Abstract:
One of the forms of Tannaka's theorem is called the reconstruction theorem for algebras and it sounds as follows: each algebra $A$ over a field $k$ can be reconstructed from the category $_A\operatorname{Vect}$ of its left modules as the algebra of endomorphisms $\operatorname{ End} F$ of the forgetful functor $F:{_A\operatorname{Vect}}\to {_k\operatorname{Vect}}$ (which on any $A$-module $X$ leaves only the structure of the vector space over the field $k$) . This result remains true in the theory of enriched categories with the following formulation: each algebra $A$ in a symmetric monoidal category $M$ with equalizers, in which the identity object $I$ is integral, can be reconstructed from the enriched category $_A{M}$ of its left modules as the algebra of endomorphisms $\operatorname{End} F$ of the forgetful functor $F:{_A{M}}\to {M}$ (which on any $A$-module $X$ leaves only the structure of the object of the category $M$). Since the category $\operatorname{Ste}$ of stereotype spaces is complete and symmetric monoidal, and the identity object $\mathbb{C}$ is integral in it, it follows that Tannaka's theorem holds in $\operatorname{Ste}$ as well: eech stereotype algebra $A$ can be recovered from the enriched category $_A\operatorname{Ste}$ of its left modules as the algebra $\operatorname{End} F$ of endomorphisms of the forgetful functor $F:{_A\operatorname{Ste}}\to {\operatorname {Ste}}$.
