Homotopy composition of cospans

TL;DR

This paper shows that replacing pushouts with homotopy pushouts in the category of finite sets and cospans yields a broader class of commutative Frobenius algebras, extending the known universal property.

Contribution

It introduces a homotopy-theoretic modification to the cospan construction, generalizing the universal property to all commutative Frobenius algebras.

Findings

Homotopy pushouts replace pushouts in cospan categories.

The construction yields all commutative Frobenius algebras.

Extends the universal property from special to general cases.

Abstract

It is well known that the category of finite sets and cospans, composed by pushout, contains the universal {\em special} commutative Frobenius algebra. In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.