# Morphisms of fibered categories which are compatible with the chosen cleavages

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 ?

• @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. Jun 29 at 4:38