The Stacks Project

Tag 0040

Lemma 4.31.3. Let $\mathcal{C}$ be a category. The $(2, 1)$-category of categories over $\mathcal{C}$ has 2-fibre products. Suppose that $F : \mathcal{X} \to \mathcal{S}$ and $G : \mathcal{Y} \to \mathcal{S}$ are morphisms of categories over $\mathcal{C}$. An explicit 2-fibre product $\mathcal{X} \times_\mathcal{S}\mathcal{Y}$ is given by the following description

1. an object of $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ is a quadruple $(U, x, y, f)$, where $U \in \mathop{\mathrm{Ob}}\nolimits(\mathcal{C})$, $x\in \mathop{\mathrm{Ob}}\nolimits(\mathcal{X}_U)$, $y\in \mathop{\mathrm{Ob}}\nolimits(\mathcal{Y}_U)$, and $f : F(x) \to G(y)$ is an isomorphism in $\mathcal{S}_U$,
2. a morphism $(U, x, y, f) \to (U', x', y', f')$ is given by a pair $(a, b)$, where $a : x \to x'$ is a morphism in $\mathcal{X}$, and $b : y \to y'$ is a morphism in $\mathcal{Y}$ such that
1. $a$ and $b$ induce the same morphism $U \to U'$, and
2. the diagram $$\xymatrix{ F(x) \ar[r]^f \ar[d]^{F(a)} & G(y) \ar[d]^{G(b)} \\ F(x') \ar[r]^{f'} & G(y') }$$ is commutative.

The functors $p : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{X}$ and $q : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{Y}$ are the forgetful functors in this case. The transformation $\psi : F \circ p \to G \circ q$ is given on the object $\xi = (U, x, y, f)$ by $\psi_\xi = f : F(p(\xi)) = F(x) \to G(y) = G(q(\xi))$.

Proof. Let us check the universal property: let $p_\mathcal{W} : \mathcal{W}\to \mathcal{C}$ be a category over $\mathcal{C}$, let $X : \mathcal{W} \to \mathcal{X}$ and $Y : \mathcal{W} \to \mathcal{Y}$ be functors over $\mathcal{C}$, and let $t : F \circ X \to G \circ Y$ be an isomorphism of functors over $\mathcal{C}$. The desired functor $\gamma : \mathcal{W} \to \mathcal{X} \times_\mathcal{S} \mathcal{Y}$ is given by $W \mapsto (p_\mathcal{W}(W), X(W), Y(W), t_W)$. Details omitted; compare with Lemma 4.30.4. $\square$

The code snippet corresponding to this tag is a part of the file categories.tex and is located in lines 5584–5622 (see updates for more information).

\begin{lemma}
\label{lemma-2-product-categories-over-C}
Let $\mathcal{C}$ be a category.
The $(2, 1)$-category of categories
over $\mathcal{C}$ has 2-fibre products.
Suppose that
$F : \mathcal{X} \to \mathcal{S}$ and
$G : \mathcal{Y} \to \mathcal{S}$ are morphisms of categories over
$\mathcal{C}$.
An explicit 2-fibre product
$\mathcal{X} \times_\mathcal{S}\mathcal{Y}$ is given by the following
description
\begin{enumerate}
\item an object of $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ is a quadruple
$(U, x, y, f)$, where $U \in \Ob(\mathcal{C})$,
$x\in \Ob(\mathcal{X}_U)$, $y\in \Ob(\mathcal{Y}_U)$,
and $f : F(x) \to G(y)$ is an isomorphism in $\mathcal{S}_U$,
\item a morphism $(U, x, y, f) \to (U', x', y', f')$ is given by a pair
$(a, b)$, where $a : x \to x'$ is a morphism in $\mathcal{X}$, and
$b : y \to y'$ is a
morphism in $\mathcal{Y}$ such that
\begin{enumerate}
\item $a$ and $b$ induce the same morphism $U \to U'$, and
\item the diagram
$$\xymatrix{ F(x) \ar[r]^f \ar[d]^{F(a)} & G(y) \ar[d]^{G(b)} \\ F(x') \ar[r]^{f'} & G(y') }$$
is commutative.
\end{enumerate}
\end{enumerate}
The functors $p : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{X}$
and $q : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{Y}$ are the
forgetful functors in this case. The transformation $\psi : F \circ p \to G \circ q$ is given on the object $\xi = (U, x, y, f)$ by
$\psi_\xi = f : F(p(\xi)) = F(x) \to G(y) = G(q(\xi))$.
\end{lemma}

\begin{proof}
Let us check the universal property: let
$p_\mathcal{W} : \mathcal{W}\to \mathcal{C}$
be a category over $\mathcal{C}$, let $X : \mathcal{W} \to \mathcal{X}$ and
$Y : \mathcal{W} \to \mathcal{Y}$ be functors over $\mathcal{C}$, and let
$t : F \circ X \to G \circ Y$ be an isomorphism of functors over $\mathcal{C}$.
The desired functor
$\gamma : \mathcal{W} \to \mathcal{X} \times_\mathcal{S} \mathcal{Y}$
is given by $W \mapsto (p_\mathcal{W}(W), X(W), Y(W), t_W)$.
Details omitted; compare with Lemma \ref{lemma-2-fibre-product-categories}.
\end{proof}

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