# Polygraphs and Discrete Conduch{\'e} $\omega$-Functors

## TL;DR

This paper introduces discrete Conduch{é} $oldsymbol{ extomega}$-functors between strict $oldsymbol{ extomega}$-categories, showing that if the target is free on a polygraph, then the source is also free, extending known results from 1-categories.

## Contribution

It defines a new class of morphisms called discrete Conduch{é} $oldsymbol{ extomega}$-functors and proves their properties related to polygraphs in strict $oldsymbol{ extomega}$-categories.

## Key findings

- Discrete Conduch{é} $oldsymbol{ extomega}$-functors generalize 1-category functors.
- If the target of such a functor is free on a polygraph, so is the source.
- The paper establishes properties connecting these functors to polygraph structures.

## Abstract

We define a class of morphisms between strict $\omega$-categories called discrete Conduch{\'e} $\omega$-functors that generalize discrete Conduch{\'e} functors between 1-categories and we study their properties related to polygraphs. The main result we prove is that for every discrete Conduch{\'e} $\omega$-functor, if its target is a free strict $\omega$-category on a polygraph then so is its source.