# 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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I'm new to coalgebras and this is a question from section 2.1 of the book "Hopf Algebras" from Davied E Radford. I tried to pick an element $u \in U$, so $\Delta(u) = u_1\otimes u_2 \in V\... 1 vote 1 answer 43 views ### Exercise 2.3.24 - Radford's Hopf Algebras I am having trouble solving the following problem, which is Exercise 2.3.24 of Radford's Hopf Algebras: Let$C = C_n(k)$, where$n \geq 1$, and let$\{e_{i,j}\}_{1 \leq i,j\leq n}$be a standard ... 1 vote 1 answer 79 views ### For a terminal coalgebra, map is isomorphism I'm trying to solve Emily Riehl's "category theory in context", question 1.6.vi (1.6.6). The exercise says: A coalgebra for an endofunctor$T: C \rightarrow C$is an object$c \in C$... 2 votes 1 answer 93 views ### "Basic" coalgebra structures on$R$-modules Lately I have been reading about coalgebras and wondering about natural ways one can make any$R$-module into an$R$-coalgebra. Two examples of such constructions are given by tensor coalgebras and ... 1 vote 1 answer 52 views ### How to compute the terminal coalgebra (or initial algebra) of a given functor This is sort of a soft question. I'll start with an example. Fix a set$A$, and consider the functor$F\colon \mathbf{Set} \to \mathbf{Set}$,$F\colon X \mapsto A \times X$. Now, we know that the ... 3 votes 1 answer 76 views ### Koszul dual cooperad of the associative operad I am trying to compute the$k$-modules of$\mathcal{As}^¡$, the Koszul dual cooperad of the associative operad$\mathcal{As}$. I am using sections 7.1.4 and 7.2.1 of Algebraic Operads to try to do ... 1 vote 1 answer 49 views ### Axioms of a coalgebra restated using Sweedler's notation I'm struggling with understanding manipulation using Sweedler's notation at a very fundamental level. I don't understand the equivalence of the axioms of coalgebras in the standard notation [Coproduct ... 1 vote 0 answers 76 views ### Hopf "algebroid" structure of a groupoid convolution algebra? To male thinks simple as possible, lets say we have a discrete group$G.$Then the then the group algebra$\mathbb{C}[G]$(of finitely supported complex valued functions on$G$) has a convolution and ... 1 vote 0 answers 35 views ### How to prove the reduced comultiplication of a coaugmented coalgebra is coassociative? a coalgebra over a field$k$is a vector space$C$over$k$together with$k$-linear maps $$\text{comultiplication } \Delta: C \to C\otimes_k C \text{ and}$$ $$\text{counit } \epsilon: C \to k$$ ... 2 votes 0 answers 90 views ### Show that a certain element is a linear combination of tensors Let$(A, \Delta: A \to A \otimes A)$be bialgebra (unital and counital) such that the map $$T: A \otimes A \to A \otimes A: a \otimes b \mapsto \Delta(a)(1 \otimes b)$$ is surjective. We write$\Delta(... 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 ...
1 vote
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Let $(S, \le)$ be a partially ordered finite set. Let $C$ the vector space with basis $\{e_{i,j} | i,j \in S, i \le j\}$, which turns out to be a coalgebra with comultiplication and counit given by: $$... 1 vote 1 answer 132 views ### Dual of an algebra is a coalgebra Let A be an algebra over the commutative unital ring k that is finitely generated and projective as a k-module. Let A^*= \operatorname{Hom}_k(A,k).  Then the natural map$$i: A^* \otimes A^* \... 35 views

### 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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### 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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Proof attempt Let $(A, \mu, \eta)$ be an algebra and $(C, \Delta, \epsilon)$ be a coalgebra, both over the same field $k$. Define the convolution product $$*: \mathrm{Hom}(C,A)\otimes \mathrm{Hom}(C,... 3 votes 1 answer 52 views ### Showing the triple (\hom(C,A),@,\mu \epsilon) defines an algebra Showing the triple (\hom(C,A),@,\mu \epsilon) defines an algebra Let (C,\Delta,\epsilon) be a colalgebra and (A, \mu, \nu) be an algebra where \Delta, \mu are the coproduct and product whilst ... 1 vote 1 answer 54 views ### Showing \lambda: A \otimes C^* \rightarrow \text{Hom}(C,A) is a morphism of algebras Show that \lambda: A \otimes C^* \rightarrow \text{Hom}(C,A) is a morphism of algebras. Let either C^* or A be finite dimensional, and let \lambda be the isomorphism \lambda: A \otimes C^* \... 0 votes 1 answer 25 views ### Prove that \gamma is a map of C^*-modules Let C be a coalgebra and M a C^*-module, where C^* is C's dual. Prove that$$\gamma: M \otimes C \rightarrow \text{Hom}_\mathbb{k}(C^*,M) \\ m \otimes c \mapsto [f \mapsto mf(c)]$$is a C^*-... 3 votes 1 answer 258 views ### Showing tensor product of coalgebras is a coalgebra. Let (C, \Delta, \epsilon) and (C',\Delta', \epsilon') be two coalgebras over the field k. I'm trying to show that C \otimes C' is a coalgebra for the comultiplication$$\overline{\Delta}:=(id_{... 134 views

### 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 ...
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