Questions tagged [tannakian-category]

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Extensions in the category $F\text{-Isoc}(X)$

Let $X$ be a smooth affine scheme over a finite field $k$, let $W(k)$ denote its Witt ring, and by $K$ its fraction field. Let $F\text{-Isoc}(X/K)$ denote the category of convergent $F$-isocrystals on ...
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Fibre functors of the category $F\text{-Isoc}(X)$

Let $X$ be a smooth affine scheme over a finite field $k$. Denote its Witt ring by $W(k)$, and the fraction field of its Witt ring by $K$. Let $F\text{-Isoc}(X)$ denote the category of convergent $F$-...
kindasorta's user avatar
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Full Tannakian subcategories and surjection of fundamental groups

Let $(\mathcal{T},w)$ be a neutral Tannakian category over a field $k$, with fundamental group $G$, and $w$ a fibre functor. Let $(\mathcal{S},w|_{\mathcal{S}})$ be a full Tannakian sub-category (i.e. ...
kindasorta's user avatar
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Tannaka duality for Hopf algebroids

Setting. Let $k$ be a field, $A$ a finite-dimensional $k$-algebra, and $H$ a Hopf algebroid over $A$ with invertible antipode. Denote by $\operatorname{mod}(H)$ the category of finite-dimensional ...
Max Demirdilek's user avatar
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1 answer
259 views

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

I am trying to understand the proof of proposition 1.3 in the following article by P. Deligne & J.S. Milne: Tannakian Categories. They reference Saavedra Rivano's 1972 "Catégories ...
Ben123's user avatar
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What exactly is a Tannakian subcategory?

I've searched all the standard references (Deligne--Milne, Saavedra-Rivano) and cannot find a definition of Tannakian subcategory. What I find is many authors who discuss the Tannakian subcategory ...
David Corwin's user avatar
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2 votes
2 answers
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Tensor functor between rigid tensor categories preserves $\text{Hom}$-objects

I was looking at these notes on Tannakian categories. Let me briefly recall the notion of tensor functors: Let $(\mathcal{C},\otimes)$ and $(\mathcal{C'},\otimes')\DeclareMathOperator{\uphom}{\...
Hajime_Saito's user avatar
5 votes
1 answer
263 views

Tannakian reconstruction for braided categories

Let $\mathcal{C}$ be a symmetric monoidal category. One can imagine a theorem Tannakian reconstruction: If $\mathcal{B}$ is a braided monoidal category and $F:\mathcal{B}\to \mathcal{C}$ is a functor ...
Pulcinella's user avatar
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4 votes
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115 views

Tannakian reconstruction and the distribution algebra

$\DeclareMathOperator\Dist{Dist}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\End{End}$Let $G$ be an affine group scheme over a commutative ring $k$ (I am mainly ...
Antoine Labelle's user avatar
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Admissible representations of an $\ell$-group are a (neutral) Tannakian category?

Let $G$ be an $\ell$-group in the sense of Bernstein/Zelevinsky (sometimes also called td-group), i.e. $G$ is a Hausdorff locally compact totally disconnected topological group. Prominent examples ...
Maty Mangoo's user avatar
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Finite groups acting on algebraic groups and representations

Let $H$ be a connected algebraic group over an algebraically closed field $k$, and $I$ a finite group which acts on $H$ through group scheme morphisms. Denote by $Rep(H)$ the category of finite ...
SoruMuz's user avatar
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166 views

Has anyone written about filtered Tannakian categories?

tl;dr Is there any source that discusses the concept of a filtered Tannakian category? I'm writing a paper with this notion and want to know if it's ever been discussed. The original book by Saavedra-...
David Corwin's user avatar
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What is a morphism of Tannakian categories?

I feel that this question is interesting but has not received enough attention; possibly because it's in MSE. So, the present question is mainly a repost, in the hopes of getting a good answer. (If ...
Gabriel's user avatar
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1 answer
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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, ...
angry_math_person's user avatar
2 votes
1 answer
118 views

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 ...
Gabriel's user avatar
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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 ...
Gabriel's user avatar
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1 answer
357 views

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 ...
THC's user avatar
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6 votes
1 answer
407 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. ...
Andrea Marino's user avatar
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0 answers
378 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 ...
pupshaw's user avatar
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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)\...
Stabilo's user avatar
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290 views

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 ...
Patrick Elliott's user avatar
11 votes
1 answer
722 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-...
Tim Campion's user avatar
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A question about subobjects of the unit in a rigid abelian tensor category

I have a question about Proposition 1.17 in Deligne and Milne, Tannakian Categories (see here), in the last 4 lines of the proof. I don't know how it follows from $U\otimes U\simeq U$ that $T=\ker(...
Lao-tzu's user avatar
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6 votes
1 answer
199 views

Tannakian criterion for reducedness of Tannakian dual group

Given an affine group scheme G over a field of positive characteristic. Question: Is there a simple criterion for G to be reduced in terms of the neutral Tannakian category of its finite dimensional ...
Christopher Marlowe's user avatar
2 votes
0 answers
181 views

Restriction to the maximal torus

$\DeclareMathOperator\ad{ad}\DeclareMathOperator\ind{ind}\DeclareMathOperator\res{res}\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Hom{Hom}$Let me say that I am kind of sure that all the things I ...
Andrea Marino's user avatar
2 votes
0 answers
210 views

Deligne's internal characterisation of Tannakian categories - glueing of algebras

I am currently reading the proof of Deligne's internal characterization of Tannakian categories (Thm. 7.1) in Deligne: Catégories tannakiennes. My question is similar to this one. Given a ...
Aaron Wild's user avatar
10 votes
1 answer
845 views

Is Tannaka theory easy?

