Double Homotopy (Co)Limits for Relative Categories

TL;DR

This paper investigates the conditions under which homotopy (co)limits in categories with weak equivalences can be interchanged, addressing the lack of a calculus for homotopy (co)limits.

Contribution

It introduces double homotopy (co)limits for relative categories and explores their properties without assuming a derivator structure.

Findings

Establishes conditions for Fubini-type interchange of homotopy (co)limits.

Develops a framework for double homotopy (co)limits in relative categories.

Provides insights into homotopy limit calculus without derivators.

Abstract

We answer the question to what extent homotopy (co)limits in categories with weak equivalences allow for a Fubini-type interchange law. The main obstacle is that we do not assume our categories with weak equivalences to come equipped with a calculus for homotopy (co)limits, such as a derivator.