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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.

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  • $\begingroup$ @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) $\endgroup$ Commented Jun 6, 2020 at 13:30
  • $\begingroup$ @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 $C$. $\endgroup$ Commented Jun 6, 2020 at 13:38
  • $\begingroup$ @PraphullaKoushik It's not only a one-one correspondence but the corresponding pullbacks are identified upto a unique choice of isomorphism... $\endgroup$ Commented Jun 6, 2020 at 13:46
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    $\begingroup$ 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.. $\endgroup$ Commented Jun 6, 2020 at 13:53
  • $\begingroup$ @PraphullaKoushik Yes exactly. $\endgroup$ Commented Jun 6, 2020 at 14:03

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Two different cleavages produce isomorphic pseudofunctors.

This follows immediately from Theorem 8.3.1 in Borceux's Handbook of Categorical Algebra 2.

Specifically, part (1) of this theorem states that for any pseudofunctors P and Q we have an isomorphism of categories PsFun(P,Q)→Cart(φ(P),φ(Q)).

Now if P and Q are two pseudofunctors produced using two different choices of a cleavage, then the Grothendieck fibrations φ(P) and φ(Q) are canonically isomorphic and this isomorphism lifts to a canonical isomorphism between P and Q.

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  • $\begingroup$ Thank You Sir very much for the answer and the reference. $\endgroup$ Commented Jun 6, 2020 at 17:59
  • $\begingroup$ 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 $\gamma \in \rm{Mor}(C)$ ]. $\endgroup$ Commented Mar 15, 2023 at 8:21
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    $\begingroup$ @AdittyaChaudhuri: The two functors will be pseudonaturally isomorphic. The naturality square will not commute strictly, but rather only up to an isomorphism, and the isomorphism itself satisfies certain coherence conditions. This isomorphism is constructed using both cleavages. $\endgroup$ Commented Mar 15, 2023 at 14:01
  • $\begingroup$ Thanks, I got your point! $\endgroup$ Commented Mar 15, 2023 at 15:48

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