This page describes some nice properties of the category $R Cocomm Coalg$ of cocommutative coalgebras over a ground ring $R$, 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 $R Cocomm Coalg$ are created (reflected) by the forgetful functor $U: R Cocomm Coalg \to R Mod$. As the codomain is cocomplete, so is the domain. Thus for example, the coproduct of coalgebras $C, D$ is the direct sum module $C \oplus D$ equipped with the evident comultiplication $C \oplus D \to (C \otimes C) \oplus (D \otimes D) \hookrightarrow (C \oplus D) \otimes (C \oplus D)$.
It is perhaps well to point out explicitly that if $f: C \to D$ is a coalgebra map, then the image $f(C)$ in $R Mod$ inherits a coalgebra structure from $D$ and provides the image in $R Cocomm Coalg$. This is an easy consequence of the fact that $f(C \otimes C) = f(C) \otimes f(C)$ as submodules of $D \otimes D$.
Most of the nice properties can be derived as consequences of the local presentability of $R Cocomm Coalg$. For this we refer to the paper by Porst.
In the case where $R$ is a field $k$, the category $k Cocomm Coalg$ is locally finitely presentable. The finitely presentable objects of $k Cocomm Coalg$ 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 $C$ is the union of its finite-dimensional subcoalgebras, i.e., is the directed colimit of the system of finite-dimensional subcoalgebras of $C$ and inclusion maps between them.
(For the nonce, we define “subcoalgebra” of $C$ to mean vector subspace $i: V \hookrightarrow C$ such that the restricted comultiplication $\delta_C \circ i$ is contained in the subspace $V \otimes V$ of $C \otimes C$. Later we will see that subcoalgebras are actually the same thing as subobjects in $k Cocomm Coalg$ in the sense of equivalence classes of monomorphisms.)
Every finite-dimensional coalgebra $C$ is finitely presentable, i.e., $\hom(C, -): k Cocomm Coalg \to Set$ preserves filtered colimits.
If $D = colim_j D_j$ is a directed colimit, then the image of a coalgebra map $f: C \to D$ is a finite-dimensional subcoalgebra inclusion $f(C) \to D$ which, as a finitely presentable vector space, is included in one of the components $D_j \to D$ of the colimit cone; this inclusion is a subcoalgebra inclusion.
It follows easily from these results and cocompleteness of $k Cocomm Coalg$ that $k Cocomm Coalg$ is locally finitely presentable. As a result we have a Gabriel-Ulmer equivalence
where the category of finite-dimensional coalgebras is dual to the category of finite-dimensional algebras, so that also
Naturally, choosing a representative of each isomorphism class of finite-dimensional coalgebras, we obtain a generating set of $k Cocomm Coalg$.
As is the case for any locally presentable category, $R Cocomm Coalg$ 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 $f, g: C \stackrel{\to}{\to} D$ is the largest subcoalgebra contained in the $R Mod$ equalizer (the latter coincides with the equalizer as computed in Set). This can be described even more explicitly in Sweedler notation: the equalizer of $f, g$ is the set
with the unique structure of coalgebra that makes it a subcoalgebra of $C$.
Assuming the construction of cofree cocommutative coalgebras, viz. the right adjoint $K: R Mod \to R Cocomm Coalg$ to the forgetful functor $U: R Cocomm Coalg \to R Mod$ (which we also touch on below), the product of a family of coalgebras $C_i$ can be described as follows. Consider the product $\prod_i C_i$ taken in $R Mod$, and let $p: U K(\prod_i C_i) \to \prod_i C_i$ be the component of the counit of the adjunction $U \dashv K$ at that product. Then the product $C$ of the $C_i$ taken in $R Cocomm Coalg$ is the largest subcoalgebra $j: C \hookrightarrow K(\prod_i C_i)$ such that each composite $\pi_i p j: C \to C_i$ is a coalgebra map. For a proof, see the article by Agore.
The cartesian product of two coalgebras $C, D$ is given by $C \otimes D$ with the evident coalgebra structure. The functor $C \otimes -: R Cocomm Coalg \to R Cocomm Coalg$ is a cocontinuous functor, since colimits are reflected from $R Mod$ and $C \otimes -: R Mod \to R Mod$ is cocontinuous there.
For locally presentable or more generally total categories $\mathbf{C}$, cocontinuity of a functor $F: \mathbf{C} \to \mathbf{C}$ is enough to guarantee that $F$ has a right adjoint. It follows that $R Cocomm Coalg$ is cartesian closed.
Again, since $U: R Cocomm Coalg \to R Mod$ is cocontinuous and $R Cocomm Coalg$ is locally presentable, $U$ has a right adjoint $K$. This is described more explicitly at cofree cocommutative coalgebra.
The comonadicity of $U$ is proven in the article by Barr, section 4.
We will show that $k Cocomm Coalg$ is a lextensive category.
Coproducts of coalgebras are disjoint.
The coproduct of coalgebras $C, D$ is $C \oplus D$, and the pullback of the two coalgebra inclusions $i_C: C \to C \oplus D, i_D: D \to C \oplus D$ is the coalgebra equalizer of the two maps
This coalgebra equalizer is constructed as the largest subcoalgebra of $C \otimes D$ contained in the $k Vect$ equalizer. But the $k Vect$ equalizer is easily seen to be $0$ (cf. the fact that $k Vect$ coproducts are themselves disjoint), and so the coalgebra equalizer must be $0$ as well, concluding the proof.
Coproducts of coalgebras are stable under pullback.
Consider the following diagram:
where $f$ is an arbitrary coalgebra map and $j: F \hookrightarrow X$ is a finite-dimensional subcoalgebra inclusion. We will show that the pullback of the coproduct diagram along $f \circ j$ is a coproduct decomposition of $F$.
Michael Barr, Coalgebras over a commutative ring, Journal of Algebra 32 3 (1974) 600-610 [doi:10.1016/0021-8693(74)90161-6, pdf, pdf]
Hans-Eberhard Porst, On corings and comodules, Archivum Mathematicum 42 4 (2006) 419-425 [dmlcz:108017, pdf]
Ana L. Agore, Limits of Coalgebras, Bialgebras and Hopf Algebras, arxiv.org/pdf/1003.0318, 2010. (pdf)
Last revised on November 6, 2023 at 07:45:20. See the history of this page for a list of all contributions to it.