Questions tagged [coalgebras]

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

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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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1answer
297 views

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)...
3
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2answers
225 views

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 $\...
6
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143 views

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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36 views

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 ...
5
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1answer
472 views

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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1answer
116 views

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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106 views

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 ...
2
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1answer
79 views

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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1answer
207 views

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^...
5
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2answers
313 views

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 ...
5
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0answers
216 views

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 ...
3
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69 views

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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114 views

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 $\...
2
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1answer
392 views

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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134 views

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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158 views

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 ...
3
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1answer
176 views

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 ...
7
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1answer
229 views

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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82 views

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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1answer
411 views

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 ...
3
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140 views

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 ...
4
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108 views

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 ...
4
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1answer
228 views

Tensor product of coaugmented conilpotent coalgebras

Let $\mathbb{K}$ be a field of char. 0. Let $\mathrm{A}, \mathrm{B}$ be conilpotent cocommutative coaugmented counital dg-coalgebras over $\mathbb{K}$ (i.e. that their corresponding cokernel of their ...
2
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1answer
100 views

Variant of co-Tor in a bimodule category

Say $\mathcal{C}$ is a strict monoidal abelian category and $A$ is a coalgebra object in $\mathcal{C}$, with left co-modules $M$ and right co-module $N$ (also in $\mathcal{C}$). Then we have a notion ...
6
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2answers
185 views

A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module

For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \...
3
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1answer
185 views

What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates?

Asked on math.stackexchange https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde but didn't get response (in fact got a ...
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1answer
201 views

Meaning of coinvariants of a comodule

Let $M$ be a comodule over the bialgebra $B$, with structure map $\rho:M \to M \otimes B$. The space of coinvariants is defined as $M^{coB}:=\{m \in M~|~\rho(m)=m\otimes 1_B\}$. A book I'm reading ...
3
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1answer
180 views

Confusion about comodules

I'm learning "An Introduction to Exterior Differential Systems" by Gregor Weingart. I have some confusions regarding comodules: In section 3, the author defines a comodule: Later, the "space of ...
5
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117 views

What are regular epimorphisms of coalgebras?

It is known that epimorphisms between coalgebras are surjective maps. My question is What are regular epimorphisms between coalgebras? I am in particular ...
5
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1answer
291 views

Category of bicomodules of a cosemisimple Hopf algebra

A cosemisimple Hopf algebra $H$ is one which is equal to the direct sum of its subcoalgebras. As is well-known, this is equivalent to its category of $H$-comodules being semisimple. Is this also true ...
5
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3answers
439 views

On a dual of Kaplansky's $2^{nd}$ conjecture: admissible algebras?

Kaplansky's second conjecture (on Hopf algebras) deals with "admissible" coalgebras: He calls a coalgebra admissible, if there is an algebra structure making it a Hopf algebra. The conjecture states ...
4
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2answers
395 views

Cocommutativity, comultiplication and coalgebra maps

Given a coalgebra $(C,\Delta,\varepsilon)$, over a field, the following is a well-known property: the comultiplication $\Delta:C\to C\otimes C$ is a coalgebra map if and only if $C$ is ...
9
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1answer
524 views

What are types of coalgebras that are more naturally described by cooperads?

Some background. Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of ...
1
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2answers
316 views

Definition of a cosemisimple Hopf algebra

A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the ...
4
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0answers
257 views

Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see ...
0
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1answer
127 views

bialgebras on quotient polynomials

Is there a general procedure for constructing a bi-algebra out of a quotient polynomial ring? In particular, how do I construct a bi-algebra corresponding to quotient polynomial ring $<x^2-x-1>$?...
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0answers
159 views

An identity satisfied by “Differentiation”

I asked this question in MSE but I did not received any answer. So I repeat it here: Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order ...
1
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1answer
148 views

Yetter-Drinfeld modules as rigid category

I'm reading a proof of the following theorem If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category. In this proof we define $V^*...
1
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1answer
256 views

Difference between two definitions of graded coalgebra

I'm reading articles about applications of Hopf algebras in physics. In one of them there is following definition of graded coalgebra: A coalgebra $C$ is called graded if $C=\bigoplus\limits_{n\ge ...
8
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3answers
423 views

Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...
17
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1answer
1k views

Lurie's approach to the bar-cobar adjunction

I've been trying to read Jacob Lurie's approach to the bar-cobar constructions (Higher algebra §5.2 in the 2014-09 version) but I don't recognize what I know about these constructions. I wonder if ...
6
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1answer
303 views

Cofree Lie Coalgebra

I have problems finding anything about the cofree Lie coalgebra functor $\mathcal{L}ie^c$ out there. Basically all I found was that it appears in Harrison cohomology and that, given a $\mathbb{Z}$-...
6
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1answer
201 views

A coalgebra structure on compact operators

Is there a coalgebra structure $\Delta_{n}$ on $M_{n}(\mathbb{C})$ which is compatible with the natural embedding $i_{n:}M_{n}(\mathbb{C})\to M_{n+1}(\mathbb{C})$ with $i_{n}(A)= A\oplus 0$. ...
4
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0answers
190 views

Explicit description of graded (counital) cofree cocommutative coalgebras

Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$. Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...
2
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1answer
427 views

When is an exponential functor a bialgebra?

My question is about the notion of exponential functors as they are frequently defined in the literature on (strict) polynomial functors, e.g., the paper "General Linear and Functor Cohomology" by ...
7
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1answer
235 views

Problem with Eisenbud's Lemma “Symmetry of Diagonalization”?

This question was first asked on MathSE but nobody answered. In his proof of Lemma A2.5 in his book Commutative Algebra with a View towards Algebraic Geometry, Prof. Eisenbud writes something like ...
17
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1answer
1k views

How is a descent datum the same as a comodule structure?

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong S\...
4
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0answers
177 views

Injectivity criterion for surjective coalgebra maps: does it hold in full generality?

Let $k$ be a commutative ring. Let $C$ be a filtered $k$-coalgebra. This means a $k$-coalgebra equipped with an increasing $k$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...
15
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2answers
2k views

What is a Homotopy between $L_\infty$-algebra morphisms

A $L_\infty$-algebra can be defined in many different ways. One common way, that gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...