# Questions tagged [coalgebras]

For questions about coalgebras, comultiplication, cocommutativity, counity, comodules, bicomodules, coactions, corepresentations, cotensor product, subcoalgebras, coideals, coradical, cosemisimplicity, ...

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### Hi... I just beginning with the study of categories so my question might seem elementary. [closed]

Please in want to know how to show that the functor $\mathcal{P}(\Sigma \times Id)$ weakly preserve pullbacks.
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### Coalgebraic logic in a general category?

Are there any works on coalgebraic logic (which is, in a way, generalization of modal logic, where we have an object $W$ in a category and a $T$-coalgebra $(W, \gamma)$; modal logic is a spacial case ...
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### Literature for modal logic and coalgebras (focused on logic and category theory)

I've been looking for literature that focuses on coalgebras and modal logic. But all literature I can find is mostly related to automata theory, or otherwise takes heavy influence from computer ...
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### What is a most elementary Coalgebra?

I just read the definition of a coalgebra, defined categorically ( reversing the arrows of the Algebra category ), the given example in the text I am referring to is the Homology on a topological ...
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### Is there some practical intuition when working with a cooperad given by cogenerators and corelations?

In the case of algebras and operads, a description by generators and relations is common practice and I have a good understanding of this. A non-symmetric operad $\mathcal{P}$ given by a linear space ...
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### If C is a coalgebra with counit € and D=ker(€), how define a coalgebra structure on tensor algebra T(D) so that C to T(D) is inclusion coalgebra map

Let C be a coalgebra with comultiplication N and counit € and let D = ker(€). Consider the tensor algebra on D , denoted by T(D). My question is : how one can define a coalgebra structure on T(D) so ...
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### Finding dual of matrix algebra

Consider $A=M_n(K)$, the algebra of matrices over the field $K$. Let $\{e_{ij}\}$ be the standard basis for $A$ and $\{X_{ij}\}$ be the basis of $A^*$ dual to $\{e_{ij}\}$. I need to find the ...
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1 vote
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1 vote
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### Isomorphism between two Hopf algebras

Let $k$ be a field, over which we consider algebras and coalgebras. A $k$-coalgebra is a comonoid object in $k$-modules, and a $k$-algebra is a monoid object in $k$-modules. A $k$-bialgebra is ...
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### Showing that $U(\mathfrak{sl}_2)$ is a coalgebra

We know that there is a coalgebra structure on $U(\mathfrak{sl}_2)$ as follows for any $z\in \mathfrak{sl}_2$: $$\Delta(z)=1\otimes z+z\otimes 1, \qquad \epsilon(z)=0.$$ Can someone be so kind to ...
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### When is there an "intuitive" functor from F-coalgebras to T-coalgebras?

Suppose $F, T : Set \rightarrow Set$ are two functors on the category of sets. Let $F^{coalg}$, $T^{coalg}$ denote the categories of $F$, respectively $T$ coalgebras. Vaguely, I'm interested in when ...
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### Do the regular monomorphisms in the category of coalgebras coincide with the injective ones?

Given a commutative ring $k$ (not necessarily a field), do the regular monomorphisms in the category of (coassociative and counital as usual) $k$-coalgebras (henceforth denoted $k-\mathrm{Coalg}$) ...
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### Can we intersect coalgebras?

Let $k$ be a commutative ring, which I am unwilling to assume is a field, and suppose $(C_i)$ is a collection of coassociative $k$-subcoalgebras1 of a coassociative $k$-coalgebra $C$. Is there always ...
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### Modules over the dual of an infinite dimensional coalgebra

Let $k$ be a field and let $A$ be a finite dimensional (unital, associative, not necessarily commutative) $k$-algebra. The $k$-linear dual of $A$ is a coalgebra, and viceversa, the $k$-linear dual of ...
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1 vote
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### Proving uniqueness of antipodes in Hopf algebras

Let $(H,\mu,\nu,\Delta,\epsilon)$ be a Bialgebra where H is the vector space, $\mu, \nu$ are the product and unit whilst $\Delta, \epsilon$ are the coproduct and counit. Now, for $f,g \in end(H)$ ...
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### A monic coalgebra morphism whose underlying $\text{Set}$ morphism is not injective

Let $F:\mathscr A\to \mathscr A$ be a functor. Consider the following category $\mathscr C$. The objects are arrows $A\to F(A)$. If $\alpha:A\to F(A)$ and $\beta:B\to F(B)$ are two objects, then a ...
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### Confusing with definition of (co)algebra

I began to read the book « Hopf algebras » by Sweedler. He gave the definitions of algebra and coalgebra using commutative diagrams. These diagrams just show (co)associativity and (co)unitary ...
1 vote
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### A question about infinitesimal bialgebra or Newtonian coalgebra

Recently, I am studing an interesting coalgebra which was called the infinitesimal bialgebra by Joni and Rota. It can be regarded as an algebraic framework for the calculus of divided differences. ...
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