Timeline for Reference request: Grothendieck construction for $\mathbb V$-distributors?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 10, 2013 at 23:35 | comment | added | Buschi Sergio | I have a answere, but (as ever) I'm no too sure, can I Email you ? (I'm [email protected]) | |
| Aug 3, 2013 at 19:26 | comment | added | Gerrit Begher | @BuschiSergio: Yep; found this one as well: The difference is that the paper deals with functors $$I\to\mathbb V\mathrm{-Cat}$$ where $I$ is an ordinary category; whereas I consider $\mathbb V$-functors $$X\to\mathbb V$$ where $X$ is a $\mathbb V$-category. | |
| Aug 3, 2013 at 18:49 | comment | added | Buschi Sergio | Hi, I have find this article: The Grothendieck Construction and Gradings for Enriched Categories (Tamaki D) front.math.ucdavis.edu/0907.0061 | |
| Jul 29, 2013 at 20:53 | comment | added | Buschi Sergio | yes right, thinking to rings I close my mind to one-object category (mistake). I seems that work, in Johnstone "TOpos theory" (1978), he do this construction in internal category context, I try to extend this to a monoidal situation, but seems that dont work (need universal property of pullbaks(?)). Anyway above was $g_*(f^*(N))$ | |
| Jul 29, 2013 at 18:44 | comment | added | Gerrit Begher | I assume $D$ is your symbol for the supposed Grothendieck construction. If so: Note that the construction I sketched above yields an Ab-Category having in general more than one object. For example the Grothendieck construction for $M=_AM_B$ yields an Ab-category with object set $M$ and morphism objects $[m,n]=\{(a,b)|am=nb\}$. | |
| Jul 29, 2013 at 18:20 | comment | added | Buschi Sergio | Ab= "commuative groups". A preaddittive category is a Ab-enriched category (abbr. "p.a.cat"), a rings R, S,.. are p.a.cat's with olny one object. a Ab-profunctor $M: A\to B$ is a $(A, B)$-bimodule. If Ab-enriched construction exist for ${}_AM_B$ then (as they did in the refernce) you have two rings morphism $f: D\to A$ $g: D\to B$ such that $N\otimes_A M = g_*(f^*(M))=(N_{|D})\otimes_D B$ but I seems the this cannot be true in general | |
| Jul 29, 2013 at 15:47 | comment | added | Gerrit Begher | Thank you for the reference! I had actually read this section a few weeks back but totally forgot about it. I don't understand your objection, though: Could you explain in a bit more detail why the case you describe is a problem? | |
| Jul 29, 2013 at 14:37 | comment | added | Buschi Sergio | I think that isn't possible in general, because this construction become naturally a category and the profunctor D is "splitted by two funtors" (see mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf §6.4). Now preaddittive categories are one object Ab-enriched one, and Ab-profuntors are bimodules, now I seem that the associated composition functor $(-){}_A\otimes B$ isn't (ever) equivalent to a extension-restriction of rings.. | |
| Jul 29, 2013 at 11:03 | history | asked | Gerrit Begher | CC BY-SA 3.0 |