Yoneda Lemma for Simplicial Spaces

TL;DR

This paper extends the Yoneda lemma to simplicial spaces by developing the covariant model structure and establishing key invariance and recognition principles.

Contribution

It introduces left fibrations for simplicial spaces and proves a recognition principle for covariant equivalences within this framework.

Findings

Established a covariant model structure for simplicial spaces.

Proved invariance of the covariant model structure under complete Segal space equivalences.

Provided a recognition principle for covariant equivalences over simplicial spaces.

Abstract

We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and and studying its associated model structure, the covariant model structure. In particular, we prove a recognition principle for covariant equivalences over an arbitrary simplicial space and invariance of the covariant model structure with respect to complete Segal space equivalences.