# Questions tagged [coalgebras]

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

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### Subcoalgebras of symmetric algebra

Consider the symmetric algebra $S(V)$, with its coalgebra structure: $\Delta(x)=1\otimes x+x\otimes1$ on $V$, extended multiplicatively. What are its subcoalgebras?
In some vague sense, they seem to ...

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### Infinite-dimensional, non-unital Frobenius algebras

A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital coassociative ...

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### Hopf algebra and coideal question

Let $A$ be a Hopf algebra (over a field). Consider a unital subalgebra $B\subseteq A$ with $\Delta(B)\subseteq B\otimes A$. Put
$$B^+:= B\cap \ker(\epsilon).$$
It can be shown that $B^+$ is a two-...

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### Is there a coalgebraic definition of filtered algebras?

If $M$ is a monoid, then an $M$-graded algebra over $k$ is the same thing as a $k[M]$ comodule algebra. To see this, if $\delta$ is a coaction of $k[M]$ on an algebra $A$, for each $m \in M$ define
$$...

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### Equivalent definitions of pro-unipotent coalgebras

I'm trying to find a reference in the literature for equivalence of the following two definitions of pro-unipotent coalgebras.
Definition Let be $H$ a coagumented coalgebra and let $\Delta \colon H \...

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### Lie coalgebra with no finite-dimensional subcoalgebras

In Walter Michaelis' paper Lie Coalgebras, he gives on page 9 an explicit example of a Lie coalgebra which is not the union of its finite-dimensional Lie subcoalgebras. In fact, Michaelis' example has ...

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### Coproduct for a Frobenius algebra

The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...

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### Inverse limit of chains of Eilenberg Mac Lane spaces

Let $... \to G_2 \to G_1$ an inverse system of abelian groups with inverse limit $G$, let $n \geq 2$ and $F$ a field. The induced inverse system $$... \to C_*(K(G_2,n);F) \to C_*(K(G_1,n);F) \ (*)$$
...

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### Homotopy fixed points vs coalgebras

Referring to the last part of this answer https://mathoverflow.net/a/225403/170683, I would like to understand how in the case of a Galois cover $f\colon X\to Y=X/G$ with Galois group $G$ (I guess ...

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### Morphism of conilpotent coalgebras

I have a stupid question about morphisms of two conilpotent coalgebras $\phi:X\to Y$. Is it a morphism of coaugmented coalgebras such that $\phi(F_n(X))\subseteq F_n(Y)$? Here $F_n$ denotes the ...

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### Conilpotent coalgebras as pushouts of trivial coalgebras

Let $K$ be a field and $C$ a non-counital conilpotent coassociative coalgebra over $K$
whose underlying $K$-vector space is finite dimensional.
Question: Can one obtain $C$ by iterately taking ...

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### Which categories of presheaves are algebraically cocomplete?

We say that a category is algebraically complete when every endofunctor has an initial algebra. Similarly, a category is algebraically cocomplete when every endofunctor has a final coalgebra.
Assuming ...

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### How do you get the rational submodule of a $C^*$-module (equivalent to a $C$-comodule)?

Let $C$ be a coalgebra over a field $K$. Let $M$ be a $C^*$-module. I am trying to understand the rational submodule of $M$, which will carry the structure of a $C$-comodule.
It seems to me that the ...

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### Different Bialgebra/Hopf algebra structures on coalgebras

Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.

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### What is a coalgebra?

A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. ...

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### Bialgebra maps and Hopf algebra maps

Let $H$ and $H'$ be two Hopf algebras, and let $\phi:H \to H'$ be an bialgebra map. Then is $\phi$ automatically a Hopf algebra map?

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### Classifying Hopf algebras that admit a single irreducible comodule

Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule
$$
k \to k \otimes H, ~~ k \mapsto k ...

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### Frobenius algebras associated to posets and coalgebra structures

Let $P$ be a finite poset that we assume for simplicity to be bounded (that is it has a global maximum M and minimum m).
Let k be a field, then the classical incidence algebra $kP$ has $k$-vector ...

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### Cotensor products (in monoidal categories) without regularity

In Internal Categories and Quantum Groups, Aguiar defines the cotensor product of two bicomodules as follows. Let
$(\mathcal{V},\otimes_{\mathcal{V}},\mathbf{1}_{\mathcal{V}})$ be a monoidal category;...

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### Reconstruction of coalgebras

In the paper Reconstruction of hidden symmetries of Bodo Pareigis in the subsection "3.1 Reconstruction of coalgebras" there is the following proposition (3.3.).
Let $\mathcal{C}$ be a ...

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### Do chains send homotopy inverse limits of spaces to homotopy inverse limits of $E_\infty$-coalgebras?

Let $X_\bullet := ... X_2 \to X_1$ be a tower of connected and simple spaces
with the following properties:
The induced tower $H_\ast(X_\bullet; \mathbb{F}_p)$ of graded $\mathbb{F}_p$-vector spaces
...

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### Show that a certain element is a linear combination of tensors

I posted this question on MSE but got no answer even after putting a bounty on it, so I figured I can try to ask here.
Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that ...

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### Contramodule as direct limit of its finitely generated subcontramodules

$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \...

