Monoids, Segal's condition and bisimplicial spaces

TL;DR

This paper characterizes simplicial objects in categories with finite products via a strictified Segal condition, extending to bisimplicial spaces to relate double loop spaces with specific components.

Contribution

It provides a strictification of Segal's condition for simplicial objects and generalizes this to bisimplicial spaces, linking geometric realizations to loop spaces.

Findings

Characterization of simplicial objects via a strict Segal condition

Extension of Segal's result to bisimplicial spaces

Conditions for homotopy equivalence of loop spaces and specific components

Abstract

A characterization of simplicial objects in categories with finite products obtained by the reduced bar construction is given. The condition that characterizes such simplicial objects is a strictification of Segal's condition guaranteeing that the loop space of the geometric realization of a simplicial space $X$ and the space $X_1$ are of the same homotopy type. A generalization of Segal's result appropriate for bisimplicial spaces is given. This generalization gives conditions guaranteing that the double loop space of the geometric realization of a bisimplicial space $X$ and the space $X_{11}$ are of the same homotopy type.