16 votes
Accepted

Tannakian Formalism for the Quaternions and Dihedral Group

Let $V_D$ and $V_Q$ be the two dimensional simple representations of $D_4$ and $Q_8$ respectively. Let $1_D$ and $1_Q$ denote their trivial representations. Suppose that there is a tensor equivalence ...
15 votes
Accepted

Conjecture of relation between residues of Feynman integrals and mixed Tate motives

1) Counterexamples were found in the paper Brown, Francis; Schnetz, Oliver: "A $K3$ in $\phi^4$". Duke Math. J. 161 (2012), no. 10, 1817–1862. It is now the general feeling that most $\phi^4$-Feynman ...
13 votes
Accepted

Why linearization leads to arithmetization?

I think: The category of varieties over $\mathbb Q$ is already very arithmetic. One reason that the linearization is considered arithmetic is that so much of the tractable arithmetic information is ...
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12 votes

Tannakian Formalism for the Quaternions and Dihedral Group

The categories ${\rm Rep}(Q_8)$ and ${\rm Rep}(D_8)$ are not equivalent as tensor categories. They have the same Grothendieck ring, but they have non equivalent associators. As far as I am aware, it ...
12 votes
Accepted

k-linear abelian categories which are not categories of modules

If $A$ is a $k$-algebra, and $M$,$N$ are finite-dimensional $A$-modules, then $$\operatorname{Ext}^i_A(M,N)\cong\operatorname{Tor}^A_i(M,N^*)^*$$ (where $*$ denotes $k$-dual). So $\operatorname{Ext}^...
11 votes
Accepted

Derived version of equivalence between motives and representations of Motivic galois groups?

Let $k$ be a field and $\operatorname{DM}_{gm}(k)_{\mathbb Q}$ the ∞-category of rational geometric motives over $k$. A mixed Weil cohomology theory induces a symmetric monoidal exact functor $$ R: \...
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11 votes
Accepted

Is the category $\operatorname{sVect}$ an "algebraic closure" of $\operatorname{Vect}$?

$\newcommand\sVec{\mathrm{sVec}}\newcommand\Vec{\mathrm{Vec}}$Yes. Over an algebraically closed field of characteristic $0$, $\sVec$ is the algebraic closure of $\Vec$. By "algebraic closure" of $K$ I ...
9 votes

Tannaka duality for semisimple groups

In order for ${\mathcal C}$ to come from an algebraic group rather than a pro-algebraic one, you want ${\mathcal C}$ to be finitely generated. And for semisimplicity, you want the group to have finite ...
  • 555
8 votes
Accepted

Tannaka duality for semisimple groups

Another criterion is that there should be only finitely many objects of bounded dimension. This condition might be easy to check in practice from abstract finiteness theorems. The proof is that, if ...
  • 122k
8 votes
Accepted

A question about the Tannaka-Krein reconstruction of finite groups

I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a ...
  • 26.8k
8 votes
Accepted

Relations between Motivic Galois groups and Motivic t-structure?

The argument sketched in Example 3.20 of [J. P. Pridham, Tannaka duality for enhanced triangulated categories, arXiv:1309.0637] demonstrates the comparison assuming the existence of the motivic t-...
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8 votes
Accepted

Establishing Duality in Tannakian Categories

As Mostafa points in the comments, it suffices to have a canonical isomorphism $$ \hom(X\otimes Y,Z) \overset?= \hom(X,Z\otimes D(Y)). $$ But if I am not mistaken, you have $$ \begin{aligned} \hom(X\...
7 votes
Accepted

Tannakian fundamental group of two explicit tensor categories

One possibly useful way of describing these groups is by their universal properties. The first group has the property that homomorphisms from it to any other pro-algebraic group $H$ are in bijection ...
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7 votes
Accepted

Exact sequences of groups and Tannakian formalism

About your main question I suggest looking at Appendix A in On Nori's Fundamental Group Scheme Hélène Esnault, Phùng Hô Hai, Xiaotao Sun Geometry and dynamics of groups and spaces, 377–398, Progr. ...
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7 votes

How much of the category of motives can be recovered from automorphisms of the Betti functor