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### $M$ comodule if and only if $N$ and $L$ comodules

Let $k$ be a field, $C$ a $k$-coalgebra, and $M$ a left $C$-comodule. Then, for a short exact sequence
$$
0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0
$$
of vector spaces, we have that $N$...

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### Comparing Hochschild (co)homology for algebras and coalgebras

Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes
$$M\otimes A^{\otimes n}$$
and
$$\...

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### Definition of subcoalgebra over a commutative ring

Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$.
Notes I'm reading give the following definition:
$D$ is called subcoalgebra of $C$ if the ...

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### Non-counital coalgebras

For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...

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### On reflexive bialgebras

Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...

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### presentability rank of categories of coalgebras

The following theorem is relatively classical:
Theorem: Given an accessible endofunctor, (co)pointed endofunctor or (co)monad $T$ on a locally presentable category $C$, then the category of $T$-(co)...

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### Universal property of the cocomplete category of models of a limit sketch

Let $\mathscr{S}$ be a limit sketch in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\...

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### Comonoids in the category of monoids

Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids?
...

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### Weakly reflexive algebra vs proper (residually finite-dimensional) algebra

Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine):
An algebra $A$ is called ...

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### Comultiplication on objects in an (abelian?) category

Looking for example at $R$-modules for some commutative $R$, we have the direct sum and the tensor product acting analogously to addition and multiplication.
After studying a little bit about co-...

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### Examples of basic coalgebras

For an algebraically closed field $k$, let $C$ be a $k$-coalgebra. Given a minimal injective cogenerator $E$, there is a so-called basic coalgebra $B_C=coend^C(E)$, s.t. the comodule categories $Mod^C$...

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### Do comodules form an exact category?

Let $R$ be a commutative ring, $C$ a coalgebra over $R$. I am asking about the category of $C$-comodules $C$-Comod.
It is clear that if $C$ is a flat $R$-module, then $C$-Comod is abelian. Hence, is ...

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### Is there a way to adjoin a counit to a non counital coalgebra?

Let $k$ be a field.
If $A$ is a non unital $k$-algebra, there is a simple way to make it unital by taking $\tilde{A}:=A\oplus k$ and setting
$$ (a+\lambda)(b+\mu):=ab+\lambda b +\mu a +\lambda \mu$$
...

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### An inner product approach to Hopf algebras

We fix the standard inner products on $\mathbb{C}^n$ and $\mathbb{C}^n\otimes \mathbb{C}^n$.
Is there an algebra structure on $\mathbb{C}^n$ with multiplication $m$ such that the adjoint operator $m^...

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### Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?

I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg ...

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### Does the associated graded functor take products of filtered k-coalgebras to graded k-coalgebras?

Let's suppose we have two noncommutative graded k-coalgebras $C_1$ and $C_2$ with respective admissible filtrations (i.e $F_{0}C_i=0$ and $\mathrm{colim}_k F_kC_i=C_i$), I would like to know if there ...

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### Simple coalgebra under base change

Let $C$ be a simple coalgebra over a field of characteristic $0$. Let $K$ be a field extension of $k$. Is the coalgebra $C\otimes_k K$ over $K$ simple?

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### Compact Generation of Co-Module Categories

Let $\mathcal{C}$ be a compactly generated stable $\infty$-category, linear over a field of characteristic $0$ (i.e., so that it is in particular a dg-category). Let $A$ be a co-monad acting on $\...

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### Coalgebras(or quantum groups) which admit a linear operator satisfying certain functional equation

What is an example of a coalgebra $C$ which admit a linear operator $T$, different from scalar operators $T=\lambda Id$, which satisfy $$(T\otimes T)\circ \Delta=\Delta \circ T^2 $$ but $C$ is not ...

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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 ...

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### Duality between coalgebras and (pseudocompact) algebras - uniqueness

The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below.
Let $k$ be a field. The ...

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### Tannaka-Krein reconstruction and rigidity

Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have ...

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### Monoidal structures on modules over derived coalgebras

Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can ...

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### Limits in subcategories of Powerset-coalgebras

Let $F:Set\to Set$ be a functor. An $F$-coalgebra is a pair $\mathcal{A}=(A,\alpha)$ where $\alpha:A\to F(A)$ is arbitrary map.
Given $F$-coalgebras $\mathcal{A}=(A,\alpha)$ and $\mathcal{B}=(B,\beta)$...

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### Why is subcoalgebra structure unique?

Sorry for breaking the harmony of MO with a easy and silly questions. I have stuck on elementary category-theoretic reasoning about subcoalgebras, namely:
Let $F:\mathcal{Set}\to \mathcal{Set}$ be a ...

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### Completion of coalgebras

Is it possible to complete commutative dg associative (conilpotent) coalgebras over $\mathbb{Q}$ in a way so that when we complete the symmetric coalgebra $Sym(V)$ it becomes completed with respect to ...

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### Building conilpotent coalgebras from co-square-zero-extensions

Let $\mathrm{K}$ be a field of char. 0.
Given a chain complex $\mathrm{X} $ over $\mathrm{K}$ denote $\mathrm{E}(\mathrm{X})$ the co-square-zero-extension on $\mathrm{X}, $ i.e. the
cocommutative ...