Timeline for What is the right notion of a functor from an internal topological category to a topologically enriched category?

7 events

when toggle format	what		by	license	comment
May 27, 2022 at 8:08	vote	accept	Niall Taggart
May 25, 2022 at 23:50	answer	added	Mike Shulman		timeline score: 5
May 25, 2022 at 10:09	comment	added	მამუკა ჯიბლაძე		- of continuously $X$-indexed families of objects of $\mathcal M$.
May 25, 2022 at 10:08	comment	added	მამუკა ჯიბლაძე		It might help to first consider two extreme particular cases: when $\mathcal C$ has only one object and when $\mathcal C$ has only identity morphisms. The first case is more or less clear: it amounts to continuous homomorphisms from a topological monoid to topological monoids $\operatorname{End}_{\mathcal M}(M)$. While the second case, I believe, makes it clear that you need additional structure on $\mathcal M$ - not only must it be $\mathsf{Top}$-enriched and cotensored but also fibered over $\mathsf{Top}$. That is, to each space $X$ one must have the category ${\mathcal M}^X$
May 25, 2022 at 9:42	comment	added	R. van Dobben de Bruyn		Hmm, I always thought a (strictly) $\mathbf{Top}$-enriched category is the same thing as a category internal to $\mathbf{Top}$ whose object space is discrete. Then you could work with functors of internal categories, which feels like the right notion to me. But I've never tried to work with this, so it's possible that I made a mistake...
May 25, 2022 at 8:24	history	edited	Niall Taggart	CC BY-SA 4.0	added 98 characters in body
Mar 29, 2022 at 11:21	history	asked	Niall Taggart	CC BY-SA 4.0