This page describes some nice properties of the category RCocommCoalgR Cocomm Coalg of cocommutative coalgebras over a ground ring RR, in particular details of the proof that it is a complete, cocomplete, lextensive, cartesian closed category with a generating set.

In the sequel, we will simply say “coalgebra”, although we really mean cocommutative (coassociative, counital) coalgebra.


Colimits in RCocommCoalgR Cocomm Coalg are created (reflected) by the forgetful functor U:RCocommCoalgRModU: R Cocomm Coalg \to R Mod. As the codomain is cocomplete, so is the domain. Thus for example, the coproduct of coalgebras C,DC, D is the direct sum module CDC \oplus D equipped with the evident comultiplication CD(CC)(DD)(CD)(CD)C \oplus D \to (C \otimes C) \oplus (D \otimes D) \hookrightarrow (C \oplus D) \otimes (C \oplus D).

It is perhaps well to point out explicitly that if f:CDf: C \to D is a coalgebra map, then the image f(C)f(C) in RModR Mod inherits a coalgebra structure from DD and provides the image in RCocommCoalgR Cocomm Coalg. This is an easy consequence of the fact that f(CC)=f(C)f(C)f(C \otimes C) = f(C) \otimes f(C) as submodules of DDD \otimes D.

Local presentability

Most of the nice properties can be derived as consequences of the local presentability of RCocommCoalgR Cocomm Coalg. For this we refer to the paper by Porst.

In the case where RR is a field kk, the category kCocommCoalgk Cocomm Coalg is locally finitely presentable. The finitely presentable objects of kCocommCoalgk Cocomm Coalg are those coalgebras that are finite-dimensional as vector spaces. The first step towards establishing locally finite presentability is the fundamental theorem of coalgebras, which guarantees that every coalgebra is a filtered colimit of finite-dimensional coalgebras:


Every coalgebra CC is the union of its finite-dimensional subcoalgebras, i.e., is the directed colimit of the system of finite-dimensional subcoalgebras of CC and inclusion maps between them.

(For the nonce, we define “subcoalgebra” of CC to mean vector subspace i:VCi: V \hookrightarrow C such that the restricted comultiplication δ Ci\delta_C \circ i is contained in the subspace VVV \otimes V of CCC \otimes C. Later we will see that subcoalgebras are actually the same thing as subobjects in kCocommCoalgk Cocomm Coalg in the sense of equivalence classes of monomorphisms.)


Every finite-dimensional coalgebra CC is finitely presentable, i.e., hom(C,):kCocommCoalgSet\hom(C, -): k Cocomm Coalg \to Set preserves filtered colimits.


If D=colim jD jD = colim_j D_j is a directed colimit, then the image of a coalgebra map f:CDf: C \to D is a finite-dimensional subcoalgebra inclusion f(C)Df(C) \to D which, as a finitely presentable vector space, is included in one of the components D jDD_j \to D of the colimit cone; this inclusion is a subcoalgebra inclusion.

It follows easily from these results and cocompleteness of kCocommCoalgk Cocomm Coalg that kCocommCoalgk Cocomm Coalg is locally finitely presentable. As a result we have a Gabriel-Ulmer equivalence

kCocommCoalgLex(kCocommCoalg fd op,Set)k Cocomm Coalg \simeq Lex(k Cocomm Coalg_{fd}^{op}, Set)

where the category of finite-dimensional coalgebras is dual to the category of finite-dimensional algebras, so that also

kCocommCoalgLex(CommAlg fd,Set).k Cocomm Coalg \simeq Lex(Comm Alg_{fd}, Set).

Naturally, choosing a representative of each isomorphism class of finite-dimensional coalgebras, we obtain a generating set of kCocommCoalgk Cocomm Coalg.

Totality and completeness

As is the case for any locally presentable category, RCocommCoalgR Cocomm Coalg is not only cocomplete but is a totally cocomplete category and is (therefore) complete as well.

