Higher Equipments, Double Colimits and Homotopy Colimits

Abstract: This document is centered around a main idea: simplicial categories, by which we mean simplicial objects in the category of categories, can be treated as a two-fold categorical structure and their double category theory is homotopically meaningful. The most well-known two-fold structures are double categories, typically used to organize bimodules in various contexts. However there is no double category of spaces even though notions of bimodule are conceivable. We first remedy this defect of double category theory by constructing a meaningful simplicial category of spaces. Then we develop the analogy with double categories by defining double colimits and by postulating an equipment property, which is promptly satisfied in the examples. As an application we prove that certain double colimits are naturally interpreted as homotopy colimits. Quite surprisingly this analogy unveils a principle: simplicial categories are to simplicially enriched categories what double categories are to 2-categories!