The tag has no wiki summary.

learn more… | top users | synonyms

3
votes
0answers
32 views

Reference for understanding coalgebra

I am trying to read this paper, but I have no knowledge of coalgebra and have just started to learn Category Theory so I am struggling to understand it. Are there any references that can explain ...
2
votes
1answer
40 views

Reference for $F$-algebras and induction?

I've been learning about $F$-coalgebras and coinduction from this fantastic paper, which has really helped me get a feel with its many examples. I'm starting to struggle with reconciling the ...
2
votes
0answers
21 views

Are there always nontrivial primitive elements in a Hopf algebra?

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $H$ be a Hopf algebra over $k$. We say $x\in H$ is a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$, where $\Delta$ ...
3
votes
1answer
39 views

Inducing a comodule structure on Hom

If C is an R-coalgebra and M is an R-module... then is it possible to endow $Hom_{_RMod}(C,M)$ or $Hom_{_RMod}(M,C)$ with the strucutre of a C-comodule?
0
votes
0answers
33 views

Comodules as a functor category

Let C be a comonoid in some preadditive monoidal category $\mathfrak{C}$, then how can we express the category of C-comodules, in terms of some sort of functor category? I mean is there a similar ...
1
vote
2answers
72 views

co-idempotents: algebraic dual of an idempotent element?

So many times you can write out the axioms for an algebraic structure (say an algebra over a ring) as commutative diagrams and then reverse all the arrows and get a new structure: say a coalgebra. ...
1
vote
1answer
29 views

Cotensor and counit?

If M is a C-bicomodule, then considering C as a $C$-bicomodule also, is $M \square_C C \cong C$, where $\square_C$ is the cotensor product in $^C\mathscr{M}^C$.
0
votes
1answer
15 views

Commutativity with cotensor

If C is a cocommutative R-coalgebra, R is some commutative semi-simple artinian ring and A and B are C-bicomodules, then is $A\square_C B \cong B \square_C A$? If not what other conditions are ...
1
vote
1answer
26 views

Commutativity of comodules

If C is a cocommutative R-coalgebra, R is some commutative semi-simple artinian ring and A and B are C-bicomodules, then is $A\otimes_R B \cong B \otimes_R A$ as $R-modules$. However, this also true ...
3
votes
1answer
25 views

cofree comodules and embedding

For an $R$-coalgebra C, is it possible for every C-comodule M to be embeded into a C-comodule of the form $\underset{i \in I}{\bigoplus} C$?
1
vote
1answer
37 views

Comultiplication of sum

If $a$ and $b$ are elements of a Hopf algebra over a field $k$ and $\alpha, \beta \in k$, then what is $\Delta(\alpha a+\beta b)$? Is it $\alpha\Delta(a)+\beta\Delta(b)$? For example if $\Delta(x)=x ...
0
votes
0answers
19 views

derived equivalence of coalgebras

let $C$ and $D$ be two coalgebras over a field, $C$ and $D$ are called derived equivalent if the derived categories $D(C-comod)$ and $D(D-comod)$ are equivalent as triangle categories. if $C$ and ...
0
votes
1answer
36 views

Structure of coalgebra on $\mathbb Z/n\mathbb Z$

Is it possible to give the structure of a $\mathbb{Z}$-coalgebra on $\mathbb{Z}/n\mathbb{Z}$? If so, how would the comultiplication and counit be defined?
2
votes
1answer
54 views

Duals of a finite dimensional eveloping coalgebra

Let $C^e$ be the enveloping $k$-coalgebra of a $k$-coalgebra $C$ and denote by ${C^e}^{\star}:=\mathrm{Hom}\,_{k}(C^e,k)$. Then is ${C^e}^{\star} \cong {C^{\star}}^e$?
2
votes
0answers
105 views

Duality between Tor and Ext?

Let $A$ be a $\mathbb{N}$-graded, locally finite $\Bbbk$-algebra, $\Bbbk$ being a field, $A=\oplus_{n \geq 0} \ A^n$, each $A^i$ being finitely dimensional as a $\Bbbk$ vector space. Assume also that ...
1
vote
0answers
73 views

Coalgebra dual to the symmetric algebra

I'm interested in the coalgebra dual to the symmetric algebra $S(V)$ on some finite dimensional vector space $V$. I'd like to know explicitly what the comultiplication map is.
3
votes
1answer
55 views

$f:C\longrightarrow D$ coalgebra homomorphism $\Rightarrow$ $f^*:D^*\longrightarrow C^*$ algebra homomorphism

I'm trying to prove the next problem using only arrows (avoiding Sweedler's notation): If $f:C\longrightarrow D$ is a coalgebra homomorphism, then $f^*:D^*\longrightarrow C^*$ is an algebra ...
1
vote
0answers
61 views

What does it mean for a coalgebra to be cogenerated by a subspace?

The usual definition of an algebra being generated by a subspace is as follows: Let $A$ be an algebra, $X \subset A$ a subspace, $\mathrm{Alg}(X)$ the free algebra generated by $X$. Then $A$ is ...
1
vote
0answers
56 views

wedge product and direct sum of sub-coalgebras

Given a coalgebra $C$ over the field $k$ and subspaces $U$ and $V$ of $C$, one can define the wedge product $U\wedge V$ of $U$ and $V$ to be $\Delta^{-1}(U\otimes C+C\otimes V)$. Now, suppose that ...
4
votes
0answers
70 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
151 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 ...
4
votes
0answers
51 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
41 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
84 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
94 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 ...
1
vote
0answers
35 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 $v \in V$, ...
5
votes
1answer
117 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 ...
3
votes
1answer
113 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 ...