Monoidal closed structure(s) on the category "bicategories, with strict functors"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:53:46Z http://mathoverflow.net/feeds/question/21418 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21418/monoidal-closed-structures-on-the-category-bicategories-with-strict-functors Monoidal closed structure(s) on the category "bicategories, with strict functors"? Peter LeFanu Lumsdaine 2010-04-15T05:01:42Z 2010-04-23T17:33:07Z <p>I'm working with globular operadic higher categories (with the Batanin/Leinster definitions) and ending up working a lot in the categories $P$-$\mathrm{Alg}$ of algebras for some globular operad $P$, together with (literal, strict, algebraic) morphisms between them. A familiar case of these is $\textrm{Bicat}_\textit{str}$: the category of bicategories and strict functors between them.</p> <p>It would be very handy if there were some kind of mapping space constructions in these categories --- that is, some reasonable monoidal closed structures on them. Does anyone know what's been shown, either to exist or not to, either in the general case, or (more likely) for $\textrm{Bicat}_\textit{str}$?</p> <p>(The most obvious specific candidate, of course, is Cartesian closure. However, even $\textrm{Bicat}_\textit{str}$ fails to be Cartesian closed: chasing through the Yoneda argument that shows what a Cartesian closure would have to look like if it did exist leads one to a counterexample showing that the product doesn't preserve pushouts. The next best hope would presumably be some sort of Gray tensor product; this is where I've not yet been able to find anything further on the closure question.)</p> http://mathoverflow.net/questions/21418/monoidal-closed-structures-on-the-category-bicategories-with-strict-functors/22373#22373 Answer by Mike Shulman for Monoidal closed structure(s) on the category "bicategories, with strict functors"? Mike Shulman 2010-04-23T17:33:07Z 2010-04-23T17:33:07Z <p>You may know this, but there are people thinking about this sort of question at least from the monoidal viewpoint. For instance, there is <a href="http://arxiv.org/abs/0909.4715" rel="nofollow">this paper</a>, which shows that if you can defined what you want to mean by "a category enriched in P-Alg," then you can automatically recover from that a corresponding monoidal structure on P-Alg. In particular, this can produce the Gray tensor product from the notion of Gray-category. Although in general what you get is only a <em>lax</em> monoidal structure, and I don't think they have (yet) asked when it will be closed.</p>