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3
votes
0answers
65 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
votes
1answer
109 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 ...
6
votes
1answer
153 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 ...
2
votes
0answers
159 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 : ...
4
votes
0answers
138 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 $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...
9
votes
2answers
461 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 ...
0
votes
1answer
166 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 ...
4
votes
1answer
99 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 ...
5
votes
2answers
281 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 ...
1
vote
3answers
313 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 $R \subseteq A \times B$ ...
1
vote
0answers
72 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
votes
1answer
239 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 ...
4
votes
3answers
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 ...
1
vote
1answer
247 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 ...
6
votes
0answers
258 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 ...
6
votes
1answer
352 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 $x\in X$. ...
4
votes
2answers
202 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 ...
5
votes
2answers
461 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 ...
1
vote
1answer
255 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):= ...
8
votes
1answer
680 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
vote
1answer
139 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 ...
25
votes
7answers
2k views

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 ...
4
votes
1answer
391 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 ...
6
votes
1answer
578 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 $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...
7
votes
1answer
577 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. ...
3
votes
2answers
385 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 ...
2
votes
1answer
156 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 ...
5
votes
1answer
774 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: ...
5
votes
2answers
620 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 ...
0
votes
1answer
353 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 ...
0
votes
1answer
240 views

Finite dual of an algebra morphism.

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?
0
votes
3answers
951 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 $\Delta^{m}: C \to C^{\otimes m}$??? ...
10
votes
1answer
530 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 ...
3
votes
1answer
129 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 ...
14
votes
1answer
785 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 ...
5
votes
0answers
184 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 ...
22
votes
5answers
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 ...
8
votes
1answer
587 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 ...
1
vote
2answers
291 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 ...
8
votes
4answers
598 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? ...
7
votes
1answer
263 views

A coalgebraic description of the hyperfinite II_1 revisited

Back here I was asking for a coalgebraic characterisation of the hyperfinite $II_1$ factor. Recall the latter's construction by forming the inductive limit of a chain of matrix algebras $R \to M_2(R) ...
8
votes
1answer
273 views

Is there a coalgebraic characterisation of the hyperfinite II_1 factor?

Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...
9
votes
1answer
307 views

cardinality of final coalgebras in Top

Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐i ≥ 0 Si × Xi where the Si are finite sets, all but finitely many of which are empty. ...
10
votes
2answers
2k views

Co-induction understanding

Hi, I am studying coinduction(not induction) as part of a class on static analysis. Rummaging around the internet, I am simply not finding a clear, concise description of: How coinduction actually ...