$\def\A{\mathcal A}\require{AMScd}$ Disclaimer 1: "Tannaka theory" is an umbrella term referring to a family of results; I might have cherry-picked a version of the theorem that is particularly ...
fosco's user avatar
  • 13k
8 votes
1 answer
583 views

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

Say we are working with schemes over a field $k\subset \mathbb{C}.$ A motive in the sense of Voevodsky is a functor $Sch\to D^bVect$ from (an appropriate category of) schemes to the DG category of ...
Dmitry Vaintrob's user avatar
9 votes
2 answers
505 views

A question about the Tannaka-Krein reconstruction of finite groups

In Chapter 15 Section 15.2.1 of Quantum Groups and Noncommutative Geometry, 2nd edition, the authors raised a question: can we reconstruct a finite group $G$ from its category of finite dimensional ...
Zhaoting Wei's user avatar
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6 votes
1 answer
277 views

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

Tannaka duality for a finite group lets us recover the group algebra $\mathbb{C}[G]$ as the endomorphisms of the forgetful functor $F:RepG\rightarrow Vect$, and taking the monoidal automorphisms ...
Chris H's user avatar
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2 votes
0 answers
118 views

Converse to Tannaka duality for rings

Let $k$ be a field. It's well known and easy to prove that a $k$-algebra $R$ may be recovered from the functor $F: R-Mod \to Vect_k$ as the endomorphisms of $F$. Now suppose $\mathcal{C}$ is a ...
Exit path's user avatar
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11 votes
0 answers
312 views

Examples and non-examples of Tannakian $\infty$-categories

My definition of a Tannakian $\infty$-category is taken from this paper ("Tannaka duality over ring spectra" by James Wallbridge). (p. $53$, definition $7.9$) Let $R$ be an $E_{\infty}$-...
Doelt_k's user avatar
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9 votes
3 answers
641 views

Tannaka duality for semisimple groups

Tannakian formalism tells us that for any rigid, symmetric monoidal, semisimple category $\mathcal{C}$ equipped with a fiber functor $F: \mathcal{C} \to Vect_k$ for a field $k$ (of characteristic $0$) ...
Exit path's user avatar
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7 votes
0 answers
249 views

Is this Related to Tannakian Formalism?

I am wondering how I might be able to express the following phenomenon, which is essentially equivalent to Artin's linear independence of characters, in Tannakian formalism. Any help would be much ...
Ronald J. Zallman's user avatar
5 votes
0 answers
230 views

Tannakian theory for Lie algebras

Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
Rosa Ivanovic's user avatar
5 votes
0 answers
130 views

What do I know about a group if its representations are filtered?

Let $G$ be an affine group scheme over a field. Say that, for every finite-dimensional representation of $G$, I have a $\mathbb{Z}$-grading on the underlying vector space, compatible with tensor ...
Julian Rosen's user avatar
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3 votes
0 answers
133 views

Representations of the intersections of two algebraic subgroups

Let $G$ be an algebraic group over a field $k$ (say of characteristic $0$) and let $H,H'$ be two closed subgroups. I would like to understand the category $Rep_k(H \cap H')$ of finite dimensional ...
bob's user avatar
  • 123
9 votes
2 answers
853 views

Tannakian Formalism for the Quaternions and Dihedral Group

It is a basic fact in representation theory of finite groups over complex numbers that the character tables of $Q_8$ and $D_8$ are identical. I believe, this implies that the corresponding categories ...
Dr. Evil's user avatar
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11 votes
3 answers
1k views

Why linearization leads to arithmetization?

Sorry for this question, but I think it is really important the intuition here. Motives can be seen as the 'best' way of linearizing the study of schemes, des-composing them into "cohomological atoms"...
tttbase's user avatar
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8 votes
0 answers
244 views

Mumford-Tate group of the Fermat curve

Let $C$ be the Fermat curve of degree $d$, defined by the equation $x^d+y^d=z^d$ in $\mathbb{P}^2$. The first cohomology group $H^1(C, \mathbb{Q})$ carries a pure Hodge structure, so it has an ...
fette91's user avatar
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3 votes
1 answer
230 views

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

The finite-dimensional representations over $\mathbb C(q)$ of $U_q(\mathfrak{sl}(2))$ are all highest weight. There are two irreducible modules of each dimension. In one, the highest weight vector $v$ ...
Theo Johnson-Freyd's user avatar
10 votes
1 answer
901 views

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

A slight variant $\tilde Mot_{num}(k,\mathbb{Q})$ of the category of pure motives $Mot_{num}(k,\mathbb{Q})$ is a Tannakian category equivalent to a category of representations of some algebraic group $...
tttbase's user avatar
  • 1,700
7 votes
0 answers
786 views

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 ...
tttbase's user avatar
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3 votes
1 answer
455 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 ...
tttbase's user avatar
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7 votes
1 answer
599 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 ...
mikis's user avatar
  • 797
5 votes
0 answers
690 views

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 ...
Dimitri Chikhladze's user avatar
3 votes
1 answer
429 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.
YoYo's user avatar
  • 325
3 votes
0 answers
132 views

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 ...
Mostafa's user avatar
  • 4,454
6 votes
0 answers
166 views

Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...
Will Sawin's user avatar
  • 135k
12 votes
1 answer
252 views

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

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors $...
Will Sawin's user avatar
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