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Let $\pi \colon F \rightarrow C$ and $\pi' \colon F' \rightarrow C$ be two fibered categories over the category $C$. A morphism from $\pi \colon F \rightarrow C$ to $\pi' \colon F' \rightarrow C$ is a functor $\phi \colon F \rightarrow F'$ such that $\pi' \circ \phi= \pi$ and $\phi$ maps any cartesian morphism to another cartesian morphism. This naturally defines a category $\rm{Fib}(C)$, the category of fibrations(fibered categories) over $C$, which is well known.

If we now define a new category $\rm{Fib}(C)_{cleavage}$, whose objects are pairs $(\pi \colon F \rightarrow C, K)$ where $\pi \colon F \rightarrow C$ is a fibration over $C$ and $K$ is a cleavage of $\pi$. A morphism from $(\pi \colon F \rightarrow C, K)$ to $(\pi' \colon F' \rightarrow C, K')$ is defined as a functor $\phi \colon F \rightarrow F'$ such that $\phi$ is a morphism of fibered categories(defined in the previous paragraph) and it is compatible with cleavages i.e $\phi(K(\gamma,p))= K'(\gamma,\phi(p))$ for all $(\gamma,p) \in \rm{Mor}(C) \times_{t,\pi_0,\pi_0} \rm{Obj}(F)$, where $t$ is the target map of $C$ and $\pi_o$ is the object level map of the functor $\pi$. It is easy to see that it defines a category.

My questions are the following:

(1) What are the advantages/disadvantages of the category $\rm{Fib}(C)_{cleavage}$ over the standard notion $\rm{Fib}(C)$?

(2) How (1) will affect the equivalence with the category of pseudofunctors over $C$?

Just to be clear:

I am aware of the notion of cloven fibrations(fibrations equipped with a choice of cleavage) over a category. But I am not sure/don't know any reference where they explicitly described the 1-morphisms and 2-morphisms of the 2-category of cloven fibrations over a category. Are 1-morphisms in this 2-category assumed to be compatible with the chosen cleavages(like the way I mentioned above) or they are just simply morphisms of fibered categories ?

Thanks in advance.

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    $\begingroup$ If you're working with choice, there shouldn't be any difference. $\endgroup$
    – Alec Rhea
    Commented Jun 29, 2023 at 1:40
  • $\begingroup$ @AlecRhea Thanks for the link. I understand the point. But my confusion is about the "morphism of fibered categories compatible with the choices of cleavages". I think the usual forgetful functor may not be a full functor $\rm{Fib}(C)_{cleavage} \rightarrow \rm{Fib}(C)$, as all morphisms of fibered categories are not compatible with the cleavage choices. $\endgroup$ Commented Jun 29, 2023 at 4:38

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I'm sure there are better references, but if you have none my notes on researchgate address these topics from pages 193 onward. They aren't complete in any sense of the word, but should provide enough framework to understand what's going on.

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  • $\begingroup$ Thanks for the notes link, I will go through it. $\endgroup$ Commented Jun 29, 2023 at 4:42

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