In page 27 in HTT of J.Lurie, the expression $$\text{Map}_S(X,Y):=Y^X\times_{S^X}\{\phi\}\in \text{Set}_\Delta$$ appears for simplicial set $X,Y,S$ in Warning 1.2.2.2. However, I couldn't understand two notation in this expression, first one is exponential of simplicial set and the second one is product which have lower index. I guess that $X^Y$ is kind of $Hom_{\text{Set}_\Delta}(X,Y)$, but I couldn't be sure about it.
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2$\begingroup$ $Y^X$ is the simplicial set whose $n$-simplices are the maps $X\times\Delta^n\to Y$, that is the internal hom in simplicial sets. $\endgroup$– Denis NardinCommented Jan 13, 2021 at 15:45
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Just think in terms of ordinary sets for the moment. We have sets and maps $X\xrightarrow{\phi}S\xleftarrow{\psi}Y$ and we want to think about the set $$ \text{Map}_S(X,Y) = \{f\colon X\to Y: \psi f=\phi\}. $$ We can think of $f$ as an element of $Y^X$, and composition with $\psi$ gives a map $\psi_*\colon Y^X\to S^X$, and the condition $\psi f=\phi$ can be written as $\psi_*(f)=\phi$, so $\text{Map}_S(X,Y)$ is the preimage under $\psi_*$ of the set $\{\phi\}\subset S^X$. In other words, the diagram $\require{AMScd}$ \begin{CD} \text{Map}_S(X,Y) @>>>\{\phi\} \\ @VVV @VVV \\ Y^X @>>> S^X \end{CD} is a pullback. This is the meaning of Lurie's notation. Everything works in essentially the same way for simplicial sets.
answered Jan 13, 2021 at 15:43