The coalgebras tag has no usage guidance.

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### 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 ...

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**0**answers

121 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 ...

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votes

**1**answer

93 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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**0**answers

145 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 ...

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votes

**1**answer

105 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 $...

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**1**answer

123 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 ...

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**1**answer

222 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 ...

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**1**answer

742 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 ...

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**0**answers

117 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 $\mathbb{Z}$-...

**5**

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**1**answer

175 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 $i_{n}(A)= A\oplus 0$. ...

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**0**answers

111 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 $C(V)$ ...

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**1**answer

226 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 ...

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**1**answer

202 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 ...

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**0**answers

179 views

### (Non trivial) coidempotents(Co-$K$-theory)

I was interested to know about coalgebraic version of "Idempotents".
So I seached the web and I found the following interesting post :
http://math.stackexchange.com/questions/689322/co-idempotents-...

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**0**answers

152 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 $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...

**12**

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**2**answers

948 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 ...

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**1**answer

197 views

### Is Khovanov's Frobenius algebra self-dual over the integers?

Khovanov's Frobenius algebra (used in the definition of Khovanov homology) is $\mathbb{Z}[X]/X^2$ with the comultiplication. $\Delta(X)=X\otimes X, \Delta(1)=1\otimes X+X\otimes 1$ and the trace $\...

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**1**answer

127 views

### A terminal coalgebra of a certain functor on Mes

Let $\mathfrak C = \mathsf{Mes}$ be the category of meausurable spaces and measurable maps. For any object $X\in \mathfrak C_0$ we assign a measurable space $\mathcal P(X)$ whose elements $\mu$ are ...

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**2**answers

322 views

### Free cocommutative commutative Hopf monoids

I have some questions about generalizations of abelian groups, relative to symmetric monoidal categories.
1) Let $C$ be a cocomplete cartesian monoidal category with equalizers. I can show that the ...

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**3**answers

360 views

### Why the preimage rather than image in Stone-type dualities.

I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.
Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ ...

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**0**answers

102 views

### Coinduction and corestriction are quasi-inverse equivalences for comodules?

I'm reading http://arxiv.org/abs/math/0310337.
There the following statement is given without proof:
Let $k$ be a field. Let $C$ be a counitary coaugmented coalgebra, i.e. there is $\eta: C\to k$ ...

**10**

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**1**answer

273 views

### Which W*-algebras are the duals of C*-coalgebras?

A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...

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**3**answers

1k views

### Mistake in Wikipedia Entry “Coalgebra”

Consider the following quote from the Wikipedia entry Coalgebra:
The kernel of every coalgebra morphism $f : C_1 \to C_2$ is a coideal in $C_1$, and the image is a subcoalgebra of $C_2$.
I can't ...

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**1**answer

276 views

### (Co)Universal Property of Quotients/Subs

I'm not completely sure if this bunch of questions is the appropriate Level of MO. However at the same time I think that it is at least slightly above the level of stackex. ...
The tensor algebra ...

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**0**answers

315 views

### Reference for Lie coalgebras needed

In the Wikipeadia article
http://en.wikipedia.org/wiki/Lie_coalgebra
on Lie coalgebras an author said:
"... Just as the exterior algebra of vector fields on a manifold form a Lie algebra [...], the ...

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votes

**1**answer

443 views

### Maps between sets and coalgebras

Hi,
To every set $X$ there corresponds a group-like coalgebra $kX$, with basis $X$. "Grouplike" means that there is a basis $X$ with $\epsilon(x)=1$ and $\Delta(x)=x\otimes x$ for all $x\in X$.
First,...

**4**

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**2**answers

215 views

### Reference for Tensors on graded spaces needed

Is there a good introduction to
1.) Tensor (co)algebras on graded vector spaces ?
2.) Tensor (co)algebras on graded modules ?
In the research field of $L_\infty$-algebras there is some stuff, but ...

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votes

**3**answers

643 views

### Primitive elements of a tensor product of bialgebras

Given a field $k$ of characteristic $0$. For every $k$-bialgebra $A$, let $\mathrm{Prim} A$ denote the $k$-vector subspace of $A$ consisting of all primitive elements of $A$.
What conditions can we ...

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**1**answer

292 views

### Reference for the fact that a coderivation of the (non reduced) tensor coalgebra is determined by its corestrictions

If $V$ is a vector space, let us consider the tensor coalgebra $TV=\bigoplus\limits_{k=0}^\infty V^{\otimes^k}$ with coproduct given by
$$\Delta (x_1\otimes \dots \otimes x_n):= \sum\limits_{i=0}^{n}(...

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**1**answer

731 views

### Connected graded involutive bialgebras (Hopf algebras) sought (for Dynkin idempotent checking)

Again, there is a general and a concrete question:
Concrete question. Let $H$ be a connected graded bialgebra over a commutative ring $k$ with unity (where graded means $\mathbb N$-graded). Then, it ...

