open import Cat.Diagram.Monad
open import Cat.Functor.Base
open import Cat.Univalent
open import Cat.Prelude

    {o } {C : Precategory o }
    (isc : is-category C)
    (M : Monad C)

Eilenberg-Moore Categories🔗

Given a base univalent category C\ca{C}, we can consider a monad MM on C\ca{C}, and its associated Eilenberg-Moore category CM\ca{C}^M, as a standard way of constructing categories of “algebraic gadgets” backed by objects of C\ca{C}. A concrete example is given by the category of monoids: A monoid (in sets) is equivalent to an algebra for the list monad.

Given that “hand-rolled” categories of this sort tend to be well-behaved, in particular when it comes to identifications (see univalence of monoids, univalence of groups, univalence of semilattices), it’s natural to ask whether all Eilenberg-Moore categories are themselves univalent, assuming that their underlying category is. Here we give a positive answer.

Fixing a monad MM on a univalent category C\ca{C}, we abbreviate its Eilenberg-Moore category CM\ca{C}^M as EM.

private EM = Eilenberg-Moore C M

import Cat.Reasoning EM as EM
import Cat.Reasoning C as C

As usual, we take the centre of contraction to be AA and the identity isomorphism AAA \cong A; The hard part is proving that, given a pair of MM-algebras AA and XX, together with a specified isomorphism f:AXf : A \cong X, we can build an identification AXA \cong X, such that over this identification, ff is the identity map.

Eilenberg-Moore-is-category : is-category EM
Eilenberg-Moore-is-category A .centre = A ,
Eilenberg-Moore-is-category (A , Am) .paths ((X , Xm) , A≅X) =
  Σ-pathp A≡M triv where

The first thing we shall note is that an algebra is given by a pair of two data: An underlying object A0A_0 (resp X0X_0), together with the structure AmA_m (resp. XmX_m) of an M-algebra on A0A_0. Hence, an identification of algebras can be broken down into an identification of their components, re-using the equality lemma for dependent pairs.

Recall that a homomorphism of M-algebras between (A0,Am)(X0,Xm)(A_0,A_m) \to (X_0,X_m) is given by a map h:A0X0h : A_0 \to X_0 in C\ca{C}, such that the diagram below commutes. By forgetting that the square commutes, algebra homomorphisms correspond faithfully to morphisms in the underlying category C\ca{C}.

Hence, given an isomorphism (A0,Am)(X0,Xm)(A_0, A_m) \cong (X_0, X_m) (let us call it ff) in CM\ca{C}^M, we can forget all of the commutativity data associated with the algebra homomorphisms, and recover an isomorphism between the underlying objects in C\ca{C}:

    A₀≅X₀ : A C.≅ X
    A₀≅X₀ = C.make-iso
      (map A≅ (map A≅X.from) (ap map A≅X.invl) (ap map A≅X.invr)

Since we assumed C\ca{C} to be univalent, this isomorphism can be made into a path A0X0A_0 \equiv X_0. This covers a third of the task: We must now show first that the algebra structures AmA_m and XmX_m are identified over A₀≡X₀, and we must prove that the resulting identification makes ff into the identity isomorphism.

    A₀≡X₀ : A  X
    A₀≡X₀ = iso→path C isc A₀≅X₀

By the characterisation of paths in algebras, it suffices to show that A₀≡X₀ transports AmA_m’s multiplication to that of XmX_m’s; Using the corresponding lemma for paths in hom-spaces of univalent categories, we can get away with (still calling our isomorphism ff) showing the square below commutes.

Since we have assumed that ff is an MM-algebra isomorphism, we can simultaneously turn the square above into one which has ff and f1f^{-1} in adjacent faces and swap AmA_m for XmX_m; A straightforward calculation then shows that the square above commutes.

    Am≡Xm : PathP  i  Algebra-on C M (A₀≡X₀ i)) Am Xm
    Am≡Xm = Algebra-on-pathp _ A₀≡X₀ same-mults′ where
        : PathP
           i  C.Hom (iso→path C isc (F-map-iso (Monad.M M) A₀≅X₀) i) (A₀≡X₀ i))
          (Am .ν) (Xm .ν)
      same-mults =
        Hom-pathp-iso C isc (
          map A≅ C.∘ Am .ν C.∘ Monad.M₁ M (map A≅X.from)                 ≡⟨ C.pulll (sq A≅ 
          (Xm .ν C.∘ Monad.M₁ M (A≅ .map)) C.∘ Monad.M₁ M (map A≅X.from) ≡⟨ C.cancelr (sym (Monad.M-∘ M _ _) ·· ap (Monad.M₁ M) (ap map A≅X.invl) ·· Monad.M-id M) 
          Xm .ν                                                              

Note, however, that the path above is not in the correct space! While it is in a space of C\ca{C}-morphisms, the source is not of the correct type. This is because functors between univalent categories can act on paths in “two” ways: One can either apply the functor’s action on isos, then take a path in the codomain category; Or take a path in the domain category, and then use the canonical action on paths. Fortunately these coincide, and we can correct the source:

      same-mults′ : PathP  i  C.Hom (Monad.M₀ M (A₀≡X₀ i)) (A₀≡X₀ i))
                      (Am .ν) (Xm .ν)
      same-mults′ =
           j  PathP
             i  C.Hom (F-map-path (Monad.M M) A₀≅X₀ isc isc (~ j) i) (A₀≡X₀ i))
            (Am .ν) (Xm .ν))

    A≡M : Path (Algebra _ M) (A , Am) (X , Xm)
    A≡M i = A₀≡X₀ i , Am≡Xm i

To finish the proof that CM\ca{C}^M is univalent, we must show that the identification we’ve built trivialises the isomorphism AXA \cong X we were given. This follows immediately from the characterisation of paths in isomorphism spaces and in Hom-spaces.

    triv : PathP  i  (A , Am) EM.≅ A≡M i) A≅X
    triv = EM.≅-pathp refl _
      (Algebra-hom-pathp _ _ _ (Hom-pathp-reflr-iso C isc (C.idr _)))