The co-algebra tag has no wiki summary.
3
votes
0answers
30 views
Question about comultiplication
I have a question about comultiplication for coalgebras:
Suppose $C$ is a coalgebra over the field $k$. How does one show that the comultiplication map $\Delta:C\to C\otimes C$ is a coalgebra map if ...
1
vote
1answer
71 views
Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.
First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes:
The motivation for ...
3
votes
0answers
33 views
Equivalence of categories of coalgebras
I'm studying monadicity and comonadicity and I´m stuck with the following:
Let $L\dashv R:X\rightarrow Y$ be an adjunction with unit $\eta$ and counit $\varepsilon$. The induced monad on $Y$ is ...
5
votes
2answers
36 views
Co-algebra structure on $k[G]$
Let $k$ be a field. Given an affine algebraic group $G$ (defined as a functor from the category of $k$-algebras to the category of sets) then we have the coordinate ring (or the $k$-algebra ...
5
votes
1answer
60 views
Geometric interpretation of the fundamental theorem for coalgebras?
Given an element $m$ in a coalgebra $C$, there always exists a finite-dimensional subcoalgebra $D \subset C$ containing $m$; this is the fundamental theorem for coalgebras. This obviously isn't the ...
2
votes
0answers
61 views
When do counital coalgebras have a basis of grouplike elements?
Question.
Under what conditions do counital coalgebras have bases consisting entirely of grouplike elements? At least in the case of finite-dimensional coalgebras, or for bialgebras (or Hopf ...
0
votes
0answers
19 views
Is a $C$-subcomodule of $C$ actually a subcoalgebra of $C$?
Let $k$ be some field and let $(C, \Delta, \epsilon)$ be a coalgebra over $k$. Then $(C, \Delta)$ may be viewed as a comodule over $C$. Suppose that $D \subseteq C$ is a subcomodule of $C$ - this ...
1
vote
0answers
27 views
Showing every finite dimensional subspace of a comodule lies in a finite dimensional subcomodule
Let $k$ be a field and let $(C, \Delta, \epsilon)$ be a vector space which is a coalgebra. Let $(M, \delta)$ be a comodule. Suppose $V \subseteq M$ is a finite dimensional space. For each $v \in V$, ...
6
votes
0answers
78 views
What would be an interesting example of a Co-algebra with a base category other than Set?
In most or perhaps all the examples of a co-algebra that I have seen, the properties of sets as the base category was used, like the existence of products and co-product and Cartesian closeness. Does ...
1
vote
0answers
76 views
Are there any known interesting F-(co)algebras where F isn't a set endofunctor?
Are there any known interesting F-(co)algebras where F isn't a $Set$ endofunctor? Every example I can think of deals with sets: an algebra of $X\mapsto 1+X$ for natural numbers, an algebra of ...
