I'm trying to understand the mapping space of two objects in an infinity category. Below is context and definitions, but please let me know if I have any misunderstandings.
The source of everything below is Jacob Lurie's Higher Topos Theory, section 1.2.2.
Let $S$ be an $\infty$-category. Let $x$ and $y$ be objects of $S$. The morphism space $\operatorname{Map}_{S}(x,y)$ is an object of the homotopy category which represents the space of maps from $x$ to $y$. It can be represented up to homotopy by the simplicial set of right morphisms $\operatorname{Hom}_{S}^{R}(x,y)$. This simplicial set is, in degree $n$, the set of all $(n+1)$-simplices in $S$ such that the $(n+1)$th vertex is $y$ and all the other vertices are $x$.
My own perspective currently is the $\infty$-category $s\mathbf{Alg}_{k}$, the simplicial $k$-algebras. I will write $R_{\bullet}$ and $S_{\bullet}$ to denote simplicial $k$-algebras. So in this case, $\operatorname{Map}_{s\mathbf{Alg}_{k}}(R_{\bullet},S_{\bullet})$ is represented by the homotopy type of a simplicial set that is in degree $n$, the collection of $(n+1)$-simplices with vertices $R_{\bullet},\ldots,R_{\bullet},S_{\bullet}$ with maps that make every triangle commute. For instance, in degree $1$, we have the set of all possible $2$-simplices, i.e., commutative triangles
\begin{align*} \begin{array}{ccc} R_{\bullet} & \rightarrow & S_{\bullet}\\ & \searrow & \uparrow\\ & & R_{\bullet} \end{array} \end{align*} with any possible maps that make the triangle commute.
My questions:
- Is my characterization of $\operatorname{Map}_{S}(x,y)$ and $\operatorname{Hom}_{S}^{R}(x,y)$ correct? I am doing a bit of decoding from Lurie, so I have written these objects in the way I understand them, not verbatim how Lurie defines them.
- Suppose $f:S_{\bullet}\to T_{\bullet}$ is a map in $s\mathbf{Alg}_{k}$. How do I understand the induced map $\operatorname{Map}_{s\mathbf{Alg}_{k}}(R_{\bullet},S_{\bullet})\to\operatorname{Map}_{s\mathbf{Alg}_{k}}(R_{\bullet},T_{\bullet})$? Can I understand it as the map which, in degree $n$, takes an $(n+1)$-simplex of the form $R_{\bullet},\ldots,R_{\bullet},S_{\bullet}$ and produces an $(n+1)$-simplex $R_{\bullet},\ldots,R_{\bullet},T_{\bullet}$ where every map $R_{\bullet}\to R_{\bullet}$ remains untouched, while a map $R_{\bullet}\to S_{\bullet}$ gets sent to the map $R_{\bullet}\to S_{\bullet}\xrightarrow{f}T_{\bullet}$?