# A category for bijective combinatorics

## TL;DR

This paper introduces a categorical framework for bijective combinatorics using cobordisms of signed sets, providing a new perspective on the involution principle.

## Contribution

It extends the category of matchings to cobordisms of signed sets, offering a novel categorical approach to bijective combinatorics.

## Key findings

- Categorical framework for matchings and cobordisms.
- Reveals the involution principle's connection to cobordisms.
- Provides a new perspective on bijective combinatorics.

## Abstract

The category of matchings between finite sets extends to the category of cobordisms of signed sets. A chain of cobordisms that starts and ends with unsigned sets A and B yields a matching from A to B. This is a convenient way to package the involution principle of Garsia and Milne, which reveals itself to have little to do with involutions.