatTrans}\, F\, G)\]
A functor \(N : 𝒞 ⟶ 𝒟 ^ 𝟚\) gives, for each object \(C : \Obj 𝒞\) an object in \(𝒟 ^ 𝟚\) which is precisely an arrow in \(𝒟\), rewrite it as \(N_C : F\,C ⟶ G\,C\) where \(F\,C \,≔\, N\, C\, 0\) and \(G\, C \,≔\, N\, C\, 1\).
Likewise, for each arrow \(f : A ⟶ B\) in 𝒞 we obtain an arrow \(N\, f \,:\, N\, A ⟶ N\, B\) in \(𝒟 ^ 𝟚\) which is precisely a commutative square in 𝒟; that is, a pair of 𝒟-arrows \((F\,f , G\,f) ≔ N\,f\) with \(N_A ⨾ G\,f \;=\; F\,f ⨾ N_B\).
Notice that we have implicitly defined two functors \(F, G : 𝒞 ⟶ 𝒟\). Their object and morphism mappings are clear, but what about functoriality? We prove it for both \(F, G\) together.
Identity:
\begin{calc} (F \,\Id \, , \, G\, \Id) \step{ definition of $F$ and $G$ } N \, \Id \step{ $N$ is a functor } \Id \,∶\, 𝒟 ^ 𝟚 \step{ identity in arrow categories } (\Id , \Id) \end{calc}Composition:
\begin{calc} ( F (f ⨾ g) , G (f ⨾ g) ) \step{ definition of $F$ and $G$ } N\, (f ⨾ g) \step{ $N$ is a functor } N\, f ⨾ N\, g \step{ definition of $F$ and $G$ } (F\, f, G\, f) ⨾ (F\,g , G\,g) \step{ composition in arrow categories } (F\,f ⨾ F\,g , G\,f ⨾ G\,g) \end{calc}Sweet!
Conversely, given a natural transformation \(η : F \overset{.}{⟶} G\) we define a functor \(N : 𝒞 ⟶ 𝒟 ^ 𝟚\) by sending objects \(C\) to \(η_C : F\, C ⟶ G\, C\), which is an object is \(𝒟 ^ 𝟚\), and sending morphisms \(f : A ⟶ B\) to pairs \((G f , F f)\), which is a morphism in \(𝒟 ^ 𝟚\) due to naturality of η; namely \(η_A ⨾ G\, f \;=\; F\, f ⨾ η_B\). It remains to show that \(N\) preserves identities and composition –Exercise!
Now it remains to show that these two processes are inverses and the isomorphism claim is complete. Woah!
Similarly, to show \[ \Func\, (𝟚 ⊗ 𝒞) \, 𝒟 \qquad≅\qquad (Σ F₀ , F₁ ∶ 𝒞 ⟶ 𝒟 • \mathsf{NatTrans}\, F₁ \, F₂)\]
By setting \(H\, i \;=\; Fᵢ\) on objects and likewise for morphisms
but with \(H(\Id, 1) \;=\; η\) where \(1 : 0 ⟶ 1\) is the non-identity arrow of 𝟚
.
(Spoilers! Alternatively: Arr (Func 𝒞 𝒟) ≅ 𝟚 ⟶ 𝒞 ^ 𝒟 ≅ 𝒞 × 𝟚 ⟶ 𝒟
since 𝒞𝒶𝓉
has exponentials,
and so the objects are isomorphic; i.e., natural transformations correspond to functors 𝒞×𝟚⟶𝒟
)
Why are we mentioning this alternative statement? Trivia knowledge of-course!
On a less relevant note, if you’re familiar with the theory of stretching-without-tearing, formally known as topology which is pretty awesome, then you might’ve heard of paths and deformations of paths are known as homotopies which are just continuous functions \(H : X × I ⟶ Y\) for topological spaces $X, Y,$ and \(I \,=\, [0,1]\) being the unit interval in ℝ. Letting \(𝒥 = 𝟚\) be the ‘categorical interval’ we have that functors \(𝒞 × 𝒥 ⟶ 𝒟\) are, by the trivia-relevant result, the same as natural transformations. That is, natural transformations extend the notion of homotopies, or path-deformations.
On mathoverflow, the above is recast succinctly as: A natural transformation from \(F\) to \(G\) is a functor, targeting an arrow category, whose ‘source’ is \(F\) and whose ‘target’ is \(G\). \[ \hspace{-2em} F \overset{.}{⟶} G : 𝒞 ⟶ 𝒟 \quad≅\quad Σ η ∶ 𝒞 ⟶ \mathsf{Arr}\, 𝒟 •\; \mathsf{Src} ∘ η = F \;\;∧\;\; \mathsf{Tgt} ∘ η = G \] Where, the projection functors
\begin{align*} \mathsf{Src}& &:& \mathsf{Arr}\, 𝒟 ⟶ 𝒟 \\ \mathsf{Src}& (A₁ , A₂ , f) &=& A₁ \\ \mathsf{Src}& (f , g , h₁ , h₂ , proofs) &=& h₁ \end{align*}with \(\mathsf{Tgt}\) returning the other indexed items.
We give an example of a functor by building on our existing graphs setup. After showing that graphs correspond to certain functors, we then mention that the notion of graph-map is nothing more than the associated natural transformations!
module graphs-as-functors where
Let us construct our formal graph category, which contains the ingredients for
a graph and a category and nothing more than the equations needed of a category.
The main ingredients of a two-sorted graph are two sort-symbols E, V
, along with
two function-symbols s, t
from E
to V
—this is also called ‘the signature
of graphs’. To make this into a category, we need function-symbols id
and a composition
for which it is a unit.
-- formal objects data 𝒢₀ : Set where E V : 𝒢₀ -- formal arrows data 𝒢₁ : 𝒢₀ → 𝒢₀ → Set where s t : 𝒢₁ E V id : ∀ {o} → 𝒢₁ o o -- (forward) composition fcmp : ∀ {a b c} → 𝒢₁ a b → 𝒢₁ b c → 𝒢₁ a c fcmp f id = f fcmp id f = f
Putting it all together,
instance 𝒢 : Category 𝒢 = record { Obj = 𝒢₀ ; _⟶_ = 𝒢₁ ; _⨾_ = fcmp ; assoc = λ {a b c d f g h} → fcmp-assoc f g h ; Id = id ; leftId = left-id ; rightId = right-id } where -- exercises: prove associativity, left and right unit laws
Now we can show that every graph G
gives rise to a functor: A semantics of 𝒢
in 𝒮e𝓉
.
toFunc : Graph → Functor 𝒢 𝒮e𝓉 toFunc G = record { obj = ⟦_⟧₀ ; mor = ⟦_⟧₁ ; id = ≡-refl ; comp = λ {x y z f g} → fcmp-⨾ {x} {y} {z} {f} {g} } where ⟦_⟧₀ : Obj 𝒢 → Obj 𝒮e𝓉 ⟦ 𝒢₀.V ⟧₀ = Graph.V G ⟦ 𝒢₀.E ⟧₀ = Σ x ∶ Graph.V G • Σ y ∶ Graph.V G • Graph._⟶_ G x y ⟦_⟧₁ : ∀ {x y : Obj 𝒢} → x ⟶ y → (⟦ x ⟧₀ → ⟦ y ⟧₀) ⟦ s ⟧₁ (src , tgt , edg) = src ⟦ t ⟧₁ (src , tgt , edg) = tgt ⟦ id ⟧₁ x = x -- Exercise: fcmp is realised as functional composition fcmp-⨾ : ∀{x y z} {f : 𝒢₁ x y} {g : 𝒢₁ y z} → ⟦ fcmp f g ⟧₁ ≡ ⟦ f ⟧₁ ⨾ ⟦ g ⟧₁
Conversely, every such functor gives a graph whose vertices and edges are the sets
associated with the sort-symbols V
and E
, respectively.
fromFunc : Functor 𝒢 𝒮e𝓉 → Graph fromFunc F = record { V = obj F 𝒢₀.V ; _⟶_ = λ x y → Σ e ∶ obj F 𝒢₀.E • src e ≡ x × tgt e ≡ y -- the type of edges whose source is x and target is y } where tgt src : obj F 𝒢₀.E → obj F 𝒢₀.V src = mor F 𝒢₁.s tgt = mor F 𝒢₁.t
It is to be noted that we can define ‘‘graphs over 𝒞’’ to be the category Func 𝒢 𝒞
.
Some consequences are as follows: Notion of graph in any category, the notion of graph-map
is the specialisation of natural transformation (!), and most importantly, all the power of functor categories
is avaiable for the study of graphs.
In some circles, you may hear people saying an ‘algebra over the signature of graphs’ is an interpretation
domain (𝒞
) and an operation (Functor 𝒢 𝒞
) interpreting the symbols. Nice!
We briefly take a pause to look at the theory of category theory.
In particular, we show a pair of constructions to get new categories from old ones,
interpret these constructions from the view of previously mentioned categories, and
discuss how to define the morphism type _⟶_
on morphisms themselves, thereby
yielding a functor.
The ‘dual’ or ‘opposite’ category 𝒞ᵒᵖ is the category constructed from 𝒞 by reversing arrows: \((A ⟶_{𝒞ᵒᵖ} B) \;≔\; (B ⟶_𝒞 A)\), then necessarily \((f ⨾_{𝒞ᵒᵖ} g) \;≔\; g ⨾_𝒞 f\). A ‘contravariant functor’, or ‘cofunctor’, is a functor F from an opposite category and so there is a reversal of compositions: \(F(f \,⨾\, g) \;=\; F g \,⨾\, F f\).
