The Stacks project

Lemma 14.17.5. Assume the category $\mathcal{C}$ has coproducts of any two objects and finite limits. Let $a : U \to V$, $b : U \to W$ be morphisms of simplicial sets. Assume $U_ n, V_ n, W_ n$ finite nonempty for all $n \geq 0$. Assume that all $n$-simplices of $U, V, W$ are degenerate for all $n \gg 0$. Let $T$ be a simplicial object of $\mathcal{C}$. Then

\[ \mathop{\mathrm{Hom}}\nolimits (V, T) \times _{\mathop{\mathrm{Hom}}\nolimits (U, T)} \mathop{\mathrm{Hom}}\nolimits (W, T) = \mathop{\mathrm{Hom}}\nolimits (V \amalg _ U W, T) \]

In other words, the fibre product on the left hand side is represented by the Hom object on the right hand side.

Proof. By Lemma 14.17.4 all the required $\mathop{\mathrm{Hom}}\nolimits $ objects exist and satisfy the correct functorial properties. Now we can identify the $n$th term on the left hand side as the object representing the functor that associates to $X$ the first set of the following sequence of functorial equalities

\begin{align*} & \mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n], \mathop{\mathrm{Hom}}\nolimits (V, T) \times _{\mathop{\mathrm{Hom}}\nolimits (U, T)} \mathop{\mathrm{Hom}}\nolimits (W, T)) \\ & = \mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n], \mathop{\mathrm{Hom}}\nolimits (V, T)) \times _{\mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n], \mathop{\mathrm{Hom}}\nolimits (U, T))} \mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n], \mathop{\mathrm{Hom}}\nolimits (W, T)) \\ & = \mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n] \times V, T) \times _{\mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n] \times U, T)} \mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n] \times W, T) \\ & = \mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n] \times (V \amalg _ U W), T)) \end{align*}

Here we have used the fact that

\[ (X \times \Delta [n] \times V) \times _{X \times \Delta [n] \times U} (X \times \Delta [n] \times W) = X \times \Delta [n] \times (V \amalg _ U W) \]

which is easy to verify term by term. The result of the lemma follows as the last term in the displayed sequence of equalities corresponds to $\mathop{\mathrm{Hom}}\nolimits (V \amalg _ U W, T)_ n$. $\square$

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