A precise way to say that "sufficiently nice functors" from schemes to complexes of vector spaces determine realization functors of Voevodsky's motives is the notion of mixed Weil cohomology. There ...
6 votes

A question about the Tannaka-Krein reconstruction of finite groups

Although you already have one sufficient answer, I thought I would add here that Etingof and Gelaki, "Isocategorical groups" considered exactly this question. For finite groups, they gave a complete ...
5 votes

Exact sequences of groups and Tannakian formalism

You can think about category $Rep(K)$ as about $G-$equivariant sheaves on $G'$. This translates into the following: let $A$ be the algebra of functions on $G'$; this is commutative algebra in the ...
5 votes

k-linear abelian categories which are not categories of modules

This is not an answer. Below "finite" means "finite-dimensional over $k$," so "profinite" means "pro-finite-dimensional" and so forth. The category of coalgebras is the ind-category of the category ...
5 votes

Why linearization leads to arithmetization?

I'm not sure if this is already implicit in your question, but: In the subsection "Artin motives" of this paper by Deligne and Milne, the authors explain how the reconstruction result Theorem 2.11 ...
5 votes

Why would the category of Motives be Tannakian?

My impression is that the language of Tannakian categories is a nice thing for itself; thus applying it to motives may (and will, see below) yield interesting applications. There are several ways of ...
4 votes
Accepted

What properties of a finite group fibre functor give its endomorphisms a hopf algebra structure?

The tensor product functor $\otimes : \text{Rep}(G) \times \text{Rep}(G) \to \text{Rep}(G)$ is "bilinear": it preserves colimits in both variables. It can therefore be regarded as being a "linear" ...
4 votes
Accepted

Tannakian formalism for topological Hopf algebras

I believe the best answer to your question at the moment is in this paper (which only treats the case of bialgebras) http://adsabs.harvard.edu/abs/2014arXiv1411.3183L Indeed, in topological setting ...
4 votes
Accepted

Relationship between $ \mathcal{D} $ - modules, Tannakian formalism and Galois theory of monodromy representations

I don't know what level of depth you want, so here is the short version. Given a smooth complex variety $X$, as Daniel Barter points out, the categories of $\mathcal{O}_X$-coherent $D$-modules, vector ...
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4 votes
Accepted

Is there a finite test for isomorphisms of rigid monoidal abelian categories?

In the situation you describe here, the category $\mathcal{C}$ will automatically be symmetric, and the functors will automatically be symmetric functors. The reason is the following: If you have two ...
  • 4,819
4 votes

Tannakian criterion for reducedness of Tannakian dual group

Yes, and one could argue as follows, at least if the group scheme is assumed to be of finite type (so it is an algebraic group), and the base field is perfect. Recall that if $k$ is a field of ...
  • 5,308
3 votes

Exact sequences of groups and Tannakian formalism

Given two elements $V,W$ of $Rep_G$, we can compute $Hom_{K} (V, W)$. It is just the maximal subobject of $V^{\vee} \otimes W$ that comes by pullback from a representation of $G'$. (Because it is the ...
  • 122k
3 votes

Faithful exact functors to tensor categories

If for example $P$ is a fusion category, then such a box product is what is usually referred to as a module category structure on $M$. In case $M$ is nice enough- for example semisimple with finitely ...
  • 4,819
3 votes
Accepted

What is the name of the Hopf algebra whose comodules are the "positive" highest weight modules of $U_{q}(sl(2))$?

It usually goes by "quantized function algebra" of $SL_2$, or "regular function algebra" of $SL_q(2)$, of course with various minor variations. In Kassel's book it's denoted by $SL_q(2)$ (see Section ...
3 votes
Accepted

Functors between module categories that comes from restriction

Let's assume that we are using right modules and that the isomorphism of underlying vector space functors is a natural isomorphism because we will just work up to isomorphism of functors. Note that $...
3 votes

Group action on fibre functor

Deligne proved that assuming that $C$ is Tannakian and $K$-linear, where $K$ is an algebraically closed field of characteristic zero, then there is a unique fiber functor from C to $Vec_K$. If you ...
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