Is there a tricategory of bicategories and biprofunctors? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:38:44Z http://mathoverflow.net/feeds/question/36830 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36830/is-there-a-tricategory-of-bicategories-and-biprofunctors Is there a tricategory of bicategories and biprofunctors? Evan Jenkins 2010-08-27T01:08:55Z 2013-02-04T23:49:04Z <h2>Background</h2> <p>There is a bicategory where the objects are categories, the 1-morphisms are profunctors, and the 2-morphisms are morphisms of profunctors. The non-obvious part of this assertion is that profunctors admit a "good" (i.e., coherently associative) composition. The way one sees this is to use the fact that the Yoneda embedding is the free cocompletion, from which we may identify profunctors from $\mathcal{C}$ to $\mathcal{D}$ with cocontinuous functors from $\operatorname{Set}^{\mathcal{C}^{\operatorname{op}}}$ to $\operatorname{Set}^{\mathcal{D}^{\operatorname{op}}}$. Over here, there is a strictly associative composition, so life is good.</p> <p>All of this extends easily to the enriched setting. In particular, if we enrich over $\mathcal{V} = \operatorname{Cat}$, we can talk about strict 2-profunctors between strict 2-categories, and we get a nice bicategory of 2-categories, strict 2-profunctors, and morphisms between these.</p> <h2>Question</h2> <p>I would like to know if there is an analogous way to obtain a tricategory of bicategories and biprofunctors. You can "follow your nose" and write down what the composition should be, but checking all the axioms of a tricategory does not seem like the simplest approach. Of course, we can strictify to get equivalent strict 2-categories and 2-profunctors, but I doubt the composition will be the same; I don't think the "strict" colimits will in general agree with the "weak" colimits. Also, using the standard enriched theory only gives a bicategory; the natural transformations are lost. It seems that an appropriate statement of the form "the bicategorical Yoneda embedding is a free cocompletion" will handle things just as it does in the setting of ordinary profunctors, but I am not sufficiently comfortable with limits and colimits in bicategories to try to figure out if such a statement makes sense. Has anybody worked all of this out?</p> <p>(The closest thing I have seen to such a result is <a href="http://front.math.ucdavis.edu/0911.4979" rel="nofollow">this paper</a> by Justin Greenough, which more or less establishes this result in the case of sufficiently nice $k$-linear monoidal categories. But his proof seems to be very specific to that setting, whereas one would hope that a proof could be established along the same lines as the result for usual profunctors.)</p> http://mathoverflow.net/questions/36830/is-there-a-tricategory-of-bicategories-and-biprofunctors/36881#36881 Answer by Urs Schreiber for Is there a tricategory of bicategories and biprofunctors? Urs Schreiber 2010-08-27T12:45:11Z 2010-08-27T12:45:11Z <p>Depending on what you actually need for your application, there might be something useful already known. Do you really need a tricategory, as opposed to some other model of weak 3-categories? Even if you had that, would you be able to do much with it?</p> <p>There are several equivalent ways to think of profunctors, and some of them lend themselves to categorification quite easily. One is this:</p> <p>A profunctor from $C$ to $D$ is the same a a colimit-preserving ordinary functor between the presheaf categories $PSh(C)$ and $PSh(D)$.</p> <p>(See here for details: <a href="http://ncatlab.org/nlab/show/profunctor#FuncsOnPresheaves" rel="nofollow">http://ncatlab.org/nlab/show/profunctor#FuncsOnPresheaves</a>)</p> <p>That's good, because a lot is known about categorifications of categories of presheaves and of morphisms between them. </p> <p>For instance it is straightforward to set this up over bicategories: take objects bicategories, and hom-bicategories to be the full sub-bicategory on bicolimit-preserving bifunctors between their bipresheaf categories. </p> <p>In case that your application is such that you only need higher categories whose higher morphisms are all invertible, one can go much further and consider the (oo,1)-category of (oo,1)-profunctors. By the above, this is simply the gadget whose objects are small (oo,1)-categories and whose hom-oo-groupoids are the full sub-oo-groupoids on the (oo,1)-colimit preserving (oo,1)-functors between the corresponding (oo,1)-presheaf categories.</p> <p>More generally, one can generalize here (oo,1)-presheaf categories and all in addition all their reflective sub-(oo,1)-categories. That structure of (oo,1)-profunctors has proven to play a major role as kind of categorification of the category of vector spaces: one thinks of a presentable (oo,1)-category as vector space, of colimits as being sums of vectors, as colimit preserving (oo,1)-functors as linear maps.</p> <p>More on this is here: <a href="http://ncatlab.org/nlab/show/Pr(infinity,1)Cat" rel="nofollow">http://ncatlab.org/nlab/show/Pr(infinity,1)Cat</a> .</p> http://mathoverflow.net/questions/36830/is-there-a-tricategory-of-bicategories-and-biprofunctors/66237#66237 Answer by Finn Lawler for Is there a tricategory of bicategories and biprofunctors? Finn Lawler 2011-05-27T21:59:59Z 2011-05-27T21:59:59Z <p>If you're still interested, I've worked this out on my personal web at nLab <a href="http://ncatlab.org/finnlawler/show/biprofunctor" rel="nofollow">here</a>, with supporting material linked to from <a href="http://ncatlab.org/finnlawler/show/2-categorical+miscellany" rel="nofollow">this page</a>.</p> http://mathoverflow.net/questions/36830/is-there-a-tricategory-of-bicategories-and-biprofunctors/120811#120811 Answer by Mike Shulman for Is there a tricategory of bicategories and biprofunctors? Mike Shulman 2013-02-04T23:49:04Z 2013-02-04T23:49:04Z <p><a href="http://arxiv.org/abs/1301.3191" rel="nofollow">This recent paper</a> comes within about $3\epsilon$ of constructing the tricategory of enriched bicategories and enriched profunctors. I'm kind of sad that we didn't get around to crossing the remaining distance.</p>