Cofibration category structures on the category of graphs

TL;DR

This paper proves that the category of finite graphs cannot be equipped with a cofibration category structure based solely on -homotopy equivalences, and enlarging the class of weak equivalences introduces non-isomorphic stiff subgraphs.

Contribution

It establishes the non-existence of a cofibration category structure on finite graphs with -homotopy equivalences and explores limitations of enlarging weak equivalences.

Findings

No cofibration category structure exists with -homotopy equivalences as weak equivalences.

Enlarging weak equivalences leads to morphisms with non-isomorphic stiff subgraphs.

The results delineate structural limitations in modeling finite graphs categorically.

Abstract

In this article, we show that there is no cofibration category structure on the category of finite graphs with $\times$-homotopy equivalences as the class of weak equivalences. Further, we show that it is not possible to enlarge the class of weak equivalences to get cofibration category structure on the category of finite graphs without including morphisms where domain and codomain have non-isomorphic stiff subgraphs.