_ᵒᵖ : ∀ {i j} → Category {i} {j} → Category 𝒞 ᵒᵖ = {! exercise !}
Notice that \((𝒞 ᵒᵖ) ᵒᵖ \;=\; 𝒞\) and \(𝒞 ᵒᵖ \;≅\; 𝒞\) –one may have an intuitive idea of what this isomorphsim means, but formally it is only meaningful in the context of an ambient category; keep reading ;)
We must admit that for categories, the notion of isomorphism is considered less useful than that of equivalence which weakens the condition of the to-from functors being inverses to just being naturally isomorphic to identities; C.f., ‘evil’ above.
Some interpretations:
𝒮e𝓉ᵒᵖ is usual sets and functions but with ‘backwards composition’:
infix 10 _∘_ _∘_ : ∀ {i j } ⦃ 𝒞 : Category {i} {j}⦄ {A B C : Obj 𝒞} → B ⟶ C → A ⟶ B → A ⟶ C f ∘ g = g ⨾ f
Indeed, we have g ⨾ f within 𝒞 = f ∘ g within 𝒞 ᵒᵖ
; which is how these composition operators
are usually related in informal mathematics (without mention of the ambient categories of course).
On a more serious note, the opposite of 𝒮e𝓉 is clearly 𝓉ℯ𝒮 haha —technically for the purposes of this pun we identify the words ‘opposite’ and ‘reverse’.
For a poset (viewed as a category), its opposite is the ‘dual poset’: \((P, ⊑)ᵒᵖ \;=\; (P, ⊒)\).
In particular, the ‘least upper bound’, or ‘supremum’ in \((P, ⊑)\) of two elements \(x,y\) is an element \(s\) with the ‘universal property’: \(∀ z •\; x ⊑ z ∧ y ⊑ z \;≡\; s ⊑ z\). However, switching ⊑ with ⊒ gives us the notion of ‘infimum’, ‘greatest upper bound’! So any theorems about supremums automatically hold for infimums since the infifum is nothing more than the supremum in the dual category of the poset.
It is not difficult to see that this idea of “2 for the price of 1” for theorems holds for all categories.
FinBoolAlg ≃ FinSets ᵒᵖ
, witnessed by considering the collection of
atoms of a Boolean Algebra in one direction and the power set in the other.
Finiteness can be removed at the cost of completeness and atomicitiy,
CompleteAtomicBoolAlg ≃ 𝒮ℯ𝓉 ᵒᵖ
.
Speaking of functors, we can change the type of a functor by ᵒᵖ
-ing its source and target,
while leaving it alone,
-- this only changes type opify : ∀ {i j i’ j’} {𝒞 : Category {i} {j}} {𝒟 : Category {i’} {j’}} → Functor 𝒞 𝒟 → Functor (𝒞 ᵒᵖ) (𝒟 ᵒᵖ) opify F = record { obj = obj F ; mor = mor F ; id = Functor.id F ; comp = Functor.comp F }
Category Theory is the ‘op’ium of the people!
— Karl Marx might say it had cats existed in his time
This two definitions seem to indicate that we have some form of opposite-functor … ;) —keep reading!
opify
seems to show that Functor 𝒞 𝒟 ≡ Functor (𝒞 ᵒᵖ) (𝒟 ᵒᵖ)
, or alternatively a
functor can have ‘two different types’ —this is akin to using the integers as reals
without writing out the inclusion formally, leaving it implicit; however, in the Agda mechanization
everything must be made explicit —the type system doesn’t let you get away with such things.
Professor Maarten Fokkinga has informed me that
the formalization allowing multiple-types is called a
pre-category.
With 𝒞𝒶𝓉
in-hand, we can formalise the opposite, or ∂ual, functor:
∂ : ∀ {i j} → Functor (𝒞𝒶𝓉 {i} {j}) 𝒞𝒶𝓉 ∂ = record { obj = _ᵒᵖ ; mor = opify ; id = ≡-refl ; comp = ≡-refl }
Conjecture: Assuming categories are equipped with a contravariant involutionary functor that is identity on objects, we can show that the identity functor is naturally isomorphic to the opposite functor.
ah-yeah : ∀ {i j} (let Cat = Obj (𝒞𝒶𝓉 {i} {j})) -- identity on objects cofunctor, sometimes denoted _˘ → (dual : ∀ (𝒞 : Cat) {x y : Obj 𝒞} → x ⟶ y ∶ 𝒞 → y ⟶ x ∶ 𝒞) → (Id˘ : ∀ ⦃ 𝒞 : Cat ⦄ {x : Obj 𝒞} → dual 𝒞 Id ≡ Id {A = x}) → (⨾-˘ : ∀ ⦃ 𝒞 : Cat ⦄ {x y z : Obj 𝒞} {f : x ⟶ y} {g : y ⟶ z} → dual 𝒞 (f ⨾ g ∶ 𝒞) ≡ (dual 𝒞 g) ⨾ (dual 𝒞 f) ∶ 𝒞) -- which is involutionary → (˘˘ : ∀ ⦃ 𝒞 : Cat ⦄ {x y : Obj 𝒞} {f : x ⟶ y} → dual 𝒞 (dual 𝒞 f) ≡ f) -- which is respected by other functors → (respect : ⦃ 𝒞 𝒟 : Cat ⦄ {F : 𝒞 ⟶ 𝒟} {x y : Obj 𝒞} {f : x ⟶ y} → mor F (dual 𝒞 f) ≡ dual 𝒟 (mor F f)) -- then → ∂ ≅ Id within Func (𝒞𝒶𝓉 {i} {j}) 𝒞𝒶𝓉
ah-yeah = {! exercise !}
Some things to note.
Categories whose morphisms are all isomorphisms are called ‘groupoids’ —groups are just one-object groupoids. Consequently, restricted to groupoids the opposite functor is naturally isomorphic to the identity functor!
In fact, the group case was the motivator for me to conjecture the theorem, which took a while to prove since I hadn’t
a clue what I needed to assume. Here we’d use a ˘ ≔ a ⁻¹
.
Consider the category Rel
whose objects are sets and whose morphisms are ‘typed-relations’ \((S, R, T)\),
where \(R\) is a relation from set \(S\) to set \(T\), and
composition is just relational composition
—the notion of ‘untyped’, or multi-typed, morphisms is formalized as pre-categories;
see Fokkinga.
Then we can define an endofunctor by taking -˘
to be relational converse: \(x \,(R ˘)\, y \;≡\; y \,R\, x\).
Consequently, restricted to the category Rel
we have that the opposite functor is naturally isomorphic to the identity functor.
The above items are instance of a more general concept, of course.
A category with an involutionary contravariant endofunctor that is the identity on objects
is known as a dagger category, an involutive/star category, or a category with converse
—and the functor is denoted as a superscript suffix by †, *, ˘
, respectively.
The dagger notation probably comes from
the Hilbert space setting while the converse notation comes from the relation algebra setting.
As far as I know, the first two names are more widely known.
A dagger category bridges the gap between arbitrary categories and groupoids.
Just as matrices with matrix multiplication do not form a monoid but rather a category, we have that not all matrices are invertible but they all admit transposition and so we have a dagger category. In the same vein, relations admit converse and so give rise to a category with converse.
Besides relations and groupoids, other examples include:
Spoilers!! Just as the category of categories is carestian closed, so too is the category of dagger
categories and dagger preserving functors –c.f.,the respect
premise above.
For any two categories 𝒞 and 𝒟 we can construct their ‘product’ category \(𝒞 ⊗ 𝒟\) whose objects and morphisms are pairs with components from 𝒞 and 𝒟: \(\Obj\, (𝒞 ⊗ 𝒟) \;\;=\;\; \Obj\, 𝒞 \,×\, \Obj\, 𝒟\) and \((A , X) ⟶_{𝒞 ⊗ 𝒟} (B , Y) \;\;=\;\; (A ⟶_𝒞 B) \,×\, (X ⟶_𝒟 Y)\).
-- we cannot overload symbols in Agda and so using ‘⊗’ in-place of more common ‘×’. _⊗_ : ∀ {i j i’ j’} → Category {i} {j} → Category {i’} {j’} → Category 𝒞 ⊗ 𝒟 = {! exercise !}
Observe that in weaker languages, a category is specified by its objects, morphisms, and composition —the proof obligations are delegated to comments, if they are realized at all. In such settings, one would need to prove that this construction actually produces a full-fledged category. Even worse, this proof may be a distance away in some documentation. With dependent types, our proof obligation is nothing more than another component of the program, a piece of the category type.
In a similar fashion we can show that the sum of two categories is again a category and in general
we have the same for quantified variants: Π 𝒞 ∶ Family • 𝒞
, likewise for ‘Σ’.
For the empty family, the empty sum yields the category 𝟘
with no objects and
the empty product yields the category 𝟙
of one object.
One can then show the usual ‘laws of arithmetic’ —i.e., ×,+ form a commutative monoid, up to isomorphism—
hold in this setting: Letting ★ ∈ {+,×}
, we have
associtivity A ★ (B ★ C) ≅ (A ★ B) ★ C
, symmetry A ★ B ≅ B ★ A
,
unit 𝟙 × A ≅ 𝟘 + A ≅ A
, and zero 𝟘 × A ≅ 𝟘
.
These notions can be defined for any category though the objects may or may not exist
— in 𝒮e𝓉
and 𝒢𝓇𝒶𝓅𝒽
, for example, they do exist ;) —and these associated arithmetical
laws also hold.
Question! What of the distributivity law,
A × (B + C) ≅ (A × B) + (A × C)
, does it hold in the mentioned cases?
Let 𝒫𝒮e𝓉
be the category of sets with a distinguished point, i.e., Σ S : Obj 𝒮e𝓉 • S
, and
functions that preserve the ‘point’, one can then show —if he or she so desires, and is not
lazy— that this category has notions of product and sum but distributivity fails.
