Join and slices for strict $\infty$-categories

TL;DR

This paper develops a join and slice theory for strict ∞-categories, establishing a monoidal structure, adjoints, and conjectures on functoriality, with applications to a Quillen Theorem A and revisiting Gray tensor products.

Contribution

It introduces a new join operation for strict ∞-categories, proving it forms a monoidal structure and exploring its properties and conjectures.

Findings

Join defines a monoidal category structure.

Existence of right adjoints leading to ∞-categorical slices.

Preliminary results on functoriality and Gray tensor product.

Abstract

The goal of this paper is to develop a theory of join and slices for strict $\infty$-categories. To any pair of strict $\infty$-categories, we associate a third one that we call their join. This operation is compatible with the usual join of categories up to truncation. We show that the join defines a monoidal category structure on the category of strict $\infty$-categories and that it respects connected inductive limits in each variable. In particular, we obtain the existence of some right adjoints; these adjoints define $\infty$-categorical slices, in a generalized sense. We state some conjectures about the functoriality of the join and the slices with respect to higher lax and oplax transformations and we prove some first results in this direction. These results are used in another paper to establish a Quillen Theorem A for strict $\infty$-categories. Finally, in an appendix, we revisit the Gray tensor product of strict $\infty$-categories. One of the main tools used in this paper is Steiner's theory of augmented directed complexes.