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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 17th 2010
   • (edited Oct 17th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexIn Cruttwell-Shulman, there is a table in the introduction showing that a monoid (this terminology is not (yet?) standard) for the identity monad on $\mathbf{V}-mat$ is a $\mathbf{V}$-enriched category (can all $\mathbf{V}$-enriched categories be realized this way?). Thinking about enrichment this way, it seems like we should need the following to define the enrichment of a quasicategory in a monoidal quasicategory: - A notion of a monoidal quasicategory - A quasicategorical analogue of $\mathbf{V}-Mat$ - A sufficiently weak notion of a monad. - A sufficiently weak notion for a $T$-monoid for a monad $T$. What are the technical barriers to using this to define the enrichment of a quasicategory? Are there any obvious pitfalls that I'm missing?

   In Cruttwell-Shulman, there is a table in the introduction showing that a monoid (this terminology is not (yet?) standard) for the identity monad on $\mathbf{V}-mat$ is a $\mathbf{V}$-enriched category (can all $\mathbf{V}$-enriched categories be realized this way?). Thinking about enrichment this way, it seems like we should need the following to define the enrichment of a quasicategory in a monoidal quasicategory:

   • A notion of a monoidal quasicategory
   • A quasicategorical analogue of $\mathbf{V}-Mat$
   • A sufficiently weak notion of a monad.
   • A sufficiently weak notion for a $T$-monoid for a monad $T$.

   What are the technical barriers to using this to define the enrichment of a quasicategory? Are there any obvious pitfalls that I’m missing?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 17th 2010
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   Author: Mike Shulman
   Format: MarkdownItexThe only technical barriers that I'm aware of involve actually defining all of those things. (-: Although of course if you just want enriched (non-multi) categories, then the monad bit is unnecessary (since it would be the identity monad). I've thought a bit off and on about "quasi- (virtual) double categories" but it never seemed worthwhile to do all the work pushing it through. PS. Yes, all V-enriched categories can be realized this way -- it's an equivalent *definition* of "V-enriched category."

   The only technical barriers that I’m aware of involve actually defining all of those things. (-: Although of course if you just want enriched (non-multi) categories, then the monad bit is unnecessary (since it would be the identity monad). I’ve thought a bit off and on about “quasi- (virtual) double categories” but it never seemed worthwhile to do all the work pushing it through.

   PS. Yes, all V-enriched categories can be realized this way – it’s an equivalent definition of “V-enriched category.”

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 18th 2010
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   Author: Harry Gindi
   Format: MarkdownItex> I've thought a bit off and on about "quasi- (virtual) double categories" but it never seemed worthwhile to do all the work pushing it through. Well, this would give a straightforward construction of so-called $(\infty,\infty)$-categories by iterative enrichment, so isn't it worthwhile?

   I’ve thought a bit off and on about “quasi- (virtual) double categories” but it never seemed worthwhile to do all the work pushing it through.

   Well, this would give a straightforward construction of so-called $(\infty,\infty)$-categories by iterative enrichment, so isn’t it worthwhile?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2010
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   Author: Mike Shulman
   Format: MarkdownItexFor a suitable definition of "straightforward," and if you manage to play the right games with terminal coalgebras to boost yourself up from (∞,n) to (∞,∞), maybe. My intuition is that it would end up being a good deal more complicated than most of the existing approaches to (∞,∞)-categories, but if you want to prove me wrong, more power to you. (-: If all you want to do is iteratively define (∞,n)-categroies starting from quasicategories, though, then I believe Joyal has suggested a way that might be simpler, using "reduced" internal categories in a quasicategory (kind of like complete Segal spaces, but done entirely in the quasicategorical world).

   For a suitable definition of “straightforward,” and if you manage to play the right games with terminal coalgebras to boost yourself up from (∞,n) to (∞,∞), maybe. My intuition is that it would end up being a good deal more complicated than most of the existing approaches to (∞,∞)-categories, but if you want to prove me wrong, more power to you. (-: If all you want to do is iteratively define (∞,n)-categroies starting from quasicategories, though, then I believe Joyal has suggested a way that might be simpler, using “reduced” internal categories in a quasicategory (kind of like complete Segal spaces, but done entirely in the quasicategorical world).

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 18th 2010
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   Author: Harry Gindi
   Format: MarkdownItexDo you have a reference?

   Do you have a reference?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2010
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   Author: Mike Shulman
   Format: MarkdownItex[1] A. Joyal. Talk given at the Fields Institute conference on Higher Categories and their Applications, January 2007. Sorry....

   [1] A. Joyal. Talk given at the Fields Institute conference on Higher Categories and their Applications, January 2007.

   Sorry….

7. • CommentRowNumber7.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 18th 2010
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   Author: Harry Gindi
   Format: MarkdownItexThanks! What are you sorry for?

   Thanks! What are you sorry for?

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2010
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   Author: Mike Shulman
   Format: MarkdownItexNot having a reference you can actually read. (-:

   Not having a reference you can actually read. (-: