Category Theory in Coq

This development encodes category theory in Coq, with the primary aim being to allow representation and manipulation of categorical terms, as well realization of those terms in various target categories.

• Versions used: Coq 8.14.1, 8.15.2, 8.16+rc1.
• Some parts depend on Coq-Equations 1.2.4, 1.3.

Usage

It is recommended to include this library in your developments by adding the following to your _CoqProject file:

-R <path to this library> Category

Then include the primary elements of the library using:

Require Import Category.Theory.

Library structure

This library is broken up into several major areas:

• Core Theory, covering topics such as categories, functors, natural transformations, adjunctions, kan extensions, etc.

• Categorical Structure, which reveals internal structure of a category by way of morphisms related to some universal property.

• Categorical Construction, which introduces external structure by building new categories out of existing ones.

• Species of different kinds of Functor, Natural.Transformation, Adjunction and Kan.Extension; for example: categories with monoidal structure give rise to monoidal functors preserving this structure, which in turn leads to monoidal transformations that transform functors while preserving their monoidal mapping property.

• The Instance directory defines various categories; some of these are fairly general, such as the category of preorders, applicable to any PreOrder relation, but in general these are either not defined in terms of other categories, or are sufficiently specific.

• When a concept, such as limits, can be defined using more fundamental terms, that version of limits can be found in a subdirectory of the derived concept, for example there is Category.Structure.Limit and Category.Limit.Kan.Extension. This is done to demonstrate the relationship of ideas; for example: Category.Construction.Comma.Natural.Transformation. As a result, files with the same name occur often, with the parent directory establishing intent.

Programming sub-library

Within Theory.Coq there is now a sub-library that continues work started in the coq-haskell library. This sub-library is specifically aimed at "applied category theory" for programmers in the category of Coq types and functions. The aim is to mimic the utility of Haskell's monad hierarchy -- but for Coq users, similar to what ext-lib achieves. I've also adjusted a few of my notations to more closely match ext-lib.

Where this library differs, and what is considered the main contribution of this work, is that laws are not defined for these structures in the sub-library. Rather, the programmer establishs that her Monad is lawful by proving that a mapping exists from that definition into the general definition of monads (found in Theory.Monad) specialized to the category Coq. In this way the rest of the category-theory library is leveraged to discharge these proof obligations, while keeping the programmatic side as simple as possible.

For example, it is trivial to define the composition of two Applicatives. What is not so trivial is proving that this is truly an Applicative, based on that simple definition. While this proof was done in coq-haskell, it requires a bit of Ltac magic to keep the size down:

Program Instance Compose_ApplicativeLaws
  `{ApplicativeLaws F} `{ApplicativeLaws G} : ApplicativeLaws (F \o G).
Obligation 2. (* ap_composition *)
  (* Discharge w *)
  rewrite <- ap_comp; f_equal.
  (* Discharge v *)
  rewrite <- !ap_fmap, <- ap_comp.
  symmetry.
  rewrite <- ap_comp; f_equal.
  (* Discharge u *)
  apply_applicative_laws.
  f_equal.
  extensionality y.
  extensionality x.
  extensionality x0.
  rewrite <- ap_comp, ap_fmap.
  reflexivity.
Qed.

What's ill-gotten about this proof is that it's confined to the very specific case of Coq applicative endo-functors. However, there is a more general truth, which is that any two lax monoidal functors in any monoidal category also compose. So why not appeal to that proof to establish that our programmatic applicative in Coq is lawful by construction?

This is what the new sub-library does. Since the more general proof is already completed (and is too large to paste here), one may appeal to it directly to establish the desired fact:

(* The composition of two applicatives is itself applicative. We establish
   this by appeal to the general proofs that:

   1. every Coq functor has strength;
   2. (also, but not needed: any two strong functors compose to a strong
      functor; if it is a Coq functor it is known to have strength); and
   3. any two lax monoidal functors compose to a lax monoidal functor. *)

Theorem Compose_IsApplicative
  `(HF : EndoFunctor F)
  `(AF : @Functor.Applicative.Applicative _ _ (FromAFunctor HF))
  `(HG : EndoFunctor G)
  `(AG : @Functor.Applicative.Applicative _ _ (FromAFunctor HG)) :
  IsApplicative (Compose_IsFunctor HF HG)
    (@Compose_Applicative
       F HF (EndoApplicative_Applicative HF AF)
       G HG (EndoApplicative_Applicative HG AG)).
Proof.
  construct.
  - apply (@Compose_LaxMonoidalFunctor _ _ _ _ _ _ _ _ AF AG).
  - apply Coq_StrongFunctor.
Qed.

This proof pulls in several instances to establish that the category Coq is cartesian, closed, and thus closed monoidal, etc.

The hope is this will become a happy marriage of simple, useful computational constructions for Coq programmers, with relevant proof results from what category theory tells us about these structures in general, such as the above fact concerning composition of monoidal functors.

Duality

The core theory is defined in such a way that "the dual of the dual" is evident to Coq by mere simplification (that is, "C^op^op = C" follows by reflexivity for the opposite of the opposite of categories, functors, natural transformation, adjunctions, isomorphisms, etc.).

For this to be true, certain artificial constructions are necessary, such as repeating the associativity law for categories in symmetric form, and likewise the naturality condition of natural transformations. This repeats proof obligations when constructing these objects, but pays off in the ability to avoid repetition when stating the dual of whole structures.

As a result, the definition of comonads, for example, is reduced to one line:

Definition Comonad `{M : C ⟶ C} := @Monad (C^op) (M^op).

