Derived profunctors

Abstract

The classical construction of the tensor product of modules, as a quotient of a free abelian group by a certain subgroup, is purely algebraic. The same is true for the left derived functors of the tensor product, the so called torsion products, that take values in the category of abelian groups. The traditional approach to the computation of torsion products involves a certain resolution of one of the arguments. Robinson gives a new interpretation of these groups by proving them to be the homotopy groups of a certain symmetric monoidal category. Later this approach was used by Retakh for Ext functors, and by Modawi for coproduct-preserving functors on a certain class of small categories. Mac Lane gives an explicit construction of Tor groups via the slide product of modules, and uses it to describe the multiple torsion products in the generalized Kunneth formulas by generators and relations. On the other hand, the zeroth right derived functor of a finitely presented functor can be expressed via its defect introduced by Auslander. Recently Martsinkovsky defined the defect for arbitrary additive functors on modules and showed that the defect and the contravariant Yoneda embedding form a right adjoint contravariant pair. We will discuss how derived functors (and torsion products, in particular) arise as weighted (co)limits and how derived functors of additive functors can be extended to profunctors, even for enriched categories. Next, we will examine a model for delooping of torsion and slide products, and use it to give a categorical description of the injective stabilization of the tensor product, introduced by Auslander and Bridger and recently developed by Martsinkovsky and Russell. We will extend the defect to arbitrary enriched profunctors, which will lead us to connections between the defect and Isbell duals and Janelidze satellites. We will also show that all concepts in category theory can be expressed as defects of profunctors, superseding Mac Lane's observation that all concepts are Kan extensions. Finally, we show how to define derived enriched functors. In particular, we recover the classical derived functors as the ones enriched in abelian groups.--Author's abstract