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

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).

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).

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 ?

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I don't really understand what you are asking in your first question. The answer to your second question is: yes, under some mild assumptions on $\mathbb{V}$. See: "Variation through enrichement" by R. Betti, A. Carboni, R. Street, R. Walters. However, it is generally not the best idea to define profunctors that way. – Michal R. Przybylek Sep 14 '13 at 22:06
One more comment (on the first question). It is better to not think of profunctors as of (discrete) fibrations (because this view is not even true for enriched categories). Instead you may think of profunctors as of (codiscrete) cofibrations. You should definitely take a look into the paper mentioned in the above. – Michal R. Przybylek Sep 14 '13 at 22:40
@Przybylek: Thank you. About the first question, If we apply the bicategorial construction to the monoidal category $Ab$ (abelian groups and tensor) we get the usual (bi)modules bicategory, but we haven't the morphisms of rings. What should we add to the initial bicategory, and how we could generalize the module bicategory construction for have modules PLUS rings (i.e. monoids) morphisms? – Buschi Sergio Sep 15 '13 at 9:15
Ach..., so your question is about "the right" categorical concept that encapsulates both modules over monoids and monoid homomorphisms. I think the closest to what you are looking for is the concept of "proarrow equipment" introduced by R. J. Wood in "Abstract pro arrows I". In a sense, it is a categorification of an allegory (notice, however, that functors are recoverable from profunctors only up to Cauchy equivalence; BTW: you need some mild conditions on your $\mathscr{C}$ to recover it from its category of relations). (cont) – Michal R. Przybylek Sep 15 '13 at 22:05
Another way to think of a proarrow equipment is via 2-profunctors with a monoidal action and satisfying some factorization properties. I think this view is close to Mike's (Shulman) "framed bicategories", but (unfortunately) I have never learnt what they are... – Michal R. Przybylek Sep 15 '13 at 22:06