# Questions tagged [tannakian-category]

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### Group action on fibre functor

(I asked this question on mathstack here: https://math.stackexchange.com/questions/4413271/group-action-on-fibre-functor. After getting no response and being suggested in the comments to post it here, ...
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### Given fiber functors on all the subcategories of the form $\langle M\rangle$, can we obtain a fiber functor on the whole category?

Let $\mathsf{T}$ be a rigid abelian tensor category and suppose that we're given fiber functors $\omega_M:\langle M\rangle \to \mathsf{vect}_k$ for every object $M$ of $\mathsf{T}$. Is there a ...
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### Every Tannakian category over an algebraically closed field is neutral

In Tannakian Categories by P. Deligne and J. S. Milne (footnote 16 of the november 2018 version), it is said that Every Tannakian category over an algebraically closed field is neutral (letter of ...
1 vote
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### Grothendieck rings and the Tannakian formalism

I understand that the Tannakian formalism (which I only "know" extremely superficially) is very important for the theory of motives. I guess "the" conjectural category of motives ...
216 views

### Functors between module categories that comes from restriction

Suppose you have two $k$ algebras $A, B$ (say also finitely generated if this helps) and a functor $F: A-mod \to B-mod$ such that $| F(M) |= |M|$. Here $|U|$ denotes the underlying $k$ vector space. ...
341 views

### Why do Coleman functions form a sheaf?

In section 4 of Ammon Besser's 2002 'Coleman Integration Using the Tannakian Formalism,' he defines abstract Coleman functions, which we can describe roughly as those functions which arise by iterated ...
69 views

### What is the isomorphism from $\operatorname{Ext}^1_{T}(Y,X)$ to $\operatorname{Ext}_T(\mathbf{1},X\otimes Y^{\vee})$?

Let $T$ be an abelian rigid monoidal category and $\mathbf{1}$ be a unit object in $T$. For two objects $X$ and $Y$ in $T$, there is a natural group isomorphism n:\operatorname{Ext}^1_{T}(Y,X)\...
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### When does a group and its pro-algebraic completion have equivalent categories of arbitrary representations?

In the following everything is over some field $k$. Let $G$ be a discrete group. We write $G^{\text{alg}}$ for its pro-algebraic completion. The latter is a pro-affine pro-algebraic group which arises ...
534 views

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

$\DeclareMathOperator\sVect{sVect}\DeclareMathOperator\Vect{Vect}$The category $\sVect_k$ of (let's say finite-dimensional) super vector spaces can be obtained from the category $\Vect_k$ of (finite-...
1 vote
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### Roadmap to study (Deligne) Algebraic geometry over Tannakian categories

I would like to know the way to proceed in the first lecture of Deligne's Le groupe fondamental de la droite projective moins trois points. General advices for reading Deligne's paper. What should I ...
407 views

### Relations between Motivic Galois groups and Motivic t-structure?

What are some relations between the existence of Motivic t-structures and Motivic galois groups? I heard that indeed the existence of the Motivic t-structure implies the isomorphism between Ayoub's ...
558 views

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

In many articles (for example in articles given by M.Marcoli) there is statement that there is the following conjecture Residues of Feynman integrals in scalar field theories are always periods of ...
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### Deligne's theorem on the characterisation of Tannakian categories

I am trying to understand the proof of the Theorem 7.1 from Catégories Tannakiennes by Pierre Deligne https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf. Essentially, it is ...
363 views

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

Is there a relationship between $\mathcal{D}$ - modules, Tannakian formalism and Galois theory of monodromy representations ? Thanks in advance for your help.
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### Chevalley property for Tannakian categories

Are all Tannakian categories have the Chevalley property? I know that the categories $\mathrm{Rep}_k(G)$ have the Chevalley property for affine algebraic groups $G$ over a field $k$, but I don't know ...
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### Faithful exact functors to tensor categories

Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P$ an additive $k$-linear exact and faithful functor. I want to ...
1 vote
145 views

### Galois group - unknown word

I found in some articles a definition of the Galois group of a differential module. In the definition appeared the expression $(Repr(Gal(M,\nabla),Forget)$. According to the the articles, this pair ...
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### What is the role of fiber functor in Deligne's theorem on Tannakian categories?

The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...