Some interpretations:
As expected, we have projections,
Fst : ∀ {i j i’ j’} {𝒞 : Category {i} {j}} {𝒟 : Category {i’} {j’}} → Functor (𝒞 ⊗ 𝒟) 𝒞 Fst = record { obj = proj₁ ; mor = proj₁ ; id = ≡-refl ; comp = ≡-refl } Snd : ∀ {i j i’ j’} {𝒞 : Category {i} {j}} {𝒟 : Category {i’} {j’}} → Functor (𝒞 ⊗ 𝒟) 𝒟 Snd = record { obj = proj₂ ; mor = proj₂ ; id = ≡-refl ; comp = ≡-refl }
For types we have \[ (𝒳 × 𝒴 ⟶ 𝒵) \quad≅\quad (𝒳 ⟶ 𝒵 ^ 𝒴) \quad≅\quad (𝒴 ⟶ 𝒵 ^ 𝒳)\] Since categories are essentially types endowed with nifty structure, we expect it to hold in that context as well.
-- Everyone usually proves currying in the first argument, -- let’s rebel and do so for the second argument curry₂ : ∀ {ix jx iy jy iz jz} {𝒳 : Category {ix} {jx}} {𝒴 : Category {iy} {jy}} {𝒵 : Category {iz} {jz}} → Functor (𝒳 ⊗ 𝒴) 𝒵 → Functor 𝒴 (Func 𝒳 𝒵) curry₂ = {! exercise !}
Just as addition can be extended to number-valued functions pointwise, \(f + g \;≔\; λ x → f x + g x\), we can do the same thing with functors.
-- For bifunctor ‘⊕’ and functors ‘F, G’, we have a functor ‘λ x → F x ⊕ G x’ pointwise : ∀ {ic jc id jd ix jx iy jy} {𝒞 : Category {ic} {jc}} {𝒟 : Category {id} {jd}} {𝒳 : Category {ix} {jx}} {𝒴 : Category {iy} {jy}} → Functor (𝒳 ⊗ 𝒴) 𝒟 → Functor 𝒞 𝒳 → Functor 𝒞 𝒴 → Functor 𝒞 𝒟 pointwise = {! exercise !}
By ‘extensionality’ p ≡ (proj₁ p , proj₂ p)
, we have that the pointwise extension along the projections
is the orginal operation.
exempli-gratia : ∀ {𝒞 𝒟 𝒳 𝒴 : Category {ℓ₀} {ℓ₀}} (⊕ : Functor (𝒳 ⊗ 𝒴) 𝒟) → let _⟨⊕⟩_ = pointwise ⊕ in Fst ⟨⊕⟩ Snd ≡ ⊕ exempli-gratia Bi = funcext (≡-cong (obj Bi) ≡-refl) (≡-cong (mor Bi) ≡-refl)
An example bifunctor is obtained by extending the ‘⟶’ to morphisms:
Given f : A ⟶ B , g : C ⟶ D
we define (f ⟶ g) : (B ⟶ C) → (A ⟶ C)
by
λ h → f ⨾ h ⨾ g
as this is the only way to define it so as to meet the type requirements.
Hom : ∀ {i j} {𝒞 : Category {i} {j} } → Functor (𝒞 ᵒᵖ ⊗ 𝒞) (𝒮e𝓉 {j}) -- hence contravariant in ‘first arg’ and covaraint in ‘second arg’ Hom {𝒞 = 𝒞} = let module 𝒞 = Category 𝒞 instance 𝒞′ : Category ; 𝒞′ = 𝒞 ⨾-cong₂ : ∀ {A B C : Obj 𝒞} {f : A 𝒞.⟶ B} {g g’ : B 𝒞.⟶ C} → g ≡ g’ → f 𝒞.⨾ g ≡ f 𝒞.⨾ g’ ⨾-cong₂ q = ≡-cong₂ 𝒞._⨾_ ≡-refl q in record { obj = λ{ (A , B) → A ⟶ B } ; mor = λ{ (f , g) → λ h → f ⨾ h ⨾ g } ; id = extensionality (λ {h} → begin Id 𝒞.⨾ h 𝒞.⨾ Id ≡⟨ leftId ⟩ h 𝒞.⨾ Id ≡⟨ rightId ⟩ h ∎) ; comp = λ {x y z fg fg’} → let (f , g) = fg ; (f’ , g’) = fg’ in extensionality (λ {h} → begin (f’ 𝒞.⨾ f) 𝒞.⨾ h 𝒞.⨾ (g 𝒞.⨾ g’) ≡⟨ assoc ⟩ f’ 𝒞.⨾ (f 𝒞.⨾ (h 𝒞.⨾ (g 𝒞.⨾ g’))) ≡⟨ ⨾-cong₂ (≡-sym assoc) ⟩ f’ 𝒞.⨾ ((f 𝒞.⨾ h) 𝒞.⨾ (g 𝒞.⨾ g’)) ≡⟨ ⨾-cong₂ (≡-sym assoc) ⟩ f’ 𝒞.⨾ ((f 𝒞.⨾ h) 𝒞.⨾ g) 𝒞.⨾ g’ ≡⟨ ⨾-cong₂ (≡-cong₂ 𝒞._⨾_ assoc ≡-refl) ⟩ f’ 𝒞.⨾ (f 𝒞.⨾ h 𝒞.⨾ g) 𝒞.⨾ g’ ∎) }
The naming probably comes from the algebra/monoid case where the functors are
monoid hom
-omorphisms. Some prefer to use the name Mor
, short for mor
-phisms,
and that’s cool too. While Haskell programmers might call this the Reader
functor.
Usual notation for this functor is Hom
, but I like Fokkinga’s much better.
He uses (_⟶_)
and writes (f ⟶ g) = λ h • f ⨾ h ⨾ g
—the first argument of Hom is the first argument of the composition and the last
argument to Hom is the last argument of the resulting composition :-)
One way is to make it so 𝒮imple that there are obviously no deficiencies, and the other way is to make it so 𝒞omplicated that there are no obvious deficiencies. The first method is far more difficult. It demands the same skill, devotion, insight, and even inspiration as the discovery of the simple physical laws which 𝒰nderlie the complex phenomena of nature.
( The 𝒞omplex philosophy behinds games such as Chess and Go arise from some 𝒮imple board game rules. )
In this section we discuss what it means to be a ‘forgetful functor’? –Also called an `𝒰nderlying functor'.
The modifier ‘forgetful’ is meaningful when there’s a notion of extra structure. Indeed any functor F : 𝒞 ⟶ 𝒮 can be thought of as forgetful by construing the objects of 𝒞 as objects of 𝒮 with extra structure. Mostly: You know it (to be forgetful) when you see it!
A common example from set theory is the ‘inclusion’ of a subset \(A\) of \(B\), the injection \(ι : A ↪ B : a ↦ a\) —it is essentially a form of ‘type casting’: \(a ∈ A\) and \(ι a \;=\; a ∈ B\). Such injections ‘forget’ the property that the argument is actually a member of a specified subset. Indeed, construing sets as categories then functions becomes functors and inclusions are then forgetful functors!
Since a functor F consists of two maps (F₀, F₁) ≔ (obj F, mor F) and some properties, we can speak about properties of the functor and about properties of either of its maps. The common definitions are a functor \(F\) is:
Now we can generalize the previous example. Every faithful functor F : 𝒞 ⟶ 𝒟 can be construed as forgetful: The 𝒞-maps can be embedded into the 𝒟-maps, since F is faithful, and so can be thought of as a special sub-collection of the 𝒟-maps; then \(F\) ‘forgets’ the property of being in this special sub-collection.
Are faithful functors in abundance? Well any functor forgetting only axioms (and/or structure) is faithful:
Now given, \(F (f , prf) = F (g , prf) \;⇔\; f = g \;⇔\; (f , prf) = (g , prf)\) – equality does not (extensionally) depend on proof components.
Hence, faithful :-)
(Likewise for forgetting extra structure).
Of course we’re not saying all forgetful functors are necessarily faithful. A simple counterexample is the absolute value function: Given a real number \(x\) it’s absolute value \(∣x∣\) is obtained by totally ignoring its sign —of course \(x\) and \(∣x∣\) are equidistant from 0, the relation equidistant-from-0 is an equivalence relation –Exercise!–, and so the the two are isomorphic in some sense.
Motivated by this, given a set \(S\) it’s size is denoted \(∣ S ∣\) which totally forgets about the elements of the set —of course it can be shown that two sets are isomorphic precisely if they are equinumerous.
I assume it is with these as motivators, some people write \(∣·∣\) for a forgetful functor.
( Exercise: A functor F : 𝒞 ≃ 𝒟
is (part of) an equivalence iff it is full,
faithful, and ‘‘essentially surjective on objects’’:
∀ D : Obj 𝒟 • Σ C : Obj 𝒞 • F C ≅ D
—note the iso. )
If you’ve ever studied abstract algebra —the math with vector spaces— then you may recall that a collection of vectors ℬ is called a ‘basis’ if every vector can be written as a linear combination of these vectors: For any vector \(v\), there are scalars \(c₁, …, cₙ\) and vectors \(b₁, …, bₙ\) in ℬ with \(v \;=\; c₁·b₁ + ⋯ + cₙ·bₙ\). That is, a basis is a collection of ‘building blocks’ for the vector space. Then any function \(f\) between basis sets immediately lifts to a linear transformation (think vector space morphism) \(F\) as follows: Given a vector \(v\), since we have a basis, we can express it as \(c₁·b₁ + ⋯ + cₙ·bₙ\), now define \(F v \;≔\; c₁·(f\, b₁) + ⋯ + cₙ·(f\, bₙ)\).
Sweet!
