From nonabelian basechange to basechange with coefficients

TL;DR

This paper investigates conditions under which basechange theorems for sheaves extend to sheaves with coefficients in various presentable $mbda$-categories, focusing on tensor product preservation of left adjointable squares.

Contribution

It provides criteria for extending basechange theorems to sheaves with coefficients, including the case of stable and compactly generated presentable $mbda$-categories.

Findings

Proper Basechange Theorem holds with coefficients in stable or compactly generated categories.

Tensor products of presentable $mbda$-categories preserve left adjointable squares.

Results on interaction between tensor products and categorical constructions.

Abstract

The goal of this paper is to explain when basechange theorems for sheaves of spaces imply basechange for sheaves with coefficients in other presentable $\infty$-categories. We accomplish this by analyzing when the tensor product of presentable $\infty$-categories preserves left adjointable squares. As a sample result, we show that the Proper Basechange Theorem in topology holds with coefficients in any presentable $\infty$-category which is compactly generated or stable. We also prove results about the interaction between tensor products of presentable $\infty$-categories and various categorical constructions that are of independent interest.