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Tag 017N

Chapter 14: Simplicial Methods > Section 14.17: Hom from simplicial sets into simplicial objects

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 $$ \mathop{\rm Hom}\nolimits(V, T) \times_{\mathop{\rm Hom}\nolimits(U, T)} \mathop{\rm Hom}\nolimits(W, T) = \mathop{\rm Hom}\nolimits(V \amalg_U W, T) $$ 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 $\mathop{\rm Hom}\nolimits$ 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{\rm Mor}\nolimits(X \times \Delta[n], \mathop{\rm Hom}\nolimits(V, T) \times_{\mathop{\rm Hom}\nolimits(U, T)} \mathop{\rm Hom}\nolimits(W, T)) \\ & = \mathop{\rm Mor}\nolimits(X \times \Delta[n], \mathop{\rm Hom}\nolimits(V, T)) \times_{\mathop{\rm Mor}\nolimits(X \times \Delta[n], \mathop{\rm Hom}\nolimits(U, T))} \mathop{\rm Mor}\nolimits(X \times \Delta[n], \mathop{\rm Hom}\nolimits(W, T)) \\ & = \mathop{\rm Mor}\nolimits(X \times \Delta[n] \times V, T) \times_{\mathop{\rm Mor}\nolimits(X \times \Delta[n] \times U, T)} \mathop{\rm Mor}\nolimits(X \times \Delta[n] \times W, T) \\ & = \mathop{\rm 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 $\mathop{\rm Hom}\nolimits(V \amalg_U W, T)_n$. $\square$

    The code snippet corresponding to this tag is a part of the file simplicial.tex and is located in lines 1674–1692 (see updates for more information).

    \begin{lemma}
    \label{lemma-hom-from-coprod}
    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
    $$
    \Hom(V, T) \times_{\Hom(U, T)} \Hom(W, T)
    =
    \Hom(V \amalg_U W, T)
    $$
    In other words, the fibre product on the left hand
    side is represented by the Hom object on the right hand side.
    \end{lemma}
    
    \begin{proof}
    By Lemma \ref{lemma-exists-hom-from-simplicial-set-finite}
    all the required $\Hom$ 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*}
    &
    \Mor(X \times \Delta[n],
    \Hom(V, T) \times_{\Hom(U, T)} \Hom(W, T)) \\
    & =
    \Mor(X \times \Delta[n], \Hom(V, T))
    \times_{\Mor(X \times \Delta[n], \Hom(U, T))}
    \Mor(X \times \Delta[n], \Hom(W, T)) \\
    & =
    \Mor(X \times \Delta[n] \times V, T)
    \times_{\Mor(X \times \Delta[n] \times U, T)}
    \Mor(X \times \Delta[n] \times W, T) \\
    & =
    \Mor(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 $\Hom(V \amalg_U W, T)_n$.
    \end{proof}

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