2-Groupoid Enrichments in Homotopy Theory and Algebra

The use of groupoid enrichments in abstract homotopy theory is well known and classical. Recently enrichments by higher-dimensional groupoids have been considered. Here we will describe enrichment by 2-groupoids with respect to the Gray tensor product and will examine several examples (2-groupoids, 2-crossed complexes, chain complexes, etc.) from an elementary view-point. The enrichment of the category of chain complexes is examined in detail and questions of the existence of analogues of classical constructions (categories over B, under A, etc.) are explored.