The contravariant mapping space represented by a homotopical classifying space (e.g. BG) In classical homotopy theory, there are a number of spaces which are important because they represent an interesting functor on $\operatorname{Ho(Top)}$; for example, $K(G,n)$ represents singular cohomology and $BG$ represents principal $G$-bundles. However, modern homotopy theorists know that the homotopy category is merely the truncation of a much richer structure, namely the $\infty$-category $\operatorname{Spaces}$, and both of these examples then yield representable functors from this $\infty$-category to itself. The conventional interpretation would appear to suggest that these spaces should be thought of as an "enriched" or "coherent" version of cohomology and vector bundles. Is this the case? What do these spaces look like, and are there any interesting results known about them?
 A: Let $G$ be a topological group and $X$ be a paracompact Hausdorff topological space. For simplicity let us assume that $G$ has the homotopy type of a CW complex, although a lot of this answer does not need it. Then we can define the simplicial category of principal $G$-bundles over $X$ in the following way:

*

*Its objects are principal $G$-bundles $p:P\to X$

*Given two objects $p:P\to X$ and $p':P'\to X$, the simplicial set $\operatorname{Map}(p,p')$ has as $n$ simplices the equivariant continuous maps $f:P\times |\Delta^n|\to P'\times|\Delta^n|$ over $X\times|\Delta^n|$, where $|\Delta^n|=\{(t_0,\dots,t_n)\mid \sum_i t_i=1\}$ is the topological $n$-simplex.

Note that all the mapping simplicial sets are Kan complexes. Therefore we can take its simplicial nerve and we get an $\infty$-category. In fact it is easily seen to be an $\infty$-groupoid, a.k.a. a space, $\operatorname{Bun}_G(X)$, which I will call the space of principal $G$-bundles over $X$. Note that it is canonically pointed by the trivial bundle.
With some effort this can be made controvariantly functorial in $X$, with the functoriality given by the pullback of bundles (as usual, this is accomplished by constructing a suitable fibration over the category of topological spaces -- the details are left as an exercise).
I am going to claim that $\operatorname{Bun}_G(X)$ is equivalent to $\operatorname{Map}(X,BG)$ where $BG$ is the classifying space of $G$.
The quickest proof I know of this fact is by proving that $\operatorname{Bun}_G(-)$ is a sheaf on paracompact Hausdorff spaces for the open covering topology (this requires some work -- in particular you have to show that the restriction map $\operatorname{Map}(p,p')\to \operatorname{Map}(p|_U,p'|_U)$ to an open subset is a Kan fibration).
Then it is easy to see that its sheaf $\pi_0$ is trivial and that $\Omega\operatorname{Bun}_G(-)$ is the constant sheaf at $G$ (this follows from the fact that if $L$ is a space homotopy equivalent to a CW-complex, the constant sheaf at the weak homotopy type of $L$ is given by the simplicial mapping space $U\mapsto \operatorname{Map}(U,L)$, by combining Corollary 7.1.4.4 and Proposition 7.1.5.1 in Higher Topos Theory). Therefore the recognition theorem for loopspaces in an $\infty$-topos shows that $\operatorname{Bun}_G(-)$ is the constant sheaf at $BG$. Finally, arguing as in [Higher Algebra Remark A.1.4] we see that the constant sheaf at $BG$ is exactly $\operatorname{Map}(-,BG)$.

A similar argument shows that the space of fiber bundles with fiber $F$ is equivalent to the space $\operatorname{Map}(X,B\operatorname{Homeo}(F))$, where $\operatorname{Homeo}(F)$ is the simplicial group of homeomorphisms of $F$.