Thus, to define a complicated linear transformation of vector spaces, it more than suffices to define a plain old simple function of basis sets. Moreover, by definition, such \(F\) maps basis vectors to basis vectors: \(f \;=\; ι ⨾ F\) where \(ι : ℬ ↪ 𝒱\) is the inclusion that realises basis vectors as just usual vectors in the vector space 𝒱. Slogan: Vector space maps are determined by where they send their basis, and basis-vectors are preserved.
In the case of (List A, ++, [])
we may consider A
to be a ‘basis’ of the monoid —indeed,
every list can be written as a linear combination of elements of A
, given list
[x₁, …, xₙ] : List A
we have [x₁, …, xₙ] = x₁ + ⋯ + xₙ
where x + y ≔ [x] ++ [y]
.
Reasoning similarly as above, or if you have familiarity with foldr , reduce
, we have a slogan:
Monoid homomorphisms from lists are determined by where they send their basis and basis-vectors are preserved.
Now the general case: \(F ⊣ U\) is a (free-forgetful) ‘adjunction’ means for functors ‘forget’ \(U : 𝒞 ⟶ 𝒮\) and ‘free’ \(F : 𝒮 → 𝒞\), we have that for a given 𝒮imple-object \(S\) there’s 𝒮imple-map \(ι : S ⟶_𝒮 U\,(F\, S)\) —a way to realise ‘basis vectors’— such that for any 𝒞omplicated-object \(C\) and 𝒮imple-maps \(φ : S ⟶_𝒮 U\, C\), there is a unique 𝒞omplicated-map \(Φ : F\, S ⟶_𝒞 C\) that preserves the basis vectors: \(φ = ι ⨾ U Φ\).
By analogy to the previous two cases, we may consider \(U\, X\) to be a ‘basis’, and make the slogan: 𝒞omplicated-maps from free objects are determined by where they send their basis and ‘basis vectors’ are preserved.
[ In more categorical lingo, one says \(ι\) is the ‘insertion of generators’.
Question: Does the way we took \(ι\) in the previous graph show that it is a natural transformation \(ι : \Id ⟶ F ⨾ U\)? —The naturality just says that a ‘homomorphism’ \(F f\) on the free object is completely determined by what \(f\) does to the generators ;-) ]
An adjunction \(L ⊣ U\), where the L
-ower adjoint is from 𝒮 to 𝒞 and the U
-pper adjoint is in
the opposite direction, lends itself to an elemntary interpretation if we consider 𝒞
to be some universe of 𝒞omplicated items of study, while 𝒮 to be a universe of 𝒮imple
items of study. Then adjointness implies that given a simple-object \(S\) and a complicated-object
\(C\), a simple-map \(X ⟶ U\, C\) corresponds to a complicated-map \(L\, S ⟶ C\). To work with
complicated-maps it is more than enough to work with simple-maps!
Formally this correspondence, saying \(F : 𝒞 ⟶ 𝒟\) is adjoint to \(G : 𝒟 ⟶ 𝒞\), written \(F ⊣ G\), holds precisely when \((F ∘ X ⟶ Y) \;≅\; (X ⟶ G ∘ Y)\) in a functor category:
_⊣₀_ : ∀ {i j} {𝒞 𝒟 : Category {i} {j}} → Functor 𝒞 𝒟 → Functor 𝒟 𝒞 → Set (i ⊍ j) _⊣₀_ {𝒞 = 𝒞} {𝒟} F G = (F ′ ∘ X ⟶ₙₐₜ Y) ≅ (X ⟶ₙₐₜ G ∘ Y) within Func (𝒞 ᵒᵖ ⊗ 𝒟) 𝒮e𝓉 where X = Fst ; Y = Snd ; _′ = opify -- only changes types infix 5 _⟶ₙₐₜ_ _⟶ₙₐₜ_ : ∀ {i j} {𝒜 : Category {i} {j}} → Functor (𝒞 ᵒᵖ ⊗ 𝒟) (𝒜 ᵒᵖ) → Functor (𝒞 ᵒᵖ ⊗ 𝒟) 𝒜 → Functor (𝒞 ᵒᵖ ⊗ 𝒟) 𝒮e𝓉 _⟶ₙₐₜ_ {i} {j} {𝒜} = pointwise (Hom {i} {j} {𝒜})
Note that if we use Agda's built-in rewrite mechanism to add the rule,
{𝒞 𝒟 : Category {ℓ₀} {ℓ₀}} → Functor (𝒞 ᵒᵖ) (𝒟 ᵒᵖ) ≡ Functor 𝒞 𝒟
then we might be able to get away without using opify
.
Anyhow, this says for any objects \(X\) and \(Y\), the collection of morphisms \((F\, A ⟶ B)\) is isomorphic to the collection \((A ⟶ G\, B)\) and naturally so in \(A\) and \(B\).
Unfolding it, we have
record _⊣_ {i j i’ j’} {𝒞 : Category {i} {j}} {𝒟 : Category {i’} {j’}} (F : Functor 𝒞 𝒟) (G : Functor 𝒟 𝒞) : Set (j’ ⊍ i’ ⊍ j ⊍ i) where open Category 𝒟 renaming (_⨾_ to _⨾₂_) open Category 𝒞 renaming (_⨾_ to _⨾₁_) field -- ‘left-adjunct’ L ≈ ⌊ and ‘right-adjunct’ r ≈ ⌈ ⌊_⌋ : ∀ {X Y} → obj F X ⟶ Y ∶ 𝒟 → X ⟶ obj G Y ∶ 𝒞 ⌈_⌉ : ∀ {X Y} → X ⟶ obj G Y ∶ 𝒞 → obj F X ⟶ Y ∶ 𝒟 -- Adjuncts are inverse operations lid : ∀ {X Y} {d : obj F X ⟶ Y ∶ 𝒟} → ⌈ ⌊ d ⌋ ⌉ ≡ d rid : ∀ {X Y} {c : X ⟶ obj G Y ∶ 𝒞} → ⌊ ⌈ c ⌉ ⌋ ≡ c -- That for a fixed argument, are natural transformations between Hom functors lfusion : ∀ {A B C D} {f : A ⟶ B ∶ 𝒞} {ψ : obj F B ⟶ C ∶ 𝒟} {g : C ⟶ D ∶ 𝒟} → ⌊ mor F f ⨾₂ ψ ⨾₂ g ⌋ ≡ f ⨾₁ ⌊ ψ ⌋ ⨾₁ mor G g rfusion : ∀ {A B C D} {f : A ⟶ B ∶ 𝒞} {ψ : B ⟶ obj G C ∶ 𝒞} {g : C ⟶ D ∶ 𝒟} → ⌈ f ⨾₁ ψ ⨾₁ mor G g ⌉ ≡ mor F f ⨾₂ ⌈ ψ ⌉ ⨾₂ g
This is easier for verifying an adjunction, while the former is easier for remembering and understanding what an adjunction actually is.
As the slogan goes ‘adjunctions are everywhere’. They can be said to capture the notions of optimization and efficiency, but also that of simplicity.
For example, the supremum of a function is defined to be an upper bound of its image set and the least such bound. Formally, this definition carries a few quantifiers and so a bit lengthy. More elegantly, we can say the supremum operation is left-adjoint to the constant function: \[ \mathsf{sup} ⊣ 𝒦 \] which means \[ ∀ z •\qquad \mathsf{sup}\, f \,≤\, z \quad⇔\quad f \overset{.}{≤} 𝒦\, z\] Where \(𝒦\, x\, y \,=\, x\) and the \(\overset{.}{≤}\) on the right is the point-wise ordering on functions. This formulation of supremum is not only shorter to write but easier to use in calculational proofs.
For the efficiency bit, recall that it is efficient to specify a 𝒮imple-map, then use the adjuction, to obtain a 𝒞omplicated-map. Recall in the last paragraph how we define the super complicated notion of supremum of a function in terms of the most elementary constant function!
Adjunctions over poset categories are called ‘Galois connections’ and a good wealth of material on them can be found in nearly any writing by Backhouse et. al., while a very accessible introduction is by Aarts, and there is also an Agda mechanisation by Kahl & Al-hassy.
Regarding forgetful functors: Generally, but not always, forgetful functors are faithful and have left adjoints —because the notion of ‘forget’ ought to have a corresponding notion of ‘free’. An exception to this is the category of fields, which has a forgetful functor to the category of sets with no left adjoint.
Another awesome thing about adjunctions L ⊣ U
is that they give us ‘representable functors’,
a.k.a. ‘the best kind of functors’, when terminal objects exist.
𝟙
is ‘terminal’ if for any object A
there is a unique morphism ! {A} : A ⟶ 𝟙
.
In 𝒮ℯ𝓉 we have (A ⟶ 𝟙) ≅ 𝟙
and (𝟙 ⟶ A) ≅ A
.U : 𝒞 ⟶ 𝒮e𝓉
, to
a given set A
and 𝟙
we obtain (L 𝟙 ⟶ A) ≅ (𝟙 ⟶ U A) ≅ U A
and so one says
‘ U
is represented by L 𝟙
’.A
then it suffices to utilise the maps L 𝟙 ⟶ A
.
In the case of a free-forgetful adjunction, this says that
a forgetful functor is represented by the free object with generator 𝟙
.
For example, for monoids the one-point monoid is the terminal object: 𝟙 ≔ ({*}, ⊕, *)
with x ⊕ y ≔ ⋆
.
Then every monoid-homomorphism from 𝟙
just picks out an element of the carrier of a monoid and so
(𝟙 ⟶ M) ≅ 𝒰 M
where 𝒰
is the forgetful functor for monoids mentioned in the introduction.
A final note about ‘free objects’ —arising from an adjoint to a forgetful functor.
‘‘The free object is generic’’: The only truths provable for the free object are precisely those that hold for every complicated-object.
