Does not exists lax.limits of lax.functors in CAT?

Premise

1) Give a lax.functor: $\textbf{C}: \mathcal{A}^{op}\to CAT$ a op.lax.cocone (resp. lax.cocone) $\check{\phi}: \textbf{C}\Rightarrow \mathcal{C}$ form $\textbf{C}$ to a category $\mathcal{C}$ is a op.lax.transformation (resp. lax.transformation) form $\textbf{C}$ to the constant lax.functor $k_\mathcal{C}$ (that send all on $\mathcal{C},\ 1_{\mathcal{C}},\ 1_{1_{\mathcal{C}}}$) (see link text).

2) Give a prefibration (see [S], p.132) $P: \mathcal{C}\to \mathcal{A}$ with a preclivage, we can associate to it the lax.functor:

$\textbf{C}: \mathcal{A}^{op}\to CAT$ with

$\textbf{C}(A)=\mathcal{C}(A)$ (the fibre of $P$ on $A$)

$\textbf{C}(f)=f^\ast: \textbf{C}(B)\to \textbf{C}(B)$ for $f: A\to B$ (the inverse image by $f$ )

with $C_{g, f}=c_{c, f}: \textbf{C}(g)\circ \textbf{C}(f)\Rightarrow\textbf{C}(g\circ f)\ g\circ f: A\to B\to C$

(where $c_{g, f}$ come from the clivage).

There exist a natural op.lax.cocone $\check{\phi}: \textbf{C}\to \mathcal{C}$ gived by

$\check{\phi}_A: \textbf{C}(A)\to \mathcal{C}$ (natural inclusion), $\check{\phi}_f: \check{\phi}_A\circ \textbf{C}(f)\Rightarrow \check{\phi}_B$ with $\check{\phi}_A(Y)=\theta_f: f^\ast(X)\to Y$ (canonical morphism of inverse image of the clivage).

And this op.lax.cocone is universal, i.e. it generalize the colimits concept for op.laxcocone (see [G] p.201), in other word $\mathcal{C}$ is a op.lax.colimit of $\textbf{C}$.

Exist a lax.colimit of a lax.functors $\textbf{C}: \mathcal{A}^{op}\to CAT$ associated to a prefibration $P: \mathcal{C}\to \mathcal{A}$?

A universal lax.cocone have to be a family:

$\check{\phi}_A: \textbf{C}(A)\to \mathcal{C}$ with

$\check{\phi}_f: \check{\phi}_B \Rightarrow \check{\phi}_A\circ \textbf{C}(f)$

I tried to define $\mathcal{C'}$ with object the couple $(X, A)\ A\in \mathcal{A}, X\in \textbf{C}(A)$ and morphisms $(\phi, f): (B, Y)\to (A, X)$ with $f: A\to B$ and $\phi: f^\ast(Y)\to X$, but there is a trouble for the definition of composition (need to reverse the arrow $c_{g, f}: f^\ast\circ g^\ast\Rightarrow (g\circ f)^\ast$). Or tried to define $\mathcal{C'}$ with objects as above but reversing $\phi$, i.e. $\phi:X\to f^\ast(Y)$ (this is the Grothendieck construction for the op.prefibration $P^{op}$) but there is a trouble for defining $\check{\phi}_A: \textbf{C}(A)\to \mathcal{C'}$ (naturally: $\textbf{C}(A)\subset \mathcal{C'}^{op}$).

[S] A. Grothendieck SGA1 cap. VI (SEMINAIRE DE GEOMETRIE ALGEBRIQUE, 1960-61) \end{document}

[G] J.w. Gray "FOrmal Category Theory I" LNM N°391

-
THe correct title is: "Does not exists lax.colimits of lax.functors in CAT ?" – Buschi Sergio Aug 13 '12 at 19:55
Cat is cocomplete as a 2-category, so it admits all lax colimits. – Mike Shulman Aug 14 '12 at 0:35
Mike Shulman. Of course CAT has op.lax.colimits and lax.colimits If we consider 2-functors (or pseudo.funtors). But what for a lax.colimit of a lax.funtor? FOr axample: consider a lax.functor associate to a prefibration, the fibrate category is a op.lax.colimit of this lax.funtor, what is its lax.limit? if it exist,, can you describe its construction? – Buschi Sergio Aug 14 '12 at 8:45
A lax functor is just a 2-functor with a different domain. See mathoverflow.net/questions/7762/… . – Mike Shulman Aug 14 '12 at 18:19