Representaility of morphism of stacks for schemes

Question

I have seen two definitions of representability of a morphism of stacks, which should be at least compatible with the definition of a morphism of categories fibered in groupoids.

1. (Representable morphism of categories fibred in groupoids). Let $\mathcal C$ be a category. Let $\mathcal X,\mathcal Y$ be categories fibred in groupoids over $\mathcal C$. Let $F:\mathcal X\to \mathcal Y$ be a $1$-morphism. We say $F$ is representable, if for every $S\in Ob(\mathcal C)$ and any $G:\mathcal C/S\to\mathcal Y$ the category fibred in groupoids $(\mathcal C/S)\times_{\mathcal Y}S\to S$ is representable.
2. (Representable morphism of stacks). We say that a morphism of stacks $ f : \mathcal X \to \mathcal Y $ is representable if for all S-schemes $ U $ and morphisms $ y : U \to \mathcal Y $, the fiber product $\mathcal X \times_{\mathcal Y} U $ is an algebraic space.
3. (Representable morphism of stacks). We say that a morphism of stacks $ f : \mathcal X \to \mathcal Y $ is representable if for all $S$-schemes $ U $ and morphisms $ y : U \to \mathcal Y $, the fiber product $ \mathcal X \times_{\mathcal Y} U $ is a scheme.

How can we reconcile these 3 definitions? Thank you for your help.

Answer 1

This is not an answer, just too long for a comment. So, writing as an answer. It turns out that, one may not be able to see the correspondence between these three definitions as one of them is stated wrongly.

You have mentioned in the question the following:

1. (Representable morphism of categories fibred in groupoids). Let $\mathcal C$ be a category. Let $\mathcal X,\mathcal Y$ be categories fibred in groupoids over $\mathcal C$. Let $F:\mathcal X\to \mathcal Y$ be a $1$-morphism. We say $F$ is representable, if for every $S\in Ob(\mathcal C)$ and any morphism of categories fibered in groupoids $G:\mathcal C/S\to\mathcal Y$, the $2$-fiber product $(\mathcal C/S)\times_{\mathcal Y}S\to S$ is representable.

This definition wrongly stated (I am hoping it was a typo). Observe that your definition of representability of $F$ depends only with the category $\mathcal{Y}$ and nothing do with neither the category $\mathcal{X}$ or the map $F:\mathcal{X}\rightarrow \mathcal{Y}$

Correct definition of representable morphism of categories fibered in groupoids is the following:

Let $\mathcal C$ be a category. Let $\mathcal X,\mathcal Y$ be categories fibred in groupoids over $\mathcal C$. Let $F:\mathcal X\to \mathcal Y$ be a $1$-morphism. We say $F$ is representable, if for every $S\in Ob(\mathcal C)$ and any morphism $G:\mathcal C/S\to\mathcal Y$ of categories fibered in groupoids, the projection map $(\mathcal C/S)\times_{G,\mathcal Y,F}\mathcal{X}\to (\mathcal C/S)$ is representable.

By "$(\mathcal C/S)\times_{G,\mathcal Y,F}\mathcal{X}\to (\mathcal C/S)$ is representable", it means that the $2$-fibered product $(\mathcal C/S)\times_{G,\mathcal Y,F}\mathcal{X}$ is representable by an object of $\mathcal{C}$; that is $\mathcal C/S)\times_{G,\mathcal Y,F}\mathcal{X}\cong (\mathcal{C}/T)$ for some $T\in \text{Ob}(\mathcal{C})$ and that the morphism $(\mathcal{C}/T)\rightarrow (\mathcal{C}/S)$ is induced from an arrow $T\rightarrow S$ (or $S\rightarrow T$??, make a guess) in $\mathcal{C}$.

I hope you will be able to prove this definition of $(1)$ is related the definition of $(3)$. I will give more details about this if there is still some difficulty.

There are two notions of representability:

1. Given a category $\mathcal{C}$ and an object $S$ of $\mathcal{C}$, one can associate a category fibered in groupoiods over $\mathcal{C}$, namely the functor $(\mathcal{C}/S)\rightarrow \mathcal{C}$. Now, an arbitrary category fibered in groupoids over $\mathcal{C}$, say $\mathcal{F}\rightarrow \mathcal{C}$ is said to be representable by an object of $\mathcal{C}$ if there is an object $S$ of $\mathcal{C}$ and an isomrophism $\mathcal{F}\cong (\mathcal{C}/S)$. This is the situtation of your point $3$.
2. It turns out that there are other interesting categories fibered in groupoids over $\mathcal{C}$. If $\mathcal{C}$ is $Sch/S$, we have algebraic spaces. If $\mathcal{C}$ is the category $\text{Man}$, we have what are called Lie groupoids. Given a Lie groupoid $\mathcal{G}$, one can assocaite a category fibered in groupoids $B\mathcal{G}\rightarrow \mathcal{C}$. Now, given a category fibered in groupoids $\mathcal{F}\rightarrow \mathcal{C}$, we can ask if this $F$ is representably by such a special category fibered in groupoids, that is a Lie groupoid $\mathcal{G}$ and an isomorphism $\mathcal{F}\cong B\mathcal{G}$. This is the situtation of your point $2$.