# A Homotopy Category for Graphs

## TL;DR

This paper develops a homotopy theory for finite graphs by constructing a 2-category with homotopies as 2-cells, introducing spider moves to explicitly describe homotopies, and establishing a homotopy category with finite stiff graphs as a skeleton.

## Contribution

It introduces a novel homotopy framework for graphs using 2-categories and spider moves, providing explicit descriptions and a skeleton for the homotopy category.

## Key findings

- Finite graphs form a homotopy category with universal properties.
- Spider moves explicitly characterize homotopies between finite graphs.
- Finite stiff graphs serve as a skeleton of the homotopy category.

## Abstract

We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call `spider moves'. We then create a category by modding out by the 2-cells of our 2-category, and use the spider moves to show that for finite graphs, this category is a homotopy category in the sense that it satisfies the universal property for localizing homotopy equivalences. We then show that finite stiff graphs form a skeleton of this homotopy category.