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Let $\mathcal{C}$ a category.

Let $Fib(\mathcal{C})$ the category of fibrations (on $\mathcal{C}$) with morphisms the cartesian functors $T: (\mathcal{A}, P)\to (\mathcal{B}, Q)$

i.e. $T: \mathcal{A}\to \mathcal{B}$ with $Q\circ T=P$ (where $P: \mathcal{A}\to \mathcal{C}$, $P: \mathcal{B}\to \mathcal{C}$ are fibrations on $\mathcal{C}$).

From literature (es. "Categorical logic and Type THeory" B.Jacobs) the pullbak of two fibration (i.e. the product in $Fib(\mathcal{C})$) is still a fibration, and this has a easy generalization to a multi-pullback i.e. a the inclusion $Fib(\mathcal{C})\subset CAT\downarrow \mathcal{C}$ create (small) product, the some is true for the kernel of a couple, then the inclusion above create all (small) limits. THe some is easly true for coproducts.

I ask: do the inclusion $Fib(\mathcal{C})\subset CAT\downarrow \mathcal{C}$ create also cokernels? Counterexamples?.

Observation $Fib(\mathcal{C})$ is equivalent to the category of pseudo-functors $P: \mathcal{C}^{op}\to CAT$ with pseudo-transformations as morphisms, and the punctual limits or colimits of pseudofunctors is still a pseudofuntors (I seems yes, anyway this is true for funtors and natural tranformations considering the category $sFib(\mathcal{C})$ of fibrations with split clevages and clevage preserving functors ), then $Fib(\mathcal{C})$ (or at least $sFib(\mathcal{C})$) is complete and cocomplete.

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1 Answer

$\mathrm{Fib}(\mathcal{C})$ is both monadic and comonadic over $\mathrm{Cat}/\mathcal{C}$, in the pseudo sense. That is, there are a monad $T$ and a comonad $G$ on $\mathrm{Cat}/\mathcal{C}$ such that $\mathrm{Fib}(\mathcal{C})$ is equivalent to the 2-category of strict $T$-algebras and pseudo $T$-morphisms, and also to the 2-category of strict $G$-coalgebras and pseudo $G$-morphisms. By definition $T$ is the span-composite with $\mathcal{C} \leftarrow \mathcal{C}^{\mathbf{2}} \to \mathcal{C}$, while $G$ is its right adjoint.

Therefore, by a theorem of Blackwell-Kelly-Power and its dual, the forgetful functor creates strict PIE-limits and strict PIE-colimits, hence also (weak) 2-limits and 2-colimits.

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Thank you very much. I guess you first answere is from the R. Street article on Sydney category seminar (LNm 420). WHere I can find references about Blackwell-Kelly-Power theorem? –  Buschi Sergio Aug 6 '12 at 21:13
    
The first observation is made in multiple places, another one is Street's "Fibrations in bicategories". The Blackwell-Kelly-Power paper is "Two-dimensional monad theory". –  Mike Shulman Aug 7 '12 at 6:35
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