Archive for July 2025

The Quill 21 ~ The Meaning of Tannakian Construction and Anveshanā July 2025

Posted on Monday, July 28, 2025

Note that this contains some incomplete accounts. 

First, we will see what a tensor category is. Actually, we are interested in a symmetric monoidal category with a tensor functor. For a category \( \mathcal{C} \), we have a bifunctor
$$ \otimes \colon \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}, $$
and for a unit object \( \mathbf{1} \in \mathcal{C} \), we have natural isomorphisms
$$ \mathbf{1} \otimes - \simeq - \simeq - \otimes \mathbf{1}. $$
Furthermore, the category is symmetric, which means that for \( M, N \in \mathcal{C} \), we have a symmetry isomorphism
$$ M \otimes N \simeq N \otimes M. $$
Anyway, we are interested in the case of schemes over a ring. For a commutative ring \( R \), the scheme is given by \( \mathrm{Spec}\, R \), which is constructed by gluing affines. But the functions on a space \( X \) in algebraic geometry are limited, unlike in differential or geometric topology, where functions like \( C^\infty \) span the whole space. So, instead, we step up to discuss stacks. They are algebraic-geometric spaces—more general than schemes and algebraic spaces—and the origin of the mathematics of stacks is in Grothendieck's work on fibered categories and descent (SGA 1).

In this Tannakian construction, one is interested in seeing whether geometric objects like schemes and stacks can be recovered from linear categories associated with them. See the paper by Lurie, Tannaka Duality for Geometric Stacks, and also Tannaka Duality Revisited by Bhatt and Halpern-Leistner.

Informally, one can say that given a commutative ring \( R \), we may view
$$ X = \varinjlim \mathrm{Spec}\, R, $$
and then we are interested in sheaves
$$ \mathrm{QC}(X) = \varprojlim R\text{-Mod}. $$
One then has to realize that
$$ \mathrm{QC}(X) \colon \text{Stacks} \to (\otimes\text{-categories})^{\mathrm{op}} $$
and that there is a right adjoint
$$ \mathrm{Spec} \colon (\otimes\text{-categories})^{\mathrm{op}} \to \text{Stacks}. $$
Given a category \( \mathcal{C} \), we are interested in defining a geometric object \( \mathrm{Spec}\, \mathcal{C} \), which will be the best way to approximate \( \mathcal{C} \) in algebraic geometry.

Now, the Tannakian reconstruction here is about building a space (a stack or scheme) from tensor categories, i.e., symmetric monoidal categories. From the adjunction map, there is a faithful embedding
$$ X \mapsto \mathrm{Spec}\, \mathrm{QC}(X), $$
where \( X \) is a geometric stack (think of an Artin stack, for instance). The above map is also called the 1-affinization or Tannakanization, and it provides the best affine approximation to \( X \) via its category of quasi-coherent sheaves. For the parts I skipped in the above discussion, please see this lecture by David Ben-Zvi.

Also sharing the July issue of Anveshanā magazine. This features interviews with Aparna Dar (IIT Kanpur), Sumathi Rao (TIFR-ICTS, Bengaluru), and Jyoti Hegde (Veenagram, Sirsi), and a selection of articles, including an essay by C.S. Aravinda on the hyphen in Harish-Chandra.  

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The Quill 20 ~ Tannaka-Krein Duality and Reconstruction

Posted on Saturday, July 12, 2025

Tannaka-Krein duality, in its classical form, says that given a compact group $G$, one can reconstruct the group $G$ from the category of the finite-dimensional complex representations $\Pi(G)$. And $\Pi(G)$ is a tensor category (thus having a monoidal structure). For a locally compact abelian group, it is a much easier case and from Pontryagin duality, one has that the dual group $\hat{G}$ is the space of 1-dimensional representations of G (namely characters), and the dual group determines the group $G$, please see this note from last year. However, for non-abelian cases, the category of representation $\Pi(G)$ contains all finite-dimensional $\mathbb{C}-$linear representations, not just 1-dimensional irreducible representations like abelian case. Actually, as I think, it is helpful to see that Pontryagin duality is an abelian prelude to Tannaka-Krein duality. The latter also generalized a lot of the framework later, which culminated in the Tannakian framework of Grothendieck, quantum groups by Drinfeld, Deligne's tensor categories, and so on.

But one can say that given
$$\Pi(G) \cong \Pi(H)$$
which is an equivalence of symmetric monoidal categories, $G \cong H$ will not always be true unless there are some conditions to observe. This was emphasized by introducing a faithful, exact, forgetful, $\mathbb{C}$-linear fiber functor which preserves the tensor products and forgets the group action $\omega$
$$\omega: \Pi(G) \rightarrow Vect_\mathbb{C}$$
which forgets the group action and remembers the underlying vector spaces. The group $G$ can now be associated with the automorphism of the fiber functor
$$G \cong \underline{Aut}^{\otimes}(\omega).$$
This essentially means that all the information about the group is encoded in $\Pi(G)$, and given this fiber functor, we can reconstruct our group. In more standard words, given any symmetrical monoidal category $\mathcal{C}$, if one has a fiber functor $\omega: \mathcal{C} \rightarrow Vect_\mathbb{C}$, then under the suitable conditions, there exists a compact group $G$ such that $\mathcal{C} \cong \Pi(G)$ and $G  \cong \underline{Aut}^{\otimes}(\omega)$.

The framework of Tannakian categories and fiber functor was introduced by Grothendieck-Deligne in the 1960s, while Tannaka originally used the ring of representative functions.

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