(Begin squinting eyes)
This follows from the
definition of adjunction which says we can construct a unique morphism between complicated-objects
from a simple-map and by naturality we may transport any proof for the free object to any
complicated object.
(Feel ‘free’ to stop squinting your eyes)
For futher reading consider reading the adjoint article at the comic book library and for more on the adjective ‘forgetful’ see ncatlab or mathworld A nice list of common free objects can be found on wikipedia.
You might be asking, musa, when am I ever going to encounter this in daily life? In a popular setting? This concept is everywhere, even inclusions as mentioned earlier are an instance. For the second question, enjoy listening to this lovely musical group –they use the words ‘forgetful functors’ ;)
The remainder of this document can be seen as one fully-worked out example of constructing a free functor for the forgetful 𝒰 defined above from 𝒞𝒶𝓉 to 𝒢𝓇𝒶𝓅𝒽.
The “right” definition is hard to obtain!
We can now define a ‘path’ of length n
in a graph G
to be a graph-map
[ n ] ⟶ G
.
Path₀ : ℕ → Graph₀ → Set (ℓsuc ℓ₀) Path₀ n G = [ n ]₀ 𝒢⟶₀ G
Unfolding the definition of graph-morphisms, this just says that a path of length n
consists of a sequence [v₀, v₁, v₂, …, vₙ]
of vertices of G
and a sequence [e₀, e₁, …, eₙ₋₁]
of edges of G
with typing eᵢ : vᵢ ⟶ vᵢ₊₁
.
The definition is pretty slick! However, as the name suggests, perhaps we can concatenate paths and it’s not at all clear how to do this for the vertex- and edge- morphisms of the graph-maps involved, whereas it’s immediately clear how to do this with sequences: We just concatenate the sequences and ensure the result is coherent.
Since the vertices can be obtained from the edges via src
and tgt
, we can dispense with them
and use the definition as outlined above.
open import Data.Vec using (Vec ; lookup) record Path₁ (n : ℕ) (G : Graph₀) : Set (ℓsuc ℓ₀) where open Graph₀ field edges : Vec (E G) (suc n) coherency : {i : Fin n} → tgt G (lookup (` i) edges) ≡ src G (lookup (fsuc i) edges)
That is, edges [e₀, …, eₙ]
with coherency tgt eᵢ ≡ src eᵢ₊₁
.
Great, we’ve cut the definition of Path₀
in half but that fact that we get a raw list of edges
and then need coherency to ensure that it is a well-formed path is still not terribly lovely.
After all, we’re in Agda, we’re among kings, we should be able to form the list in such a way that
the end result is a path. Let’s do that!
Enough of this repetition, let us fix a graph G
,
module Path-definition-2 (G : Graph₀) where open Graph₀ G mutual data Path₂ : Set where _! : V → Path₂ cons : (v : V) (e : E) (ps : Path₂) (s : v ≡ src e) (t : tgt e ≡ head₂ ps) → Path₂ head₂ : Path₂ → V head₂ (v !) = v head₂ (cons v e p s t) = v
Defining paths for the parallel-pair approach to graphs leaves us with the need to carry proofs around, and this is a tad too clunky in this case. Let's try yet again.
module Path-definition-3 (G : Graph) where open Graph G -- handy dandy syntax infixr 5 cons syntax cons v ps e = v ⟶[ e ]⟶ ps -- v goes, by e, onto path ps -- we want well-formed paths mutual data Path₃ : Set where _! : (v : V) → Path₃ cons : (v : V) (ps : Path₃) (e : v ⟶ head₃ ps) → Path₃ head₃ : Path₃ → V head₃ (v !) = v head₃ (v ⟶[ e ]⟶ ps) = v -- motivation for the syntax declaration above example : (v₁ v₂ v₃ : V) (e₁ : v₁ ⟶ v₂) (e₂ : v₂ ⟶ v₃) → Path₃ example v₁ v₂ v₃ e₁ e₂ = v₁ ⟶[ e₁ ]⟶ v₂ ⟶[ e₂ ]⟶ v₃ ! end₃ : Path₃ → V end₃ (v !) = v end₃ (v ⟶[ e ]⟶ ps) = end₃ ps -- typed paths; squigarrowright record _⇝_ (x y : V) : Set where field path : Path₃ start : head₃ path ≡ x finish : end₃ path ≡ y
This seems great, but there’s always room for improvement:
Since the cons
constructor's third argument depends on its first, we must
use a syntax declaration to get the desired look. Such aesthetic is not only
pleasing but reminiscent of diagrammatic paths;
moreover, it’s guaranteed to be an actual path and not just an
alternating lists of vertices and edges.
Using the clunky Path₂
, we’d write
v₁ ⟶[ v₁≈se₁ , e₁ , te₁≈v₂ ]⟶ v₂ ⟶[ v₂≈se₂ , e₂ , te₂≈v₃ ]⟶ v₃ ! where syntax cons v e ps s t = v ⟶[ s , e , t ]⟶ ps
yuck!
Finally, the syntax-declaration does not make the emacs agda-mode auto-case using the syntax, and so I have to write it out by hand, each time I want to use the syntax.
cons
's third argument depends on the second argument, we need a mutual
definition to extract the item of the dependence. Perhaps if we embed this item at
the type level we may avoid the need of an auxiliary mutually-defined function.Graph₀
, which we argued is less preferable to the typed-approach to graphs.
Perhaps defining paths with types by default, we can reap the benefits and simplicity
of the typed-approach to graphs.module TypedPaths (G : Graph) where open Graph G hiding(V) open Graph using (V) data _⇝_ : V G → V G → Set where _! : ∀ x → x ⇝ x _⟶[_]⟶_ : ∀ x {y ω} (e : x ⟶ y) (ps : y ⇝ ω) → x ⇝ ω
One might think that since we can write
src : {x y : V G} (e : x ⟶ y) → V G src {x} {y} e = x
we can again ignore vertices and it suffices to just keep a coherent list of edges. Then what is an empty path at a vertex? This’ enough to keep vertices around —moreover, the ensuing terms look like diagrammatic paths! Cool!
Finding this definitional form was a major hurdle in this endeavour.
With paths in hand, we can now consider a neat sequence of exercises :-)
Show that graphmaps preserve paths: (f : G ⟶ H) → x ⇝ y → fᵥ x ⇝ fᵥ y
;
this is nothing more than type-preservation for f
to be a functor 𝒫G ⟶ 𝒫H
;)
Hint: This is lift
from the next section.
Define
a connected b ≡ (a ⇝ b) ⊎ (b ⇝ a) -- path “between” a and b; not ‘from a to b’.
𝒦G
.𝒞
, define 𝒦 𝒞 ≔ 𝒦 (𝒰₀ 𝒞)
which is a subcategory of 𝒞
.𝒦f : 𝒦G ⟶ 𝒦H : (connected component of x) ↦ (connected component of fᵥ x)
.𝒦 : Graph ⟶ Set
.
Then there is a natural transformation 𝒱 ⟶ 𝒦
, where 𝒱 is the vertices functor.
Hint: Such a transformation means we can realise vertices as connected components and this suggests taking assigning a vertex to the connected-component block that contains it.
yeah!
Finally, if we let 𝒟 : 𝒮ℯ𝓉 → 𝒞𝒶𝓉
be the free category functor that associates each set with
the discrete category over it, then we have 𝒦
is the associated forgetful functor.
Here's a handy-dandy combinator for forming certain equality proofs of paths.
-- Preprend preserves path equality ⟶-≡ : ∀{x y ω} {e : x ⟶ y} {ps qs : y ⇝ ω} → ps ≡ qs → (x ⟶[ e ]⟶ ps) ≡ (x ⟶[ e ]⟶ qs) ⟶-≡ {x} {y} {ω} {e} {ps} {qs} eq = ≡-cong (λ r → x ⟶[ e ]⟶ r) eq
Less usefully, we leave as exercises:
edges : ∀ {x ω} (p : x ⇝ ω) → List (Σ s ∶ V G • Σ t ∶ V G • s ⟶ t) edges = {! exercise !} path-eq : ∀ {x y} {p q : x ⇝ y} → edges p ≡ edges q → p ≡ q path-eq = {! exercise !}
Given time, path-eq
could be rewritten so as to be more easily applicable.
For now, two path equality proofs occur in the document and both are realised by
quick-and-easy induction.
Now we turn back to the problem of catenating two paths.
infixr 5 _++_ _++_ : ∀{x y z} → x ⇝ y → y ⇝ z → x ⇝ z x ! ++ q = q -- left unit (x ⟶[ e ]⟶ p) ++ q = x ⟶[ e ]⟶ (p ++ q) -- mutual-associativity
Notice that the the base case indicate that !
forms a left-unit for ++
,
while the inductive case says that path-formation associates with path catenation.
Both observations also hold for the definition of list catenation ;-)
If we had not typed our paths, as in Path₂
, we would need to carry around a
proof that paths are compatible for concatenation:
catenate : (p q : Path) (coh : end p ≡ head q) → Path syntax catenate p q compatibility = p ++[ compatibility ] q
Even worse, to show p ++[ coh ] q ≡ p ++[ coh’ ] q
we need to invoke proof-irrelevance of
identity proofs to obtain coh ≡ coh’
, each time we want such an equality! Moving the proof
obligation to the type level removes this need.
As can be seen, being type-less is a terrible ordeal.
