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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 14th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 Schreiber|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 Shulman|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 category|Segal categories]] in any homotopical category with finite products. You can define [[complete Segal space|complete Segal spaces]] in any model category, at least, and less may suffice. And you can define $A_\infty$-[[A-infinity-category|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]

   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]

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 12th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * John D. Berman, *Enriched infinity categories I: enriched presheaves* ([arXiv:2008.11323](https://arxiv.org/abs/2008.11323)) <a href="https://ncatlab.org/nlab/revision/diff/enriched+%28infinity%2C1%29-category/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+%28infinity%2C1%29-category/11">v11</a>, <a href="https://ncatlab.org/nlab/show/enriched+%28infinity%2C1%29-category">current</a>

   added pointer to:

   • John D. Berman, Enriched infinity categories I: enriched presheaves (arXiv:2008.11323)

   diff, v11, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the following pointer (and will also add it at *[[Day convolution]]*): * [[Vladimir Hinich]], *Colimits in enriched ∞-categories and Day convolution*, Theory and Applications of Categories **39** 12 (2023) 365-422 &lbrack;[tac:39-12](http://www.tac.mta.ca/tac/volumes/39/12/39-12abs.html), [arXiv:2101.09538](https://arxiv.org/abs/2101.09538)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/enriched+%28infinity%2C1%29-category/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+%28infinity%2C1%29-category/13">v13</a>, <a href="https://ncatlab.org/nlab/show/enriched+%28infinity%2C1%29-category">current</a>

   added the following pointer (and will also add it at Day convolution):

   • Vladimir Hinich, Colimits in enriched ∞-categories and Day convolution, Theory and Applications of Categories 39 12 (2023) 365-422 [tac:39-12, arXiv:2101.09538]

   diff, v13, current