On 2-final 2-functors

TL;DR

This paper establishes a criterion for 2-final (2,1)-functors, paralleling classical results, and provides a combinatorial framework for understanding paths and homotopies in (2,1)-categories.

Contribution

It introduces a new criterion for 2-final (2,1)-functors and offers a combinatorial approach to paths and homotopies in (2,1)-categories.

Findings

A criterion for 2-final (2,1)-functors based on slice categories.

A combinatorial presentation of paths in (2,1)-categories.

Insights into homotopies of paths in (2,1)-categories.

Abstract

We present a criterion for $2$-final $(2,1)$-functors, analoguous to the classical one for final $1$-functor: a $(2,1)$-functor $F \colon A \to B$ is $2$-final if and only if, for any object $b$ of $B$, the slice $(2,1)$-category $b / F$ is nonempty, connected and simply connected. We also give a combinatorial presentation of paths and homotopies of paths in a $(2,1)$-category.