Just as the first clause of _++_
indicates _!
is a left unit,
leftId : ∀ {x y} {p : x ⇝ y} → x ! ++ p ≡ p leftId = ≡-refl
Is it also a right identity?
rightId : ∀ {x y} {p : x ⇝ y} → p ++ y ! ≡ p rightId {x} {.x} {.x !} = ≡-refl rightId {x} {y } {.x ⟶[ e ]⟶ ps} = ≡-cong (λ q → x ⟶[ e ]⟶ q) rightId
Is this operation associative?
assoc : ∀{x y z ω} {p : x ⇝ y} {q : y ⇝ z} {r : z ⇝ ω} → (p ++ q) ++ r ≡ p ++ (q ++ r) assoc {x} {p = .x !} = ≡-refl assoc {x} {p = .x ⟶[ e ]⟶ ps} {q} {r} = ≡-cong (λ s → x ⟶[ e ]⟶ s) (assoc {p = ps})
Hence, we’ve shown that the paths over a graph G
constitute a category! Let’s call it 𝒫 G
.
In the last section, we showed that the paths over a graph make a category,
𝒫₀ : Graph → Category 𝒫₀ G = let open TypedPaths G in record { Obj = Graph.V G ; _⟶_ = _⇝_ ; _⨾_ = _++_ ; assoc = λ {x y z ω p q r} → assoc {p = p} ; Id = λ {x} → x ! ; leftId = leftId ; rightId = rightId }
Can we make 𝒫
into a functor by defining it on morphisms?
That is, to lift graph-maps to category-maps, i.e., functors.
𝒫₁ : ∀ {G H} → GraphMap G H → Functor (𝒫₀ G) (𝒫₀ H) 𝒫₁ {G} {H} f = record { obj = ver f ; mor = amore ; id = ≡-refl ; comp = λ {x} {y} {z} {p} → comp {p = p} } where open TypedPaths ⦃...⦄ public instance G' : Graph ; G' = G H' : Graph ; H' = H amore : {x y : Graph.V G} → x ⇝ y → (ver f x) ⇝ (ver f y) amore (x !) = ver f x ! amore (x ⟶[ e ]⟶ p) = ver f x ⟶[ edge f e ]⟶ amore p comp : {x y z : Graph.V G} {p : x ⇝ y} {q : y ⇝ z} → amore (p ++ q) ≡ amore p ++ amore q comp {x} {p = .x !} = ≡-refl -- since ! is left unit of ++ comp {x} {p = .x ⟶[ e ]⟶ ps} = ⟶-≡ (comp {p = ps})
Sweet!
With these two together, we have that 𝒫
is a functor.
𝒫 : Functor 𝒢𝓇𝒶𝓅𝒽 𝒞𝒶𝓉 𝒫 = record { obj = 𝒫₀ ; mor = 𝒫₁ ; id = λ {G} → funcext ≡-refl (id ⦃ G ⦄) ; comp = funcext ≡-refl comp } where open TypedPaths ⦃...⦄ open Category ⦃...⦄ module 𝒞𝒶𝓉 = Category 𝒞𝒶𝓉 module 𝒢𝓇𝒶𝓅𝒽 = Category 𝒢𝓇𝒶𝓅𝒽 id : ∀ ⦃ G ⦄ {x y} {p : x ⇝ y} → mor (𝒞𝒶𝓉.Id {𝒫₀ G}) p ≡ mor (𝒫₁ (𝒢𝓇𝒶𝓅𝒽.Id)) p id {p = x !} = ≡-refl id {p = x ⟶[ e ]⟶ ps} = ⟶-≡ (id {p = ps}) comp : {G H K : Graph} {f : GraphMap G H} {g : GraphMap H K} → {x y : Graph.V G} {p : TypedPaths._⇝_ G x y} → mor (𝒫₁ f 𝒞𝒶𝓉.⨾ 𝒫₁ g) p ≡ mor (𝒫₁ (f 𝒢𝓇𝒶𝓅𝒽.⨾ g)) p comp {p = x !} = ≡-refl comp {p = x ⟶[ e ]⟶ ps} = ⟶-≡ (comp {p = ps})
It seemed prudent in this case to explicitly delimit where the compositions lives —this is for clarity, since Agda can quickly resolve the appropriate category instances.
Exercise: Show that we have a natural transformation Id ⟶ 𝒰 ∘ 𝒫
.
Free at last, free at last, thank God almighty we are free at last.
– Martin Luther King Jr.
Recall why lists give the ‘free monoid’: We can embed a type \(A\) into \(\List A\) by the map \([\_{}]\), and we can lift any map \(f : A ⟶ B\) to a monoid map \[\foldr \; (λ a b → f\, a ⊕ b)\; e \;:\; (\List A ,\_{}++\_{} , []) \,⟶\, (B,\_{}⊕\_{} , e)\] I.e., \([a₁, …, aₖ] \;↦\; f\, a₁ ⊕ ⋯ ⊕ f\, aₖ\). Moreover this ‘preserves the basis’ \(A\) – i.e., \(∀ a •\; f\, a \,=\, \foldr \,f \,e \, [ a ]\) – and this lifted map is unique.
Likewise, let us show that \(𝒫G\) is the ‘free category’ over the graph \(G\). This amounts to saying that there is a way, a graph-map, say \(ι\), that embeds \(G\) into \(𝒫G\), and a way to lift any graph-map \(f \,:\, G \,𝒢⟶\, 𝒰₀ 𝒞\) to a functor \(\mathsf{lift}\, f : 𝒫G ⟶ 𝒞\) that ‘preserves the basis’ \(f \;=\; ι ⨾ 𝒰₁ (\mathsf{lift}\, f)\) and uniquely so.
Let’s begin!
module freedom (G : Obj 𝒢𝓇𝒶𝓅𝒽) {𝒞 : Category {ℓ₀} {ℓ₀} } where open TypedPaths G using (_! ; _⟶[_]⟶_ ; _⇝_ ; _++_) open Category ⦃...⦄ module 𝒢𝓇𝒶𝓅𝒽 = Category 𝒢𝓇𝒶𝓅𝒽 module 𝒮ℯ𝓉 = Category (𝒮e𝓉 {ℓ₀}) module 𝒞 = Category 𝒞 instance 𝒞′ : Category ; 𝒞′ = 𝒞
The only obvious, and most natural, way to embed a graph into its ‘graph of paths’ is to send vertices to vertices and edges to paths of length 1.
ι : G ⟶ 𝒰₀ (𝒫₀ G) ι = record { ver = Id ; edge = λ {x} {y} e → x ⟶[ e ]⟶ (y !) }
Given a graph map \(f\), following the list-analagoue of \([a₁, …, aₖ] \;↦\; f\, a₁ ⊕ ⋯ ⊕ f\, aₖ\) we attempt to lift the map onto paths by taking the edges \(e₁, …, eₖ\) of a path to a morphism \(\edge\, f\, e₁ ⨾ ⋯ ⨾ \edge\, f\, eₖ\). That is, a path of the form \[x_0 \xrightarrow{e_1} x_1 \xrightarrow{e_2} x_2 \xrightarrow{e_3} ⋯ \xrightarrow{e_k} x_k \] Is lifted to the composition of morphisms \[\mathsf{ver}\, f\, x_0 \xrightarrow{\edge\, f\, e_1} \mathsf{ver}\, f\, x_1 \xrightarrow{\edge\, f\, e_2} \mathsf{ver}\, f\, x_2 \xrightarrow{\edge\, f\, e_3} ⋯ \xrightarrow{\edge\, f\, e_k} \mathsf{ver}\, f\, x_k \]
Of course, we then need to verify that this construction is indeed functorial.
lift : G ⟶ 𝒰₀ 𝒞 → 𝒫₀ G ⟶ 𝒞 lift f = record { obj = λ v → ver f v -- Only way to obtain an object of 𝒞; hope it works! ; mor = fmap ; id = ≡-refl ; comp = λ {x y z p q} → proof {x} {y} {z} {p} {q} } where fmap : ∀ {x y} → x ⇝ y → ver f x 𝒞.⟶ ver f y fmap (y !) = 𝒞.Id fmap (x ⟶[ e ]⟶ p) = edge f e 𝒞.⨾ fmap p -- homomorphism property proof : ∀{x y z} {p : x ⇝ y} {q : y ⇝ z} → fmap (p ++ q) ≡ fmap p 𝒞.⨾ fmap q proof {p = ._ !} = ≡-sym 𝒞.leftId proof {p = ._ ⟶[ e ]⟶ ps} = ≡-cong (λ m → edge f e 𝒞.⨾ m) (proof {p = ps}) ⟨≡≡⟩ ≡-sym assoc -- Exercise: Rewrite this calculationally!
Now we have the embedding and the lifting, it remains to show that the aforementioned ‘preserves basis’ property holds as does uniqueness.
Let's begin with the ‘basis preservation’ property:
property : ∀{f : G ⟶ 𝒰₀ 𝒞} → f ≡ (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ (lift f)) property {f} = graphmapext -- Proving: ∀ {v} → ver f v ≡ ver (ι 𝒞.⨾ 𝒰₁ (lift f)) v -- by starting at the complicated side and simplifying (λ {v} → ≡-sym (begin ver (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ (lift f)) v ≡⟨" definition of ver on composition "⟩′ (ver ι 𝒮ℯ𝓉.⨾ ver (𝒰₁ (lift f))) v ≡⟨" definition of 𝒰₁ says: ver (𝒰₁ F) = obj F "⟩′ (ver ι 𝒮ℯ𝓉.⨾ obj (lift f)) v ≡⟨" definition of lift says: obj (lift f) = ver f "⟩′ (ver ι 𝒮ℯ𝓉.⨾ ver f) v ≡⟨" definition of ι on vertices is identity "⟩′ ver f v ∎)) -- Proving: edge (ι ⨾g 𝒰₁ (lift f)) e ≡ edge f e (λ {x} {y} {e} → begin edge (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ (lift f)) e ≡⟨" definition of edge on composition "⟩′ (edge ι 𝒮ℯ𝓉.⨾ edge (𝒰₁ (lift f))) e ≡⟨" definition of 𝒰 says: edge (𝒰₁ F) = mor F "⟩′ (edge ι 𝒮ℯ𝓉.⨾ mor (lift f)) e ≡⟨" definition chasing gives: mor (lift f) (edge ι e) = edge f e ⨾ Id "⟩′ edge f e 𝒞.⨾ Id ≡⟨ 𝒞.rightId ⟩ edge f e ∎)
Observe that we simply chased definitions and as such graphmapext ≡-refl rightId
suffices as a proof,
but it’s not terribly clear why, for human consumption, and so we choose to elaborate with the
detail.
