The Stacks project

Lemma 89.16.12. Let

\[ \xymatrix{ \mathcal{H} \times _\mathcal {F} \mathcal{G} \ar[r] \ar[d] & \mathcal{G} \ar[d]^ g \\ \mathcal{H} \ar[r]^ f & \mathcal{F} } \]

be $2$-fibre product of categories cofibered in groupoids over $\mathcal{C}_\Lambda $. If $\mathcal{F}, \mathcal{G}, \mathcal{H}$ all satisfy (RS), then $\mathcal{H} \times _\mathcal {F} \mathcal{G}$ satisfies (RS).

Proof. If $A$ is an object of $\mathcal{C}_\Lambda $, then an object of the fiber category of $\mathcal{H} \times _\mathcal {F} \mathcal{G}$ over $A$ is a triple $(u, v, a)$ where $u \in \mathcal{H}(A)$, $v \in \mathcal{G}(A)$, and $a : f(u) \to g(v)$ is a morphism in $\mathcal{F}(A)$. Consider a diagram in $\mathcal{H} \times _\mathcal {F} \mathcal{G}$

\[ \vcenter { \xymatrix{ & (u_2, v_2, a_2) \ar[d] \\ (u_1, v_1, a_1) \ar[r] & (u, v, a) } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & A_2 \ar[d] \\ A_1 \ar[r] & A } } \]

in $\mathcal{C}_\Lambda $ with $A_2 \to A$ surjective. Since $\mathcal{H}$ and $\mathcal{G}$ satisfy (RS), there are fiber products $u_1 \times _ u u_2$ and $v_1 \times _ v v_2$ lying over $A_1 \times _ A A_2$. Since $\mathcal{F}$ satisfies (RS), Lemma 89.16.2 shows

\[ \vcenter { \xymatrix{ f(u_1 \times _ u u_2) \ar[r] \ar[d] & f(u_2) \ar[d] \\ f(u_1) \ar[r] & f(u) } } \quad \text{and}\quad \vcenter { \xymatrix{ g(v_1 \times _ v v_2) \ar[r] \ar[d] & g(v_2) \ar[d] \\ g(v_1) \ar[r] & g(v) } } \]

are both fiber squares in $\mathcal{F}$. Thus we can view $a_1 \times _ a a_2$ as a morphism from $f(u_1 \times _ u u_2)$ to $g(v_1 \times _ v v_2)$ over $A_1 \times _ A A_2$. It follows that

\[ \xymatrix{ (u_1 \times _ u u_2, v_1 \times _ v v_2, a_{1} \times _ a a_2) \ar[d] \ar[r] & (u_2, v_2, a_2) \ar[d] \\ (u_1, v_1, a_1) \ar[r] & (u, v, a) } \]

is a fiber square in $\mathcal{H} \times _\mathcal {F} \mathcal{G}$ as desired. $\square$


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