n-quasi-abelian categories

Abstract: Given an abelian category A its derived category D(A) admits a natural t-structure whose heart is A. Moreover by the Auslander’s Formula A is equivalent to the quotient category of coherent functors by the Serre subcategory of effecable functors.

We can associate to an exact category E its derived category D(E), does D(E) admit a canonical t-structure? If such a t-structure exists, is it possible to describe its heart in terms of coherent functors?

Testing this problem on a quasi-abelian category E we get: its derived category D(E) admits two canonical t-structures (left and right) whose hearts L and R are derived equivalent and their intersection in D(E) is E. If E is quasi-abelian but not abelian the "distance" between these two t-structures is 1, while in the abelian case these two t-structures coincides and their "distance" is 0. Moreover L (resp. R) can be described by the Auslander’s Formula as the quotient category of contravariant (resp. covariant) coherent functors by the Serre subcategory of effecable functors.

We extend this picture into a hierarchy of n-quasi-abelian categories: n=0 are abelian categories, n=1 are quasi-abelian categories, n=2 are pre-abelian categories....

This talk is based on the paper arxiv.org/abs/1602.08253v3

category theoryfunctional analysisrepresentation theory

Audience: researchers in the topic

( paper | slides | video )

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Additive categories between algebra and functional analysis

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