The construction of limits can be described explicitly. The equalizer of two coalgebra maps f,g:CDf, g: C \stackrel{\to}{\to} D is the largest subcoalgebra contained in the RModR Mod equalizer (the latter coincides with the equalizer as computed in Set). This can be described even more explicitly in Sweedler notation: the equalizer of f,gf, g is the set

E={cC:c (1)f(c (2))c (3)=c (1)g(c (2))c (3)}E = \{c \in C: c_{(1)} \otimes f(c_{(2)}) \otimes c_{(3)} = c_{(1)} \otimes g(c_{(2)}) \otimes c_{(3)}\}

with the unique structure of coalgebra that makes it a subcoalgebra of CC.

Assuming the construction of cofree cocommutative coalgebras, viz. the right adjoint K:RModRCocommCoalgK: R Mod \to R Cocomm Coalg to the forgetful functor U:RCocommCoalgRModU: R Cocomm Coalg \to R Mod (which we also touch on below), the product of a family of coalgebras C iC_i can be described as follows. Consider the product iC i\prod_i C_i taken in RModR Mod, and let p:UK( iC i) iC ip: U K(\prod_i C_i) \to \prod_i C_i be the component of the counit of the adjunction UKU \dashv K at that product. Then the product CC of the C iC_i taken in RCocommCoalgR Cocomm Coalg is the largest subcoalgebra j:CK( iC i)j: C \hookrightarrow K(\prod_i C_i) such that each composite π ipj:CC i\pi_i p j: C \to C_i is a coalgebra map. For a proof, see the article by Agore.

Cartesian closure

The cartesian product of two coalgebras C,DC, D is given by CDC \otimes D with the evident coalgebra structure. The functor C:RCocommCoalgRCocommCoalgC \otimes -: R Cocomm Coalg \to R Cocomm Coalg is a cocontinuous functor, since colimits are reflected from RModR Mod and C:RModRModC \otimes -: R Mod \to R Mod is cocontinuous there.

For locally presentable or more generally total categories C\mathbf{C}, cocontinuity of a functor F:CCF: \mathbf{C} \to \mathbf{C} is enough to guarantee that FF has a right adjoint. It follows that RCocommCoalgR Cocomm Coalg is cartesian closed.

Comonadicity over RModR Mod

Again, since U:RCocommCoalgRModU: R Cocomm Coalg \to R Mod is cocontinuous and RCocommCoalgR Cocomm Coalg is locally presentable, UU has a right adjoint KK. This is described more explicitly at cofree cocommutative coalgebra.

The comonadicity of UU is proven in the article by Barr, section 4.


We will show that kCocommCoalgk Cocomm Coalg is a lextensive category.


Coproducts of coalgebras are disjoint.


The coproduct of coalgebras C,DC, D is CDC \oplus D, and the pullback of the two coalgebra inclusions i C:CCD,i D:DCDi_C: C \to C \oplus D, i_D: D \to C \oplus D is the coalgebra equalizer of the two maps

C π C i C CD CD π D i D D \array{ & & C & & \\ & _\mathllap{\pi_C} \nearrow & & \searrow _\mathrlap{i_C} \\ C \otimes D & & & & C \oplus D \\ & _\mathllap{\pi_D} \searrow & & \nearrow _\mathrlap{i_D} \\ & & D & & }

This coalgebra equalizer is constructed as the largest subcoalgebra of CDC \otimes D contained in the kVectk Vect equalizer. But the kVectk Vect equalizer is easily seen to be 00 (cf. the fact that kVectk Vect coproducts are themselves disjoint), and so the coalgebra equalizer must be 00 as well, concluding the proof.


Coproducts of coalgebras are stable under pullback.


Consider the following diagram:

F j X f C i C CD i D D\array{ & & F & & \\ & & \downarrow j & & \\ & & X & & \\ & & \downarrow f & & \\ C & \underset{i_C}{\to} & C \oplus D & \underset{i_D}{\leftarrow} & D }

where ff is an arbitrary coalgebra map and j:FXj: F \hookrightarrow X is a finite-dimensional subcoalgebra inclusion. We will show that the pullback of the coproduct diagram along fjf \circ j is a coproduct decomposition of FF.


  • Ana L. Agore, Limits of Coalgebras, Bialgebras and Hopf Algebras,, 2010. (pdf)

category: category

Revised on September 4, 2015 19:55:42 by Todd Trimble (