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.



question about notation in HTT of J.Lurie
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.


