On semisimplicial sets satisfying the Kan condition

TL;DR

This paper provides a combinatorial proof that semisimplicial sets satisfying the Kan condition can be extended to simplicial sets, and generalizes this result to multisemisimplicial sets, enhancing understanding of their structure.

Contribution

It offers a new combinatorial proof of a known fact and extends the result to multisemisimplicial sets, broadening the theoretical framework.

Findings

Semisimplicial sets satisfying the Kan condition can be given a simplicial structure.

The proof is combinatorial, providing an alternative to existing proofs.

The results are generalized to multisemisimplicial sets.

Abstract

A semisimplicial set has face maps but not degeneracies. A basic fact, due to Rourke and Sanderson, is that a semisimplicial set satisfying the Kan condition can be given a simplicial structure. The present paper gives a combinatorial proof of this fact and a generalization to multisemisimplicial sets.