Skip to main content

Questions tagged [tannakian-category]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
8 votes
0 answers
205 views

Tannaka reconstruction for homotopy types

All sorts of things can be reconstructed from their "linear representations". One example is Tannaka (Deligne, Tannaka-Krein, etc.) reconstruction where a group is recovered from its ...
Bugs Bunny's user avatar
  • 12.2k
7 votes
1 answer
552 views

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

Let $X$ be a variety over a field $k$. The étale fundamental group of $X$ fits into the exact sequence: $$1 \to \pi_1^{\text{geom}}(X) \to \pi_1^{\text{arith}}(X) \to \text{Gal}(\overline{k}/k) \to 1,$...
kindasorta's user avatar
  • 2,103
2 votes
0 answers
94 views

Trace morphism in Deligne/Milne's "Tannakian categories"

I originally posted this on MSE, but only got a comment linking an article (Bontea and Nikshych's "Pointed braided tensor categories"). So I'll repost the question in full here: Is there a ...
Ben123's user avatar
  • 203
4 votes
2 answers
145 views

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

Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group). By the Tannakian formalism, $G(k)$ can be ...
Antoine Labelle's user avatar
2 votes
1 answer
116 views

Galois action on étale path torsors

TLDR: How is the Galois action on étale path torsors defined? Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\...
kindasorta's user avatar
  • 2,103
1 vote
0 answers
82 views

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 ...
kindasorta's user avatar
  • 2,103
2 votes
0 answers
61 views

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
  • 2,103
1 vote
1 answer
105 views

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
  • 2,103
3 votes
0 answers
120 views

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
4 votes
1 answer
299 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
  • 203
7 votes
0 answers
269 views

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
  • 15.5k
2 votes
2 answers
361 views

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
279 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
  • 5,565
4 votes
0 answers
120 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
2 votes
0 answers
106 views

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
1 vote
0 answers
103 views

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
  • 11
7 votes
0 answers
173 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
  • 15.5k
7 votes
0 answers
346 views

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
  • 1,139
3 votes
1 answer
213 views

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
127 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
  • 1,139
4 votes
0 answers
168 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 ...
Gabriel's user avatar
  • 1,139
1 vote
1 answer
371 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
  • 4,503
6 votes
1 answer
467 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
9 votes
0 answers
381 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
  • 848
2 votes
0 answers
80 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)\...
Stabilo's user avatar
  • 1,479
6 votes
0 answers
306 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
12 votes
1 answer
770 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
  • 62.6k
1 vote
0 answers
144 views

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
  • 1,876
6 votes
1 answer
201 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
191 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
216 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
853 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
  • 13.3k
8 votes
1 answer
591 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
514 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
  • 8,767
6 votes
1 answer
283 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
  • 1,939
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
  • 2,969
11 votes
0 answers
318 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
  • 419
9 votes
3 answers
656 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
  • 2,969
7 votes
0 answers
251 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 ...
user avatar
5 votes
0 answers
240 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
131 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
  • 8,961
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
867 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
  • 2,711
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
  • 1,700
8 votes
0 answers
263 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
  • 81
3 votes
1 answer
233 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
920 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
795 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
  • 1,700
3 votes
1 answer
459 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
  • 1,700
7 votes
1 answer
610 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