Questions tagged [tannakian-category]
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115 questions
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Implication of equivalence of categories on exact sequences, Tannakian formalism
Let $F:C_1\to C_2$ be an equivalence of categories. Suppose that $C_1, C_2$ are both exact categories and $F$ sends short exact sequences (SES) in $C_1$ to SES in $C_2$. Let $G$ be the quasi-inverse ...
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Tannakian reconstruction over non-fields
In the original works of Saavedra-Rivano and Deligne, a Tannakian category is defined as a $k$-linear rigid abelian tensor category, where $k$ is required to be a field. The main theorem, in its more ...
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Is every Tannakian category equivalent to a Tannakian subcategory of a hereditary Tannakian category?
Let $\mathcal{T}$ be a Tannakian category. Is it true that we can always find a hereditary Tannakian category $\mathcal{T}'$ such that $\mathcal{T}$ is equivalent (as a Tannakian category) to a ...
- 16.1k
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Is a Tannakian subcategory necessarily closed under subquotients?
This is a slightly nitpicky question and somewhat related to a previous question of mine: If $T'$ is a Tannakian subcategory of a Tannakian category $T$, then is $T'$ necessarily closed under ...
- 16.1k
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Constructing tensor isomorphism between fibre functors
Let $\mathcal{T}$ denote a Tannakian category over a field $k$, and let $\mathcal{V}$ denote a simple object. Let $\omega,\omega'$ be a pair of fibre functors, and write $\alpha:\mathcal{V}\to \...
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To what extent can you reconstruct finite groups from their representations?
I would like to understand to what extent we can reconstruct a finite group $G$ from
its category $\mathsf{Rep}(G)$ of finite-dimensional complex representations, possibly enhanced with extra ...
- 25.1k
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Equality of functions on Tannakian fundamental group
Suppose $T$ is a neutral Tannakian category over a field $k$, with fiber functor $\omega$. The fundamental group $G_\omega:=\underline{\mathrm{Aut}}^{\otimes}(\omega)$ is an affine group-scheme. Given ...
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Subquotients of tensor products of extensions
Let $G$ be an affine group scheme and let $\mathcal{T}$ denote the Tannakian category of its representations.
Let $V$ be a semisimple object of $\mathcal{T}$. Fix $\mathcal{T}_0$ to be the full ...
- 3,416
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Is a pro-algebraic group over $\mathbb{Q}_p$ with Galois action the inverse limit of Galois-equivariant quotients?
Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...
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Pro-algebraic fundamental groups
Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic zero and fix a point $x\in X(K)$.
We can associate to $X$ two Tannakian categories: the category of ...
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2
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Tannakian duality and the equivalence of $\alpha_p$- and $\mathbb{Z}/p\mathbb{Z}$-representation Categories
Let $G = \operatorname{Spec} R$ be a finite group scheme, with $R$ a finite-dimensional Hopf algebra. By Tannakian duality, we should be able to reconstruct $G$ from the category of $G$-...
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Why do monoidal functors between categories of quasicoherent sheaves commute with external tensor products
I'm reading Lurie's paper on Tannaka duality for geometric stacks. Very roughly, my question is, why do monoidal functors, from which we try to build geometric morphisms, commute with certain algebra ...
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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 ...
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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,$...
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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 ...
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