About a Double-pseudo-category generalization of the module bicategory construction

Question

To a category with finite limits $\mathscr{C}$, it is associated the bicategory of its spans $Span(\mathscr{C})$. Furthermore the bicategory of (bi)modules (and monoids) on $Span(\mathscr{C})$ is the bicategory $Prof(\mathscr{C})$ of internal categories of $\mathscr{C}$ and profunctors. (see for example first pages of "Fibrations and Yoneda's lemma in a 2-category" by R.Street and the definitions of internal profunctor in 'Topos theory' of P. Johnstone 1978).

Now given a morphism $f: X \to Y$ in $\mathscr{C}$ and a monoid $B \rightrightarrows Y$ (we call it also a internal category on $Y$) the pullback by $f$ give a monoid $f^*B$ on $X$, furthermore given a internal functors $(f_1, f): (A \rightrightarrows X) \to (B \rightrightarrows Y)$, it correspond equivalently to a morphism of monoids (i.e. a internal functor) $A \to f^*B$ on $X$. Then the category of internal funtors in $\mathscr{C}$ is equivalent to a fibred category $\mathscr{F}$ on $\mathscr{C}$ with fibres $\mathscr{F}(X)=$''internal categories (and functors) on $X$''.

From 'Bicategories of spans and relations' A. Carboni, S. Kasangian (JPAA 33, 1984) the bicategory $Span(\mathscr{C})$ and the category $\mathscr{C}$ are strictly related ( I seems that $\mathscr{C}$ is 2-equivalent (as locally discrete 2-category) to the maps (morphisms that are right adjoint) of $Span(\mathscr{C})$).

Now, considering internal profunctors (horizontally) and internal functors (vertically) we get a pseudo-double-category (for definition see for example http://arxiv.org/abs/math/0604549). The some if $\mathscr{C}$ has one object (i.e. is a monoidal category) and consider the (bi)modules (horizontally) and morphisms of monoids (vertically).

1) I ask:

If the usual construction of the bicategory of modules (from a bicategory $\mathscr{S}$ ) is generalizable to a construction of a pseudo-double-category of modules as horizontal arrows, and monoids morphisms (of a different type from the modules) as vertical arrows, this from a bicategory $\mathscr{S}$ and some fibration on some category $\mathscr{C}$ related to $\mathscr{S}$ (may be $\mathscr{S}$ more general than a spans bicategory).

And if (possibly) we can get decomposition theorems similar to that of Benabou about the decomposition of a profunctor (see 'Topos Theory' P. Johnstone Th.2.48 page 63.)

2) Given a monoidal category $\mathbf{V}$ (symmetrical, closed if you want).

I ask if:

Is knowed in mathematical literature a construction of a bicategory $\mathscr{S}$ such that its bicategory of bimodule $Mod(\mathscr{S})$ is (equivalent to) the bicategory of $\mathbf{V}$-profuntors on (small) $\mathbf{V}$-categories ?

Answer 1

As Michal mentioned in a comment, I think you would benefit from reading my paper Framed bicategories and monoidal fibrations. Among the results therein is an extension of Mod to an operation acting on well-behaved pseudo double categories (section 11). I think this is a sort of answer to your question (1), where in place of "a bicategory and a fibration on a category related to it" we have a well-behaved pseudo double category. Note that the well-behavedness includes a "fibration" condition, so it may be what you are trying to get at.

If I understand your question 2 correctly (my understanding is very different from Michal's), then the answer is yes, S should be the bicategory of "V-matrices", whose objects are sets and whose 1-cells are "matrices" (doubly indexed families) of objects of V, with composition by "matrix multiplication". This also extends to a pseudo double category, and when Mod is applied to it it produces the pseudo double category of V-enriched categories, functors, and profunctors.