Most dual constructions are similarly defined, with the exception of Initial and Cocartesian structures. Although the core classes are indeed defined in terms of their dual construction, an alternate surface syntax and set of theorems is provided (for example, using a + b to indicate coproducts) to make life is a little less confusing for the reader. For instance, it follows from duality that 0 + X ≅ X is just 1 × X ≅ X in the opposite category, but using separate notations makes it easier to see that these two isomorphisms in the same category are not identical. This is especially important because Coq hides implicit parameters that would usually let you know duality is involved.

Design decisions

Some features and choices made in this library:

• Type classes are used throughout to present concepts. When a type class instance would be too general -- and thus overlap with other instances -- it is presented as a definition that can later be added to instance resolution with Existing Instance.

• All definitions are in Type, so that Prop is not used anywhere except specific category instances defined over Prop, such as the category Rel with heterogeneous relations as arrows.

• No axioms are used anywhere in the core theory; they appear only at times when defining specific category instances, mostly for the Coq category.

• Homsets are defined as computationally-relevant homsetoids (that is, using crelation). This is done using a variant of the Setoid type class defined for this library; likewise, the category of Sets is actually the category of such setoids. This increases the proof work necessary to establish definitions -- since preservation of the equivalence relation is required at all levels -- but allows categories to be flexible in what it means for two arrows to be equivalent.

Notations

There are many notations used through the library, which are chosen in an attempt to make complex constructions appear familiar to those reading modern texts on category theory. Some of the key notations are:

• ≈ is equivalence; equality is almost never used.
• ≈[Cat] is equivalence between arrows of some category, here Cat; this is only needed when type inference fails because it tries to find both the type of the arguments, and the type class instance for the equivalence
• ≅ is isomorphism; apply it with to or from
• ≊ is used specifically for isomorphisms between homsets in Sets
• iso⁻¹ also indicates the reverse direction of an isomorphism
• X ~> Y: a squiggly arrow between objects is a morphism
• X ~{C}~> Y: squiggly arrows may also specify the intended category
• id[C]: many known morphisms allow specifying the intended category; sometimes this is even used in the printing format
• C ⟶ D: a long right arrow between categories is a functor
• F ⟹ G: a long right double arrow between functors is a natural transformation
• f ∘ g: a small centered circle is composition of morphisms, or horizontal composition generally
• f ∘[Cat] g: composition can specify the intended category, as an aid to type inference composition generally
• f ○ g: a larger hollow circle is composition of functors
• f ⊙ g: a larger circle with a dot is composition of isomorphisms
• f ∙ g: a solid composition dot is composition of natural transformations, or vertical composition generally
• f ⊚ g: a larger circle with a smaller circle is composition of adjunctions
• ([C, D]): A pair of categories in square brackets is another way to give the type of a functor, being an object of the category Fun(C, D)
• F ~{[C, D]}~> G: An arrow in a functor category is a natural transformation
• F ⊣ G: the turnstile is used for adjunctions
• Cartesian categories use △ as the fork operation and × for products
• Cocartesian categories use ▽ as the merge operation and + for coproducts
• Closed categories use ^ for exponents and ≈> for the internal hom
• As objects, the numbers 0 and 1 refer to initial and terminal objects
• As categories, the numbers 0 and 1 refer to the initial and terminal objects of Cat
• Products of categories can be specified using ∏, which does not require pulling in the cartesian definition of Cat
• Coproducts of categories can be specified using ∐, which does not require pulling in the cocartesian definition of Cat
• Products of functors are given with F ∏⟶ G, combining product and functor notations; the same for ∐⟶
• Comma categories of two functors are given by (F ↓ G)
• Likewise, the arrow category of C⃗
• Slice categories use a unicode forward slash C̸c, since the normal forward slash is not considered an operator
• Coslice categories use c ̸co C, to avoid ambiguity

Future directions

Computational Solver

There are some equivalences in category-theory that are very easily expressed and proven, but slow to establish in Coq using only symbolic term rewriting. For example:

(f ∘ g) △ (h ∘ i) ≈ split f h ∘ g △ i

This is solved by unfolding the definition of split, and then rewriting so that the fork operation (here given by △) absorbs the terms to its left, followed by observing the associativity of composition, and then simplify based on the universal properties of products. This is repeated several times until the prove is trivially completed.

Although this is easy to state, and even to write a tactic for, it can be extremely slow, especially when the types of the terms involved becomes large. A single rewrite can take several seconds to complete for some terms, in my experience.

The goal of this solver is to reify the above equivalence in terms of its fundamental operations, and then, using what we know about the laws of category theory, to compute the equivalence down to an equation on indices between the reduced terms. This is called computational reflection, and encodes the fact that our solution only depends on the categorical structure of the terms, and not their type.

That is, an incorrectly-built structure will simply fail to solve; but since we're reflecting over well-typed expressions to build the structure we pass to the solver, combined with a proof of soundness for that solver, we can know that solvable, well-typed, terms always give correct solutions. In this way, we transfer the problem to a domain without types, only indices, solve the structural problem there, and then bring the solution back to the domain of full types by way of the soundness proof.

Compiling to categories

Work has started in Tools/Abstraction for compiling monomorphic Gallina functions into "categorical terms" that can then be instantiated in any supporting target category using Coq's type class instance resolution.

This is as a Coq implementation of an idea developed by Conal Elliott, which he first implemented in and for Haskell. It is hoped that the medium of categories may provide a sound means for transporting Gallina terms into other syntactic domains, without relying on Coq's extraction mechanism.

License

This library is made available under the MIT license, a copy of which is included in the file LICENSE. Basically: you are free to use it for any purpose, personal or commercial (including proprietary derivates), so long as a copy of the license file is maintained in the derived work. Further, any acknowledgement referring back to this repository, while not necessary, is certainly appreciated.

John Wiegley