When studying homotopy theory in an enriched setting, e.g. in an enriched model category, one may wonder when a functor between underlying categories is equivalent to an enriched one. I will discuss conditions under which this is possible for several bases of enrichment, including simplicial sets, topological spaces and various models of spectra, and illustrate these with examples. Finally, I will discuss my own motivation for studying this question; namely, that it can be used to tackle certain problems appearing in Goodwillie's calculus of functors.

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