# Is there any relation between two pseudofunctors associated to two different cleavages of the same fibered category?

It is well known that given a Fibered category $$P_F: E \rightarrow C$$ with a cleavage $$K$$ we can construct a pseudofunctor $$F_K: C^{op} \rightarrow Cat$$. Now if one chooses a different cleavage $$L$$ but consider the same fibered category $$P_F$$ then how do $$F_K$$ and $$F_L$$ are related? (Note here $$F_L$$ is the pseudofunctor associated to the fibered category $$P_F$$ with the cleavage $$L$$).

Are they equivalent as objects in the 2 category of pseudofunctors over the category $$C$$?

I would be grateful if someone can refer any literature in this direction.

Thank you.

• @PraphullaKoushik Let $f: v \rightarrow u$ be a morphism in $C$ $\eta$ be an object in the fibre over $u$ . Let $\zeta_K$ and $\zeta_L$ be two corresponding Pull-Backs of $\eta_K$ and $\eta_L$ corresponding to the cleavage $K$ and $L$ respectively. Then from the definition of Cartesian Arrow there exists a unique isomorphism between $\zeta_K$ and $\zeta_L$. (Continued in the next comment) Commented Jun 6, 2020 at 13:30
• @PraphullaKoushik Now every element in a cleavage is a cartesian lift of some morphism in $C$. Hence by the definition of cleavage, for every element $\phi$ in $K$ there exists a unique element $\psi$ in $L$ such that there exists a unique isomorphism between the corresponding pullbacks. From this observation I guessed (mentioned in the question) that the corresponding pseudofunctors may be equivalent as objects in the 2 category of pseudofunctors over $C$. Commented Jun 6, 2020 at 13:38
• @PraphullaKoushik It's not only a one-one correspondence but the corresponding pullbacks are identified upto a unique choice of isomorphism... Commented Jun 6, 2020 at 13:46
• Ok. Because there is a relation between the collections $K$ and $L$ (as you mentioned in your previous comment), you are expecting some relation between the associated constructions $F_K,F_L:\mathcal{C}\rightarrow \text{Cat}$. Fair enough.. Commented Jun 6, 2020 at 13:53
• @PraphullaKoushik Yes exactly. Commented Jun 6, 2020 at 14:03

• Sir, though it's been a long time since I asked the question, but I want to clarify something here. Say, we consider category valued presheaves $F,F': C^{\rm{op}} \rightarrow \rm{Cat}$ arising from two choices of split cleavages $K$ and $K'$ of some fibered category $\pi : E \rightarrow C$. Then will $F$ and $F'$ be naturally isomorphic as functors in the usual sense? Somehow, I am not able to produce the canonical isomorphism. [What I could only show is that $F(x)=F'(x)$ for all $x \in \rm{obj}(C)$ and $F(\gamma)$ is naturally isomorphic to $F'(\gamma)$ for all $\gamma \in \rm{Mor}(C)$ ]. Commented Mar 15, 2023 at 8:21