Non-unital polygraphs form a presheaf category

TL;DR

This paper proves that the category of non-unital polygraphs is a presheaf category, introduces a new criterion for such proofs, and discusses implications for higher category theory and related conjectures.

Contribution

It establishes that non-unital polygraphs form a presheaf category and develops a new criterion for proving presheaf structures in polygraph classes.

Findings

Non-unital polygraphs form a presheaf category.

A new criterion for presheaf category proofs is introduced.

Application of the criterion to various classes of polygraphs.

Abstract

We prove, as claimed by A.Carboni and P.T.Johnstone, that the category of non-unital polygraphs, i.e. polygraphs where the source and target of each generator are not identity arrows, is a presheaf category. More generally we develop a new criterion for proving that certain classes of polygraphs are presheaf categories. This criterion also applies to the larger class of polygraphs where only the source of each generator is not an identity, and to the class of "many-to-one polygraphs", producing a new, more direct, proof that this is a presheaf category. The criterion itself seems to be extendable to more general type of operads over possibly different combinatorics, but we leave this question for future work. In an appendix we explain why this result is relevant if one wants to fix the arguments of a famous paper of M.Kapranov and V.Voevodsky and make them into a proof of C.Simpson's semi-strictification conjecture. We present a program aiming at proving this conjecture, which will be continued in subsequent papers.