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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2 votes
1 answer
89 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 ...
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4 votes
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139 views

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 ...
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1 vote
1 answer
297 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 ...
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5 votes
1 answer
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. ...
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9 votes
0 answers
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 ...
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2 votes
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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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6 votes
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222 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 ...
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10 votes
1 answer
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-...
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1 vote
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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(...
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6 votes
1 answer
181 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 ...
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2 votes
0 answers
101 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 ...
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2 votes
0 answers
189 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 ...
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8 votes
1 answer
753 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 ...
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7 votes
1 answer
436 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 ...
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8 votes
2 answers
400 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 ...
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5 votes
1 answer
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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 ...
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2 votes
0 answers
114 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 ...
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11 votes
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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 $E_{\infty}$-...
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9 votes
3 answers
586 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 $0$) ...
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7 votes
0 answers
218 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 ...
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5 votes
0 answers
195 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 ...
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5 votes
0 answers
128 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 ...
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3 votes
0 answers
130 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 ...
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8 votes
2 answers
749 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 ...
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  • 2,041
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"...
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  • 1,560
8 votes
0 answers
210 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 ...
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3 votes
1 answer
204 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 $v$ ...
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10 votes
1 answer
783 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 $...
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7 votes
0 answers
707 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 ...
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3 votes
1 answer
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 ...
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  • 1,560
7 votes
1 answer
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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5 votes
0 answers
634 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 ...
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3 votes
1 answer
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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3 votes
0 answers
120 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 ...
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6 votes
0 answers
162 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 ...
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12 votes
1 answer
237 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 $...
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  • 114k
4 votes
1 answer
415 views

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 ...
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  • 4,406
1 vote
1 answer
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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11 votes
0 answers
601 views

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 ...
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9 votes
3 answers
648 views

Exact sequences of groups and Tannakian formalism

By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered ...
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12 votes
2 answers
1k views

k-linear abelian categories which are not categories of modules

According to Joyal, Street ("An Introduction to Tannaka Duality and Quantum Groups"), any $k$-linear abelian category $\mathcal{C}$ admitting a faithful, exact functor $U: \mathcal{C} \rightarrow \...
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55 votes
0 answers
3k views

Grothendieck's Period Conjecture and the missing p-adic Hodge Theories

Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the ...
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6 votes
1 answer
316 views

Establishing Duality in Tannakian Categories

I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid. However, I find the definition of rigid categories somewhat difficult. I don't ...
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  • 114k
6 votes
1 answer
673 views

Tannakian fundamental group of two explicit tensor categories

Let $K/k$ is a field extension and $G$ an affine group scheme over $K$. What are the Tannakian fundamental groups of these two $k$-tensor categories (with trivial fiber functors over $k$): 1. The ...
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  • 4,406
6 votes
0 answers
1k views

Torsors and twists of algebraic groups

Let $G/S$ be an affine group scheme. Then the automorphism group of every $G$-torsor over $S$ is a twist of $G$, but it this functor isn't essentially surjective in general (It may be not full nor ...
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  • 4,406
4 votes
1 answer
393 views

Applying Lemma 2.12 of Deligne's/ Milne's "Tannakian Categories" on an irreducible representation

since this is my first question here, I'm not very certain whether I state it properly. I'm thankful for any helpful remarks. Currently I'm trying to understand the above-mentioned article which can ...
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10 votes
1 answer
426 views

Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of Etingof-...
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  • 1,559
4 votes
0 answers
299 views

Tannaka categories and reductive groups

The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple. Pro-reductive groups make sense over any scheme. Is there an extension of the theory ...
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5 votes
0 answers
186 views

Algebraic fundamental group without regularity at infinity

Suppose $X$ is a smooth (connected) variety over $\mathbb{C}$. Let $\mathscr{C}$ be the category of finite rank vector bundles on $X$ equipped with an integrable connection, and let $\mathscr{C}'$ be ...
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