A Quillen's Theorem A for strict $\infty$-categories I: the simplicial proof

TL;DR

This paper generalizes Quillen's Theorem A to strict ∞-categories using a simplicial proof based on Steiner's theory, advancing the homotopy theory of strict ∞-categories.

Contribution

It introduces a simplicial proof of a generalized Quillen's Theorem A for strict ∞-categories, expanding the homotopical understanding of these structures.

Findings

Proves a generalized Theorem A for strict ∞-categories

Uses Steiner's theory of augmented directed complexes in the proof

Sets the stage for a subsequent purely ∞-categorical proof

Abstract

The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely $\infty$-categorical methods.

Full text

{altabstract}

The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner’s theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely $\infty$-categorical methods.

\altkeywords

strict $\infty$-categories, augmented directed complexes, simplicial sets, join, Street’s nerve, orientals, Gray tensor product, Theorem A, slices, oplax transformations