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 ...

16
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 ...

14
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 ...

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 ...

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 ...

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: \...

11
votes

Accepted

### Are all representations of the geometric étale fundamental group subquotients of representations of the arithmetic étale fundamental group?

This is very far from true. In fact, a representation extending to the arithmetic fundamental group is such a severe restriction that these satisfy strong finiteness results; see [Litt21].
An easier (...

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 ...

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 ...

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-...

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 ...

8
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 ...

6
votes

### Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$

For a reductive group the category of representations is semisimple so the algebra of endomorphisms of the fiber functor is just a product of matrix algebras, one for each irreducible representation.
...

5
votes

### Tensor functor between rigid tensor categories preserves $\text{Hom}$-objects

Your $F$ preserves dual objects because it preserves the duality situations: quadruples $(X,Y,\alpha : X\otimes Y \rightarrow I, \beta: I \rightarrow Y \otimes X)$.
It remains to recall the formula ...

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 ...

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

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" ...

5
votes

Accepted

### Question about references for proof of Proposition 1.3 in P. Deligne & J.S. Milne's article on "Tannakian Categories"

$\require{AMScd}$Let's prove that the left square commutes (I will only give a rough sketch). This means we want to prove that the following diagram commutes
$$
\begin{CD}
X\otimes Y @>l_{X\otimes ...

4
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 ...

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

### 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 $...

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 ...

3
votes

### Tensor functor between rigid tensor categories preserves $\text{Hom}$-objects

$\DeclareMathOperator{\uphom}{\underline{Hom}}$
Looking at the notes, I can see that $\uphom(X,Y)$ is the internal hom of the category.
This has the universal property that that maps into it $ Z\to \...

3
votes

Accepted

### Tannakian reconstruction for braided categories

Beware that the forgetful functor on the category of representations of a quasi-triangular Hopf algebra is typically not a braided functor. Rather, the general pattern is that you can reconstruct a ...

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 ...

2
votes

### Tannaka duality for semisimple groups

I've decided to turn my comments into an answer.
(a) The conditions characterizing the Tannakian categories attached to connected reductive groups can be found in Chapter 2 (2.20, 2.22, 2.23) of the ...

2
votes

### Grothendieck rings and the Tannakian formalism

The conjectural category of motives, being an abelian category, has its own notion of $K$-theory, $K_0 ( \operatorname{Mot}_k)$.
This would be the quotient of the free abelian group generated by ...

2
votes

### Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$

In general I believe this should be the full dual algebra of $\mathcal{O}(G)$, or equivalently the profinite completion of the group algebra $k[G]$. If $G$ is simply connected, not necessarily ...

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