Definition of descent with pseudofunctors

In all the reference I found, stacks are defined using fibrations (or fibered categories). So a stack is a fibered category $\pi:\mathcal F\rightarrow\mathcal C$ over a site $(\mathcal C,J)$ such that for every object $X\in\mathcal C$ and every covering $\mathcal U=\{X_j\rightarrow X\}_j$, the functor $\mathcal F_X=\pi^{-1}(X)\rightarrow \mathcal F_{desc}(\mathcal U)$ is an equivalence of categories.

I would like to define the notion of descent data using the following concepts

• Pseudofunctors $\mathcal C^{op}\rightarrow\mathcal Cat$ instead of fibations.
• Using the definition of a covering sieve $\mathcal U$ as a subheaf of $h_X=\mathcal Hom(-,X):\mathcal C^{op}\rightarrow\mathcal Set$

So, given a pseudofunctor $F:\mathcal C^{op}\rightarrow\mathcal Cat$ over a site $(\mathcal C,J)$. Given an object $X\in\mathcal C$ and a covering sieve $\mathcal U\hookrightarrow h_X$, we call descent data a family given by

• For every $(f:Y\rightarrow X)\in\mathcal U(Y)$ an object $s_f\in F(Y)$
• For every $(f:Y\rightarrow X)\in\mathcal U(Y)$ and $g:Z\rightarrow Y$, and isomorphism $\xi_{f,g}:s_{fg}\rightarrow g^*(s_f)$ (where $g^*$ is the functor $F(Y)\rightarrow F(Z)$ corresponding to $g$)

We ask that for every $f:Y\rightarrow X$, if $1$ is the identity of the domain of $f$, then $\xi_{f,1}:s_f\rightarrow 1^*(s_f)$ is the inverse of $1^*(s_f)\rightarrow s_f$ coming from the natural isomorphism $1^*\rightarrow 1_{F(Y)}$

Also, for every chain $S\xrightarrow hZ\xrightarrow gY\xrightarrow fX$, we require the following identity $$(s_{fgh}\xrightarrow{\xi_{f,gh}}(gh)^*(s_f)\simeq h^*g^*(s_f))=(s_{fgh}\xrightarrow{\xi_{fg,h}}h^*s_{fg}\xrightarrow{h^*\xi_{f,g}}h^*g^*(s_f))$$ where the second isomorphism on the left hand side comes from the natural isomorphism $(gh)^*\rightarrow h^*g^*$. Can this definition be found anywhere? Is it good in the following sense?

• If there is a set $\{X_j\rightarrow X\}_j$ such that every map $(f:Y\rightarrow X)\in\mathcal U(Y)$ factors through some $X_j\rightarrow X$, is my definition equivalent to the one linked above?
• If I take a fibration $\pi:\mathcal F\rightarrow\mathcal C$ and associate to it a pseudofunctor $F_\pi:\mathcal C^{op}\rightarrow\mathcal Cat$, is the descent data in my definition for $F_\pi$ equivalent to the descent data for $\pi$? (I take the definition of descent data in terms of fibered categories from Angelo Vistoli, "Notes on Grothendieck Topologies, Fibered categories and Descent Theory")

I would like any and all feedback and to know if my definition even makes sense or if it has some flaw somewhere (p.s. I didn't mention above the "size" problem of having a family $\{s_f\}$ indexed over all the maps $Y\rightarrow X$, but I think, correct me if I'm wrong, that it can be solved with an argument by Grothendieck's Universes)

Solution 1:

The theory of pseudo-functors is equivalent to the theory of fibered categories with a choice of cleavage. Roughly, if $\pi \colon \mathcal{F} \to \mathcal{C}$ is a fibred category, then I associate to it a pseudo-functor $F \colon \mathcal{C}^{\mathrm{op}}\to \mathsf{Cat}$ by sending $c \in \mathcal{C}$ to the fibre $\pi^{-1}(c)$. See Sections 3.1.2 and 3.1.3 of Vistoli's notes for more precise formulations. In particular, then, the descent condition defining stacks can be translated directly into the language of pseudo-functors.

The same is true when trying to rephrase the coverings in terms of sieves. You can define Grothendieck topologies in terms of sieves or in terms of coverings --- it's really just a different of language. For instance, if $h_{\mathcal{U}}$ is a sieve associated to a covering $\mathcal{U} = \{X_i \to X\}$ of $X$, and $F$ is the pseudo-functor associated to your fibred category $\mathcal{F} \to \mathcal{C}$, then there is an isomorphism between $\operatorname{Hom}(h_{\mathcal{U}},F)$ and the category $\mathcal{F}_{\mathrm{desc}}(\mathcal{U})$ of descent data associated to the covering $\mathcal{U}$. I think these same notes by Vistoli treat the translation to sieve in quite a bit of detail --- see, for instance, Section 4.1.5.