**1**

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**1**answer

153 views

### Linear functional kills all primitives of a connected filtered coalgebra => it lies in m^2?

Let $C$ be a connected filtered coalgebra over a field $k$. Maybe $k$ has characteristic $0$ (though I don't know where this can be of use). Let $1$ denote the unique element of $C_0$ mapping to $1\in ...

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### What is a coalgebra intuitively?

How to think about coalgebras? Are there geometric interpretations of coalgebras?
If I think of algebras and modules as spaces and vectorbundles, what are coalgebras and comodules? What basic ...

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**1**answer

582 views

### Coderivations of S(V) correspond to linear maps S(V) -> V. Only over characteristic 0?

Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
$\Delta ...

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**1**answer

795 views

### Distributions on product spaces

I hope this is suitable to MO.
Question. Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...

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**1**answer

678 views

### What extra conditions are necessary for the following version of Koszul duality?

Conventions: So that I don't have to worry about, fix a field $k$ of characteristic zero, and always work over it. Categories of modules, etc., are always $\infty$-categories of dg modules. ...

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**2**answers

450 views

### Analogue Bialgeras vs Lie bialgebras

I once thought that the analogue of bialgebras and Lie bialegras is similar to that of (associative) algebras and Lie algebras, but it seems not that trivial.
Recall the definitions: a) bialgebra $A$...

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**1**answer

178 views

### Is there a simple description of the indecomposable dg cocommutative coalgebras?

Let $\mathcal C$ be a closed symmetric monoidal category. By $\operatorname{Cog}(\mathcal C)$ I denote the category of cocommutative (counital coassociative) coalgebras in $\mathcal C$. Suppose that ...

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**1**answer

1k views

### What kind of structures allow Galois descent?

EDIT: Question solved.
Let me explain what I mean.
The classical formulation of Galois descent, e. g. in Crawley-Boevey's "Cohomology and central simple algebras", uses the following notion:
...

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**2**answers

660 views

### A special class of regular languages: “circular” languages. Is it known?

We can define a subclass of the regular languages. Fix an alphabet $\Sigma$. Define the "circular" languages (actually, the name already exists to denote a different thing it seems, used in the field ...

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**1**answer

510 views

### How is the coradical filtration defined?

I have seen the coradical filtration of a coalgebra $C$ defined as follows:
$C_0 = \text{sum of all simple subcoalgebras of }C$;
for any $n\geq 1$, let $C_n$ be $\Delta^{-1}\left(C\otimes C_{n-1}+C_{...

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**1**answer

259 views

### Finite dual of an algebra morphism. [closed]

Is finite dual of an algebra morphism a morphism of coalgebras? Does taking finite dual preserves exactness of an exact sequence of algebra morphisms? When is this possible?

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**3**answers

1k views

### How to work with co-multiplication?

Let $C$ be a coalgebra and $\Delta: C \to C\otimes C$ a co-multiplication map. Then, due the co-associative property we can consider $\Delta^m$. But how is defined $\Delta^{m}: C \to C^{\otimes m}$???
...

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**1**answer

612 views

### Why not _co_free modules?

Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...

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**1**answer

134 views

### Maximal subcoalgebras of an $F+1$-coalgebra corresponding to an $F$-coalgebra

This context of this question is Rutten's Universal Coalgebra, used for modelling systems. I'm interested in finding a description of a functor between different types of coalgebras corresponding to ...

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**1**answer

909 views

### Comodule exercises desired

This Question is inspired by a Quote of Moore's
"There are two ‘evil’ influences at work here:
1. we are toilet trained with algebras not coalgebras
2. some of us are addicted to manifolds and so ...

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**0**answers

190 views

### “Question-answer” bisimulation

I often come across relations that would be defined as a bisimulation, except that the label match can be "inexact", that is, in the bisimulation game, a move labelled with "a" can be replied to with "...

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**5**answers

2k views

### Is there an explicit construction of a free coalgebra?

I am interested in the differences between algebras and coalgebras. Naively, it does not seem as though there is much difference: after all, all you have done is to reverse the arrows in the ...

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**1**answer

715 views

### Does the Tannaka-Krein theorem come from an equivalence of 2-categories?

Possibly the correct answer to this question is simply a pointer towards some recent literature on Tannaka-Krein-type theorems. The best article I know on the subject is the excellent
André Joyal ...

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318 views

### In which categories is every coalgebra a sum of its finite-dimensional subcoalgebras?

Let $\mathcal V$ be a reasonably nice category — I'm interested in the case when $\mathcal V$ is $\mathbb K$-linear for some field $\mathbb K$, abelian, and has all products and coproducts (...

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635 views

### Algebraic geometry for cocommutative corings with counit.

Is there a notion of algebraic geometry for these objects? If we take the dual category of the category of cocommutative corings with counit, is there geometry in it in a sense dual to affine schemes? ...