grothendieck construction for profunctors

Question

Given categories $X$ and $Y$ and a strong functor

$$D:X^{op}\times Y\to Cat$$

we can of course build the oplax colimit

$$\mathrm{colim}^{oplax}_{X^{op}\times Y}D$$

via the usual (covariant) grothendieck construction:

Objects are triples $(x,d,y)$ with $d\in D(x,y)$.

Morphisms $(x,d,y)\to (x',d',y')$ are triples $(f,\phi, g)$ with $f:x'\to x$, $g:y\to y'$ and $$\varphi: f^*x_*g\to x'.$$

There is however another possible construction that corresponds better to the slogan "presheaves are distributors into the one-point category and copresheaves are distributors out of the one-point category":

Objects are the same triples as above.

Morphisms however are now triples $(f,\varphi, g)$ with $f:x\to x'$ (the direction changed!) and $g:y\to y'$ and

$$\varphi:x_*g \to f^*x'.$$

Why does it correspond better to the slogan stated above? Taking distributors having at one side the one-point category then specialises to the usual grothendieck constructions yielding fibered and opfibered categories respectively.

Question: What can be said about the second construction? Is it some kind of colimit as well?

Related question

Answer 1

[I guess that by $x_*g$ you mean $D(\mathit{id}, g)(x)$ and by $f^*y$ you mean $D(f, \mathit{id})(y)$.]

Actually, your second construction is the usual Grothendieck construction for "$\mathbf{Cat}$-valued distributors" (BTW, this term may be misleading a bit, because in a $\mathbf{Cat}$-valued distributor $\mathbb{X}^{op} \times \mathbb{Y} \rightarrow \mathbf{Cat}$ 2-categories $\mathbb{X}$ and $\mathbb{Y}$ are not necessary degenerated). It may be universally characterised as a $(- \downarrow \mathbb{Y}) \times (\mathbb{X} \downarrow =)$-weighted colimit of $D \colon \mathbb{X}^{op}\times \mathbb{Y} \rightarrow \mathbf{Cat}$.

See: "Cosmoi of Internal Categories" by Ross Street, Transactions American Math. Soc. 258 (1980) 271-318; MR82a:18007.

Answer 2

Let $f^*:= D(f, 1) $, $g_*:= D(1, g)$ then $f^* g_*= g_*f^*$, the lax.colimit is the Grothendieck construction i.e. the cofibrated category (with base $\textbf{X}^{op}\times \textbf{Y}$) associated to $D$, its morphisms are $(f, g, \phi : (x, d, y) \to (x', d', y')$ with $f \in \textbf{X}^{op}( x, x')$ i.e. $f \in \textbf{X}( x', x)$, $g\in \textbf{Y}(y , y')$, and $\phi: g_*f^*(d) \to d' $. Now there is a second point of view: the functor $D$ describe a "span fibration" from $\textbf{X}$ to $\textbf{Y}$

(see R.Street article cited above, or Categorical Logic and Type Theory (B. JAcobs) p.517) or "Bifibration Induced Adjoint Pairs" (M. Bunge) Reports of the Midwest Category Seminar V - Lnm 195)

and (the associated category of) this span-fibration has morphisms $(f, g, \psi : (x, d, y) \to (x', d', y')$ with $f \in \textbf{X}( x', x)$, $g\in \textbf{Y}(y , y')$ and $\psi: g_*(d) \to f^*(d')$.

PS. excuse my poor English