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Proposition 2.2. The realization functor is left adjoint to the singular functor in the sense that there is an isomorphism

$$\text{hom}_\mathsf{Top}(|X|, Y) \cong \text{hom}_\mathsf{S}(X, SY)$$

which is natural in simplicial sets $X$ and topological spaces $Y$.

Proof: There are isomorphisms $$\begin{align} \text{hom}_\mathsf{Top}(|X|,Y) &\cong \varprojlim_{\Delta^n \to X} \text{hom}_\mathsf{Top}(|\Delta^n|, Y) \\ &\cong \varprojlim_{\Delta^n \to X} \text{hom}_\mathsf{S}(\Delta^n, S(Y) \\ &\cong \text{hom}_\mathsf{S}(X, SY). \end{align} $$

This is Proposition 2.2, page 7, (Simplicial Homotopy Theory, Paul G. Goerss & John F. Jardine). I understand the reasons for the first and the third isomorphisms, but I could not figure out the reason for the second isomorphism. Any help is very much appreciated.

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1 Answer 1

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That's the definition of $S(Y)$: its $n$-simplices are the maps $|\Delta^n|\to Y$. Note there's a Yoneda lemma in here too: $n$-simplices of a simplicial set are morphisms from $\Delta^n$.

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    $\begingroup$ ... and the trick of presheaves that $F \cong \operatorname{Nat}(\mathbf{y}(-), F)$. $\endgroup$
    – user14972
    Commented Jan 23, 2017 at 0:31

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