Finally, it remains to show that there is a unique way to preserve ‘basis’:
uniqueness : ∀{f : G ⟶ 𝒰₀ 𝒞} {F : 𝒫₀ G ⟶ 𝒞} → f ≡ (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F) → lift f ≡ F uniqueness {.(ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F)} {F} ≡-refl = funcext ≡-refl (≡-sym pf) where pf : ∀{x y} {p : x ⇝ y} → mor (lift (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F)) p ≡ mor F p pf {x} {.x} {p = .x !} = ≡-sym (Functor.id F) pf {x} {y} {p = .x ⟶[ e ]⟶ ps} = begin mor (lift (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F)) (x ⟶[ e ]⟶ ps) ≡⟨" definition of mor on lift, the inductive clause "⟩′ edge (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F) e 𝒞.⨾ mor (lift (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F)) ps ≡⟨ ≡-cong₂ 𝒞._⨾_ ≡-refl (pf {p = ps}) ⟩ -- inductive step edge (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F) e 𝒞.⨾ mor F ps ≡⟨" definition of edge says it preserves composition "⟩′ (edge ι 𝒮ℯ𝓉.⨾ edge (𝒰₁ F)) e 𝒞.⨾ mor F ps ≡⟨" definition of 𝒰 gives: edge (𝒰₁ F) = mor F "⟩′ (edge ι 𝒮ℯ𝓉.⨾ mor F) e 𝒞.⨾ mor F ps ≡⟨" definition of functional composition 𝒮ℯ𝓉 "⟩′ mor F (edge ι e) 𝒞.⨾ mor F ps ≡⟨ ≡-sym (Functor.comp F) {- i.e., functors preserve composition -} ⟩ mor F (edge ι e ++ ps) ≡⟨" definition of embedding and concatenation "⟩′ mor F (x ⟶[ e ]⟶ ps) ∎
Challenge: Define graph-map equality ‘≈g’ by extensionality –two graph maps are equal iff their vertex and edge maps are extensionally equal. This is far more relaxed than using propositional equality ‘≡’. Now show the stronger uniqueness claim:
∀{f : G ⟶ 𝒰₀ 𝒞} {F : 𝒫₀ G ⟶ 𝒞} → f ≈g (ι ⨾ 𝒰₁ F) → lift f ≡ F
However, saying each graph-map gives rise to exactly one unique functor is tantamount to
saying the type GraphMap G (𝒰₀ 𝒞)
is isomorphic to Functor (𝒫₀ G) 𝒞
, that is
(𝒫₀ G ⟶ 𝒞) ≅ (G ⟶ 𝒰₀ 𝒞)
—observe that this says we can ‘move’ 𝒫₀
from the left to
the right of an arrow at the cost of it (and the arrow) changing.
A few healthy exercises,
lift˘ : Functor 𝒫G 𝒞 → GraphMap G (𝒰₀ 𝒞) lift˘ F = ι ⨾g 𝒰₁ F -- i.e., record {ver = obj F , edge = mor F ∘ edge ι} rid : ∀{f : GraphMap G (𝒰₀ 𝒞)} → ∀{x y} {e : x ⟶g y} → lift˘ (lift f) ≡ f rid = {! exercise !} lid : ∀{F : Functor 𝒫G 𝒞} → lift (lift˘ F) ≡ F lid = {! exercise !}
One can of course obtain these proofs from the other ones without recourse to definitions, however for comprehension one would do well to prove them from first principles. The worked out solutions are available in the literate source file of this document.
We can then provide an alternative, and more succinct, proof of uniqueness for ‘basis preservation’:
uniqueness’ : ∀{f h} → f ≡ (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ h) → lift f ≡ h uniqueness’ {f} {h} f≈ι⨾𝒰₁h = begin lift f ≡⟨ ≡-cong lift f≈ι⨾𝒰₁h ⟩ lift (ι 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ h) ≡⟨" definition of lift˘ "⟩′ lift (lift˘ h) ≡⟨ lid ⟩ h ∎
The difference between this proof and the original one is akin to the difference between heaven and earth! That or it's much more elegant ;-)
𝒫 ⊣ 𝒰
Thus far, we have essentially shown \[(𝒫₀\, G \,⟶\, 𝒞) \quad≅\quad (G \,⟶\, 𝒰₀\, 𝒞)\] We did so by finding a pair of inverse maps:
lift : ( G ⟶ 𝒰₀ 𝒞) → (𝒫₀ G ⟶ 𝒞) lift˘ : (𝒫₀ G ⟶ 𝒞) → ( G ⟶ 𝒰₀ 𝒞)
This is nearly 𝒫 ⊣ 𝒰
which implies 𝒫
is a ‘free-functor’ since it is left-adjoint to a forgetful-functor.
‘Nearly’ since we need to exhibit naturality:
For every graph map g
and functors F, k
we have
lift˘ (𝒫 g ⨾ k ⨾ F) ≈ g ⨾ lift˘ k ⨾ 𝒰 F
in the category of graphs.
Fokkinga (Theorem A.4), among others, would call these laws ‘fusion’
instead since they inform us how to compose, or ‘fuse’, a morphism with a
lift˘
-ed morphism: Taking F
to be the identity and remembering that functors preserve
identities, we have that g ⨾ lift˘ K ≡ lift˘( 𝒫₁ g ⨾ K)
–we can push a morphism into a lift˘
at the cost of introducing a 𝒫₁
; dually for lift
-ed morphisms.
First the setup,
module _ {G H : Graph} {𝒞 𝒟 : Category {ℓ₀} {ℓ₀}} (g : GraphMap G H) (F : Functor 𝒞 𝒟) where private lift˘ = λ {A} {C} B → freedom.lift˘ A {C} B lift = λ {A} {C} B → freedom.lift A {C} B open Category ⦃...⦄ module 𝒞 = Category 𝒞 module 𝒟 = Category 𝒟 module 𝒢𝓇𝒶𝓅𝒽 = Category 𝒢𝓇𝒶𝓅𝒽 module 𝒞𝒶𝓉 = Category (𝒞𝒶𝓉 {ℓ₀} {ℓ₀}) module 𝒮ℯ𝓉 = Category (𝒮e𝓉 {ℓ₀})
Just as we needed to prove two inverse laws for lift
and lift˘
,
we need two naturality proofs.
naturality˘ : ∀ {K : Functor (𝒫₀ H) 𝒞} → lift˘ (𝒫₁ g 𝒞𝒶𝓉.⨾ K 𝒞𝒶𝓉.⨾ F) ≡ (g 𝒢𝓇𝒶𝓅𝒽.⨾ lift˘ K 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F) naturality˘ = graphmapext ≡-refl ≡-refl
That was easier than assumed!
Hahaha: Hard to formalise but so easy to prove lolz!
It says we can ‘shunt’ lift˘
into certain compositions at the cost
of replacing functor instances.
