Replacing functors with enriched ones

TL;DR

This paper provides criteria for when a functor can be replaced by an enriched functor across various bases of enrichment, offering new proofs and insights into localization and equivalence.

Contribution

It introduces simple criteria for functor enrichment equivalence and applies them to multiple bases, including simplicial sets, topological spaces, and spectra.

Findings

Criteria for functor enrichment equivalence established

New proof of Lurie's result on Dwyer-Kan localizations

Applicable to multiple enrichment bases

Abstract

We describe simple criteria under which a given functor is naturally equivalent to an enriched one. We do this for several bases of enrichment, namely (pointed) simplicial sets, (pointed) topological spaces and orthogonal spectra. We also describe a few corollaries, such as a new proof of a result of Lurie on Dwyer-Kan localizations.