Fibered Categories in Groupoids

Question

I'm reading an article of Aaron Mazel-Gee about Fibered categories in grupoids and there is an example which I don't understand. Here the full article: https://etale.site/writing/stax-seminar-talk.pdf

...and the excerpt containing the example:

The category $\mathcal{C}$ is fixed and we consider a pair $(X_0,X_1)$. So this means for me that $X_i \in \mathcal{C}$ (eg they are objects in $\mathcal{C}$).

On the other hand we consider $U \in \mathcal{C}$.

From this context I don't see what is here $X_0(U)$ and $X_1(U)$?

Answer 1

From Section 4.3.1:

$\text{Given a ring $\it R$, we define the groupoid $\it Q(R)$ of (monic) quadratic expressions and changes of variable by} $ $$\text{$\it Q(R)$} = \left\{ob(Q(R)) = \text{{$x^2 + bx + c : b,c \in R$} $\cong R \times R$ }, \\ \text{$Hom_{Q(R)}((b',c'),(b,c)) = \text{{r $\in R$ : $(x + r)^2 + b'(x + r) + c' = x^2 + bx + c$}}$}\right\}$$

$\text{A ring homomorphism R$\to S$ determines a functor $\it Q(R) \to Q(S)$}.$

$\text{So, these constructions assemble into a functor Q : Rings $\to$ Groupoids.}$

So essentially $X_0$ and $X_1$ are just functors and $X_0(U)$ is a groupoid.

c.f. page 3 of your source

Edit:

...Applying Spec everywhere, we get a groupoid (pair) (Spec $A$, Spec $\Gamma$) in AffSch.

...More explicitly and more generally, any pair of objects $(X_0,X_1)$ in a category $C$ is called a groupoid in $C$ if it has the maps

$$s : X_1 \to X_0, t : X_1 \to X_0, \epsilon : X_0 \to X_1, i : X_1 \to X_1, m : X_1 \times_{s,X_0,t} X_1 \to X_1$$ that satisfy the obvious identities coming from the definition of a groupoid.

(This is a generalization of the notion of a $\it group \space object$ in a category, whose contravariant Yoneda functor lands in Groups.)

So, rather than specifically using the functor $Spec$ (yielding a contravariant equivalence between the categories CRing and AffSch), the author generalized this to a pair of objects $(X_0,X_1)$ in an arbitrary category $C$ that have the structure maps mentioned above. i.e., yes, $X_0$ and $X_1$ are called objects since $X_0,X_1 \in ob(C)$.

c.f. page 4

Answer 2

There are two notions of internal groupoid at play here. One notion of a groupoid internal to a category is simply a functor $\mathcal G$ from $\mathcal C^{\mathbf{op}}$ into groupoids. For instance, any ordinary groupoid $\mathcal G=G_1\rightrightarrows G_0$ gives rise to a groupoid internal to the category of sets given by the functor $$S\mapsto \mathbf{Set}(S,G_1)\rightrightarrows \mathbf{Set}(S,G_0)$$ However, in general an "internal groupoid" in this sense, which really doesn't deserve the name, is a pretty worthless notion. Instead, by an internal groupoid one really means an "internal groupoid" $\mathcal G$ in the above sense such that the functors $\mathcal{C}^{\mathrm{op}}\to \mathbf{Set}$ given by composing $\mathcal G$ with the functors $0,1:\mathbf{Gpd}\to \mathbf{Set}$ taking a (small) groupoid to its set of objects, respectively, morphisms, are both representable. This seems to be the way that the Grothendieck school liked to define such things.

However, Mazel-Gee is beginning with a more elementary notion: a groupoid in a sufficiently reasonable category is defined in a diagrammatically identical way to the definition of an ordinary groupoid. The point missing in your excerpt is that such an internal groupoid is exactly the same thing as an internal groupoid in the above sense: a functor into groupoids with representable components. This identification is probably being made unconsciously here. But the point is that a groupoid $\mathcal G$ in $\mathcal C$ gives rise to a function $\hat{\mathcal G}:\mathcal{C}^{\mathrm{op}}\to \mathbf{Gpd}$ by mapping each object in $\mathcal C$ into the entire diagram $\mathcal G$. Formally, $$\hat{\mathcal G}(c)=\mathcal C(c,\mathcal G_1)\rightrightarrows \mathcal C(c,\mathcal G_0)$$ You can and should check that this really gives a completely ordinary small groupoid for each $c$, which is natural in $c$. The other direction of the equivalence is also not too difficult: once you assume that $\mathcal G_0$ and $\mathcal G_1$ are representable, the source, target, composition, and identity morphisms become representable by the full faithfulness of the Yoneda embedding, together with the fact that it preserves limits (including pullbacks.)