Now for the other proof:
naturality : ∀ {k : GraphMap H (𝒰₀ 𝒞)} → lift (g 𝒢𝓇𝒶𝓅𝒽.⨾ k 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F) ≡ (𝒫₁ g 𝒞𝒶𝓉.⨾ lift k 𝒞𝒶𝓉.⨾ F) naturality {k} = funcext ≡-refl (λ {x y p} → proof {x} {y} {p}) where open TypedPaths ⦃...⦄ instance G′ : Graph ; G′ = G H′ : Graph ; H′ = H proof : ∀ {x y : Graph.V G} {p : x ⇝ y} → mor (𝒫₁ g 𝒞𝒶𝓉.⨾ lift {H} {𝒞} k 𝒞𝒶𝓉.⨾ F) p ≡ mor (lift {G} {𝒟} (g 𝒢𝓇𝒶𝓅𝒽.⨾ k 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F)) p proof {p = _ !} = functor (𝒫₁ g 𝒞𝒶𝓉.⨾ lift {H} {𝒞} k 𝒞𝒶𝓉.⨾ F) preserves-identities proof {p = x ⟶[ e ]⟶ ps} = begin mor (𝒫₁ g 𝒞𝒶𝓉.⨾ lift {H} {𝒞} k 𝒞𝒶𝓉.⨾ F) (x ⟶[ e ]⟶ ps) ≡⟨" By definition, “mor” distributes over composition "⟩′ (mor (𝒫₁ g) 𝒮ℯ𝓉.⨾ mor (lift {H} {𝒞} k) 𝒮ℯ𝓉.⨾ mor F) (x ⟶[ e ]⟶ ps) ≡⟨" Definitions of function composition and “𝒫₁ ⨾ mor” "⟩′ mor F (mor (lift {H} {𝒞} k) (mor (𝒫₁ g) (x ⟶[ e ]⟶ ps))) -- This explicit path is in G ≡⟨" Lifting graph-map “g” onto a path "⟩′ mor F (mor (lift {H} {𝒞} k) (ver g x ⟶[ edge g e ]⟶ mor (𝒫₁ g) ps)) -- This explicit path is in H ≡⟨" Definition of “lift ⨾ mor” on inductive case for paths "⟩′ mor F (edge k (edge g e) 𝒞.⨾ mor (lift {H} {𝒞} k) (mor (𝒫₁ g) ps)) ≡⟨ functor F preserves-composition ⟩ mor F (edge k (edge g e)) 𝒟.⨾ mor F (mor (lift {H} {𝒞} k) (mor (𝒫₁ g) ps)) ≡⟨" Definition of function composition, for top part "⟩′ (edge g 𝒮ℯ𝓉.⨾ edge k 𝒮ℯ𝓉.⨾ mor F) e -- ≈ mor F ∘ edge k ∘ edge g 𝒟.⨾ (mor (𝒫₁ g) 𝒮ℯ𝓉.⨾ mor (lift {H} {𝒞} k) 𝒮ℯ𝓉.⨾ mor F) ps ≡⟨" “𝒰₁ ⨾ edge = mor” and “edge” and “mor” are functorial by definition "⟩′ edge (g 𝒢𝓇𝒶𝓅𝒽.⨾ k 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F) e 𝒟.⨾ mor (𝒫₁ g 𝒞𝒶𝓉.⨾ lift {H} {𝒞} k 𝒞𝒶𝓉.⨾ F) ps ≡⟨ {- Inductive Hypothesis: -} ≡-cong₂ 𝒟._⨾_ ≡-refl (proof {p = ps}) ⟩ edge (g 𝒢𝓇𝒶𝓅𝒽.⨾ k 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F) e 𝒟.⨾ mor (lift {G} {𝒟} (g 𝒢𝓇𝒶𝓅𝒽.⨾ k 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F)) ps ≡⟨" Definition of “lift ⨾ mor” on inductive case for paths "⟩′ mor (lift {G} {𝒟} (g 𝒢𝓇𝒶𝓅𝒽.⨾ k 𝒢𝓇𝒶𝓅𝒽.⨾ 𝒰₁ F)) (x ⟶[ e ]⟶ ps) ∎
Formally, we now have an adjunction:
𝒫⊣𝒰 : 𝒫 ⊣ 𝒰 𝒫⊣𝒰 = record{ ⌊_⌋ = lift˘ ; ⌈_⌉ = lift ; lid = lid ; rid = λ {G 𝒞 c} → rid {G} {𝒞} {c} ; lfusion = λ {G H 𝒞 𝒟 f F K} → naturality˘ {G} {H} {𝒞} {𝒟} f K {F} ; rfusion = λ {G H 𝒞 𝒟 f k F} → naturality {G} {H} {𝒞} {𝒟} f F {k} } where module _ {G : Graph} {𝒞 : Category} where open freedom G {𝒞} public
Observe that for the freedom proof we recalled
that ists determine a form of quantification, ‘folding’:
given an operation ⊕, we may form the operation [x₁, …, xₖ] ↦ x₁ ⊕ ⋯ ⊕ xₖ
.
Then used that to define our operation lift
, whose core was essentially,
module folding (G : Graph) where open TypedPaths G open Graph G -- Given: fold : {X : Set} (v : V → X) -- realise G's vertices as X elements (f : ∀ x {y} (e : x ⟶ y) → X → X) -- realise paths as X elements → (∀ {a b} → a ⇝ b → X) -- Then: Any path is an X value fold v f (b !) = v b fold v f (x ⟶[ e ]⟶ ps) = f x e (fold v f ps)
For example, what is the length of a path?
length : ∀{x y} → x ⇝ y → ℕ length = fold (λ _ → 0) -- single walks are length 0. (λ _ _ n → 1 + n) -- edges are one more than the -- length of the remaining walk
Let’s verify that this is actually what we intend by the length of a path.
length-! : ∀{x} → length (x !) ≡ 0 length-! = ≡-refl -- True by definition of “length”: The first argument to the “fold” length-ind : ∀ {x y ω} {e : x ⟶ y} {ps : y ⇝ ω} → length (x ⟶[ e ]⟶ ps) ≡ 1 + length ps length-ind = ≡-refl -- True by definition of “length”: The second-argument to the “fold”
Generalising on length
, suppose we have a ‘cost function’ c
that assigns a cost of traversing
an edge. Then we can ask what is the total cost of a path:
path-cost : (c : ∀{x y}(e : x ⟶ y) → ℕ) → ∀{x y}(ps : x ⇝ y) → ℕ path-cost c = fold (λ _ → 0) -- No cost on an empty path. (λ x e n → c e + n) -- Cost of current edge plus -- cost of remainder of path.
Now, we have length = path-cost (λ _ → 1)
: Length is just assigning a cost of 1 to each edge.
Under suitable conditions, list fold distributes over list catenation, can we find an analogue for paths? Yes. Yes, we can:
fold-++ : ∀{X : Set} {v : V → X} {g : ∀ x {y} (e : x ⟶ y) → X} → (_⊕_ : X → X → X) → ∀{x y z : V} {p : x ⇝ y} {q : y ⇝ z} → (unitl : ∀{x y} → y ≡ v x ⊕ y) -- Image of ‘v’ is left unit of ⊕ → (assoc : ∀ {x y z} → x ⊕ (y ⊕ z) ≡ (x ⊕ y) ⊕ z ) -- ⊕ is associative → let f : ∀ x {y} (e : x ⟶ y) → X → X f = λ x e ps → g x e ⊕ ps in fold v f (p ++ q) ≡ fold v f p ⊕ fold v f q fold-++ {g = g} _⊕_ {x = x} {p = .x !} unitl assoc = unitl fold-++ {g = g} _⊕_ {x = x} {p = .x ⟶[ e ]⟶ ps} unitl assoc = ≡-cong (λ exp → g x e ⊕ exp) (fold-++ _⊕_ {p = ps} unitl assoc) ⟨≡≡⟩ assoc
Compare this with the proof-obligation of lift
.
We called our path catenation _++_
, why the same symbol as that for
list catenation?
How do we interpret a list over \(A\) as a graph? Well the vertices can be any element of \(A\) and an edge \(x ⟶ y\) merely indicates that ‘‘the item after \(x\) in the list is the element \(y\)’’, so we want it to be always true; or always inhabited without distinction of the inhabitant: So we might as well use a unit type.
module lists (A : Set) where open import Data.Unit listGraph : Graph listGraph = record { V = A ; _⟶_ = λ a a’ → ⊤ }
I haven’t a clue if this works, you read my reasoning above.
The only thing we can do is test our hypothesis by looking at the
typed paths over this graph. In particular, we attempt to show every
non-empty list of \(A\)’s corresponds to a path. Since a typed path needs
a priori the start and end vertes, let us construe
List A ≅ Σ n ∶ ℕ • Fin n → A
–later note that Path G ≅ Σ n ∶ ℕ • [n] 𝒢⟶ G
.
open TypedPaths listGraph open folding listGraph -- Every non-empty list [x₀, …, xₖ] of A’s corresonds to a path x₀ ⇝ xₖ. toPath : ∀{n} (list : Fin (suc n) → A) → list fzero ⇝ list (fromℕ n) toPath {zero} list = list fzero ! toPath {suc n} list = list fzero ⟶[ tt ]⟶ toPath {n} (λ i → list(fsuc i)) -- Note that in the inductive case, “list : Fin (suc (suc n)) → A” -- whereas “suc ⨾ list : Fin (suc n) → A”. -- -- For example, if “list ≈ [x , y , z]” yields -- “fsuc ⨾ list ≈ [y , z ]” and -- “fsuc ⨾ fsuc ⨾ list ≈ [z]”.
Hm! Look at that, first guess and it worked! Sweet.
Now let’s realize the list fold as an instance of path fold,
-- List type former List = λ (X : Set) → Σ n ∶ ℕ • (Fin n → X) -- Usual list folding, but it's in terms of path folding. foldr : ∀{B : Set} (f : A → B → B) (e : B) → List A → B foldr f e (zero , l) = e foldr f e (suc n , l) = fold (λ a → f a e) (λ a _ rem → f a rem) (toPath l) -- example listLength : List A → ℕ -- result should clearly be “proj₁” of the list, anyhow: listLength = foldr (λ a rem → 1 + rem) -- Non-empty list has length 1 more than the remainder. 0 -- Empty list has length 0.
Let’s prepare for a more useful example
-- Empty list [] : ∀{X : Set} → List X [] = 0 , λ () -- Cons for lists _∷_ : ∀{X : Set} → X → List X → List X _∷_ {X} x (n , l) = 1 + n , cons x l where -- “cons a l ≈ λ i : Fin (1 + n) → if i ≈ 0 then a else l i” cons : ∀{n} → X → (Fin n → X) → (Fin (suc n) → X) cons x l fzero = x cons x l (fsuc i) = l i map : ∀ {B} (f : A → B) → List A → List B map f = foldr (λ a rem → f a ∷ rem) [] -- looks like the usual map don’t it ;) -- list concatenation _++ℓ_ : List A → List A → List A l ++ℓ r = foldr _∷_ r l -- fold over ‘l’ by consing its elements infront of ‘r’ -- Exercise: Write path catenation as a path-fold.
These few adventures would suggest that much of list theory can be brought over to the world of paths. It looks promising, let me know dear reader if you make progress on related explorations!
This note took longer to write than I had initally assumed; perhaps I should have taken into account
It always takes longer than you expect, even when you take into account Hofstadter’s Law.
Lessons learned:
The astute reader may have noticed that the tone of writing sometimes changes drastically. This is because some of this article was written by me in March 2016 and I wished to preserve interesting writing style I then had –if anything to contrast with my now somewhat semi-formal style.
This article was motivated while I was reading Conceptual Mathematics for fun. One of the problems was to show that paths over a graph form a category and do so freely. It took me about 20 minutes on paper and pencil, but this resulting mechanisation took much more time –but it was also much more fun!
I had fun writing this up & I hope you enjoy it too :-)
( This article is not yet ‘done’, but good enough for now. )