An Alternative Presentation of the Symmetric-Simplicial Category

TL;DR

This paper introduces an alternative way to present the symmetric-simplicial category by replacing codegeneracies with quasi-codegeneracies, providing a new algebraic framework for modeling homotopy types.

Contribution

It establishes a novel presentation of the symmetric-simplicial category using quasi-codegeneracies and proves a unique factorization theorem for its morphisms.

Findings

Introduces quasi-codegeneracies as replacements for codegeneracies.

Proves a unique factorization theorem for morphisms.

Lays groundwork for symmetric hypercrossed complexes as models for homotopy types.

Abstract

The category Fin of symmetric-simplicial operators is obtained by enlarging the category Ord of monotonic functions between the sets {0,1,...,n} to include all functions between the same sets. Marco Grandis has given a presentation of Fin using the standard generators Ord (cofaces and codegeneracies) as well as the adjacent transpositions which generate the permutations in Fin. The purpose of this note is to establish an alternative presentation of Fin in which the codegeneracies are replaced by special maps which we call quasi-codegeneracies. We also prove a unique factorization theorem for products of cofaces and quasi-codegeneracies analogous to the standard unique factorizations in Ord. This presentation has been used by the author to construct symmetric hypercrossed complexes (to be published elsewhere) which are algebraic models for homotopy types of spaces based on the hypercrossed complexes of Carrasco and Cegarra.