# Questions tagged [coalgebras]

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

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18 views

### Hopf Algebra Structure of fixed ring of group algebra

In Lindsay Childs' "Taming Wild extensions, Hopf algebras and local Galois module theory" , theorem 6.8 [part of Greither and Pareigis' Hopf Galois correspondence theorem] we have a ...
58 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 $c \in C$ ...
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### “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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 $$\text{comultiplication } \Delta: C \to C\otimes_k C \text{ and}$$ $$\text{counit } \epsilon: C \to k$$ ...
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### Coalgebras & Coideals: Why does $ker(\pi \otimes id_C ) = I\otimes C$ hold?

In a proof on comodules and coideals I found the following passage: “Let $C$ be a coalgebra, and $I \subset C$ a vector subspace. Let $\pi$: $C \rightarrow C/I$ be the canonical projection. Consider ...
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### Coideals in the grouplike colagebra are spanned by differences

Let $k$ be a field, and let $S$ be a nonempty set. Let $k[S]$ be the grouplike coalgebra of $S$ over $k$, i.e. the free vector space with basis $S$ equipped with the coproduct $\Delta(s)=s\otimes s$ ...
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### Matrix Coefficients Map a Coalgebra Morphism

Simple question: let $C$ be a coalgebra over a field $k$. Given a finite-dimensional right $C$-comodule $V$ determined by the structure morphism $$a:V\to V\otimes C,$$ we have a natural 'matrix ...
78 views

I'm having trouble proving the following statement: If a monad $T$ has a right adjoint $K$, then $K$ is a comonad and the categories of $T$-algebras and $K$-coalgebras are isomorphic. So far I've ...
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### The take lemma needs a coinductive proof

In Are coinductive proofs necessary?, the answerer claimed that we cannot prove inductively the take lemma: Two streams that agree on all initial subsequences of given length are the same. I was ...
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### Products In the Categary of Skew-Cocommutative Coalgebras Are Skew Tensor Products

Let $A = \bigoplus^n_{i=0} A_i,B = \bigoplus^b_{i=0} B_i$ be a graded modules over the same commutative ring $R$ . The twisting isomotphism $\tau_{A,B} : A \otimes B \to B \otimes A$ is defined on ...
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### How to get a Hopf algebra from a monoid?

Let $k$ be a field. Let $S$ be a monoid with netural element $e$. Suppose that all $s \in S$ have only finitely many factoriztions $s=ab$, where $a,b \in S$. Then the free k-module $k[S]$ has a ...
78 views

### Confusion about dual of comodule

I am confused about something. Please help! :) All objects are vector spaces over a fixed field $k$. Let $C$ be a coalgebra with comultiplication $m:C\to C\otimes C$. Let $M$ be a left comodule ...
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### Counterexample to the fundamental theorem of comodules

The Fundamental Theorem of Comodules (aka the Finiteness Theorem for Comodules) states that if $\Bbbk$ is a field, then any element of a comodule over a $\Bbbk$-coalgebra lies in a finite-dimensional ...
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### Sweedler Notation: limitations and left comodule version.

I have two questions: In the nlab entry about the sweedler notation it reads "One can formalize in fact which manipulations are allowed with such a reduced notation." Has anyone done that/an idea ...
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### $\times$ versus $\otimes$ in Definition of $K$-Algebra

While reading about algebras and coalgebras, I keep coming across two definitions of an algebra $A$. One definition uses the Cartesian product $\times$, while another uses the tensor product $\otimes$....
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### if $C$ is a filtered coalgebra, does Gr($B\Omega C)\backsimeq B\Omega ($Gr $C)$ hold?

I have heard that under some assumptions, the functor 'Gr' from filtered graded objects with exhaustive filtration to graded objects $X\rightarrow$ Gr$(X)$ commutes with direct sums (this seems to be ...
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### final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...
Let $M$ be a finite dimensional Hopf algebra with antipode $S$ and let $M^*$ be the dual bialgebra. Recalls that $x^* \in M^*$ is a primitive in $M^*$, then $x^*$ is a derivation of $M$. '' I am ...