The homotopy category of flat functors

TL;DR

This paper develops a framework for understanding the homotopy category of flat functors from a small category to a tensor Grothendieck category, introducing new adjoint functors and a replacement for a certain quotient category.

Contribution

It defines a notion of atness in functor categories and establishes the existence of a right adjoint to the inclusion of complexes of at functors, along with a new construction for a quotient of triangulated categories.

Findings

Established a right adjoint to the inclusion K(FlatA) → K(A).

Introduced a replacement for the quotient Dpac(FlatA).

Provided new tools for studying homotopy categories of flat functors.

Abstract

Let C be a small category and G be a tensor Grothendieck category. We define a notion of atness in the category Fun(C; G) of all covariant functors from C to G and show that the inclusion K(FlatA) ---> K(A) has a right adjoint where K(A) is the homotopy category of A and K(FlatA) its subcategory consisting of complexes of at functors. In addition, we find a replacement for the quotient Dpac(FlatA) = K(FlatA)Kp(FlatA) of triangulated categories where Kp(FlatA) is the homotopy category of all pure acyclic complexes of at functors.