Function spaces and classifying spaces of algebras over a prop

TL;DR

This paper establishes a method to determine the classifying spaces of categories of algebras over a prop using function spaces, generalizing Rezk's theorem to the prop setting.

Contribution

It introduces a novel approach to compute classifying spaces of algebra categories over props via function spaces, including a universal diagram construction.

Findings

Classifying spaces can be described as homotopy fibers of forgetful maps.

A new functorial method for constructing universal diagrams in algebra categories over props.

Generalization of Rezk's theorem to the context of props.

Abstract

The goal of this paper is to prove that the classifying spaces of categories of algebras governed by a prop can be determined by using function spaces on the category of props. We first consider a function space of props to define the moduli space of algebra structures over this prop on an object of the base category. Then we mainly prove that this moduli space is the homotopy fiber of a forgetful map of classifying spaces, generalizing to the prop setting a theorem of Rezk. The crux of our proof lies in the construction of certain universal diagrams in categories of algebras over a prop. We introduce a general method to carry out such constructions in a functorial way.