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.  

Available at: https://anveshanamagazine.github.io