On the equvialence of colimits and 2-colimits

TL;DR

This paper investigates the conditions under which colimits and 2-colimits of strict 2-functors in the 2-category of groupoids are equivalent, especially over certain posets related to topological coverings, and optimizes these conditions for complexity.

Contribution

It establishes specific conditions for the equivalence of colimits and 2-colimits in a 2-category of groupoids and improves these conditions from exponential to polynomial complexity.

Findings

Identifies conditions for colimit and 2-colimit equivalence in groupoid 2-categories.

Shows any 2-functor can be deformed to satisfy these conditions.

Reduces complexity of conditions from exponential to polynomial.

Abstract

We compare the colimit and 2-colimit of strict 2-functors in the 2-category of groupoids, over a certain type of posets. These posets are of special importance, as they correspond to coverings of a topological space. The main result of this paper gives conditions on the 2-functor $\mathfrak{F}$, for which $\mathsf{colim}\mathfrak{F}\simeq2\mathsf{colim}\mathfrak{F}$. One can easily see that any 2-functor $\mathfrak{F}$ can be deformed to a 2-functor $\mathfrak{F}'$, which satisfied the conditions of the theorem. At last, we also optimise our conditions, reducing from exponential to polynomial complexity.