Mapping spaces in homotopy coherent nerves

TL;DR

This paper proves that the homotopy types of middle mapping spaces in coherent nerves of Kan enriched categories are equivalent to the original mapping spaces, providing a direct and clear comparison.

Contribution

It offers a direct proof establishing the homotopy equivalence between mapping spaces in coherent nerves and the original spaces in Kan enriched categories.

Findings

Middle mapping spaces have the same homotopy type as original spaces.

Provides a direct proof method for homotopy equivalence.

Clarifies the relationship between coherent nerves and original mapping spaces.

Abstract

We give a direct proof that middle mapping spaces in coherent nerves of Kan enriched categories have the same homotopy type as the original mapping spaces.