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I thought we needed an entry enriched (infinity,1)-category, so I created one. Added an Idea-section that mentions the evident general abstract definition (which hasn’t been worked out) and mentions the evident model (which has).
I have used links to this now at table - models for (infinity,1)-operads in an attempt to clarify the “general pattern” of the table (now the first part of the table itself).
I notice/rememberd that we have two equally orphaned and equally stubby entries, titled weak enrichment and titled homotopical enrichment. Something should be done about that unfortunate state of affairs, but for the moment I just added more links between these.
There was also this ancient discussion, which we don’t need to keep there:
[begin old forwarded discussion]
Urs: can anyone point me to – or write an entry containing – a discussion of systematical “homotopical enrichment” where we enrich over a homotopical category systematically weakening everything up to coherent homotopy. If/when we have this we should also link it to (infinity,n)-category, as that is built by iteratively doing homotopical enrichement starting with Top.
Mike: If anyone ever does anything like that, I would love to see it. As far as I know there is no general theory. You can define Segal categories in any homotopical category with finite products. You can define complete Segal spaces in any model category, at least, and less may suffice. And you can define $A_\infty$-categories in any monoidal homotopical category. But the problem is finding some way to get a handle on them, like lifting a model structure to them. Of course, people have iterated the existing definitions to get notions of $n$-category and of $(\infty,n)$-category (Simpson-Tamsamani, Trimble, Barwick, Lurie, etc), but I’ve never seen a general theory. Peter May and I have been planning for a while to think about iterating enriched $A_\infty$-categories.
[end old forwarded discussion]
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