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I'm currently working with an analogue of the Grothendieck construction for enriched categories:

Given a distributor a.k.a. $\mathbb V$-functor $D:X^\mathrm{op}\otimes Y\to \mathbb V$ there is a structure of a $\mathbb V$-category on the set $$ \coprod_{X\times Y}\mathbb V(I, D(x,y))=\{\,(x,\phi,y)\,\,|\,\, \phi:I\to D(x,y),x\in X,y\in Y\,\} $$ with $[(x,\phi,y),(x',\phi',y')]$ given by the pullback of the cospan $$ [x,x']\to [x,x']\otimes D(x',y') \to D(x,y') \leftarrow D(x,y)\otimes [y,y'] \leftarrow [y,y']. $$ This construction directly generalises the Grothendieck construction for $\mathrm{Set}$-valued distributors between ordinary categories.

Question: What literature does allready exist on this subject?

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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.. –  Buschi Sergio Jul 29 '13 at 14:37
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? –  Garlef Wegart Jul 29 '13 at 15:47
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 –  Buschi Sergio Jul 29 '13 at 18:20
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 $[m,n]=\{(a,b)|am=nb\}$. –  Garlef Wegart Jul 29 '13 at 18:44
Hi, I have find this article: The Grothendieck Construction and Gradings for Enriched Categories (Tamaki D) front.math.ucdavis.edu/0907.0061 –  Buschi Sergio Aug 3 '13 at 18:49
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