# question about notation in HTT of J.Lurie

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.

• $Y^X$ is the simplicial set whose $n$-simplices are the maps $X\times\Delta^n\to Y$, that is the internal hom in simplicial sets. – Denis Nardin Jan 13 at 15:45

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.