This page describes some nice properties of the category of cocommutative coalgebras over a ground ring , 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 are created (reflected) by the forgetful functor . As the codomain is cocomplete, so is the domain. Thus for example, the coproduct of coalgebras is the direct sum module equipped with the evident comultiplication .
It is perhaps well to point out explicitly that if is a coalgebra map, then the image in inherits a coalgebra structure from and provides the image in . This is an easy consequence of the fact that as submodules of .
In the case where is a field , the category is locally finitely presentable. The finitely presentable objects of 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 is the union of its finite-dimensional subcoalgebras, i.e., is the directed colimit of the system of finite-dimensional subcoalgebras of and inclusion maps between them.
(For the nonce, we define “subcoalgebra” of to mean vector subspace such that the restricted comultiplication is contained in the subspace of . Later we will see that subcoalgebras are actually the same thing as subobjects in in the sense of equivalence classes of monomorphisms.)
Every finite-dimensional coalgebra is finitely presentable, i.e., preserves filtered colimits.
If is a directed colimit, then the image of a coalgebra map is a finite-dimensional subcoalgebra inclusion which, as a finitely presentable vector space, is included in one of the components of the colimit cone; this inclusion is a subcoalgebra inclusion.
It follows easily from these results and cocompleteness of that 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 .
The construction of limits can be described explicitly. The equalizer of two coalgebra maps is the largest subcoalgebra contained in the equalizer (the latter coincides with the equalizer as computed in Set). This can be described even more explicitly in Sweedler notation: the equalizer of is the set
with the unique structure of coalgebra that makes it a subcoalgebra of .
Assuming the construction of cofree cocommutative coalgebras, viz. the right adjoint to the forgetful functor (which we also touch on below), the product of a family of coalgebras can be described as follows. Consider the product taken in , and let be the component of the counit of the adjunction at that product. Then the product of the taken in is the largest subcoalgebra such that each composite is a coalgebra map. For a proof, see the article by Agore.
For locally presentable or more generally total categories , cocontinuity of a functor is enough to guarantee that has a right adjoint. It follows that is cartesian closed.
Again, since is cocontinuous and is locally presentable, has a right adjoint . This is described more explicitly at cofree cocommutative coalgebra.
The comonadicity of is proven in the article by Barr, section 4.
We will show that is a lextensive category.
Coproducts of coalgebras are disjoint.
The coproduct of coalgebras is , and the pullback of the two coalgebra inclusions is the coalgebra equalizer of the two maps
This coalgebra equalizer is constructed as the largest subcoalgebra of contained in the equalizer. But the equalizer is easily seen to be (cf. the fact that coproducts are themselves disjoint), and so the coalgebra equalizer must be as well, concluding the proof.
Coproducts of coalgebras are stable under pullback.
Consider the following diagram:
where is an arbitrary coalgebra map and is a finite-dimensional subcoalgebra inclusion. We will show that the pullback of the coproduct diagram along is a coproduct decomposition of .