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I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important:

When we take the loop-space of a (connected) homotopy type, we get a homotopy type group object (since as a suspension object, the circle is canonically a cogroup object). When we forget the group structure, we are left with the normal loop-space endofunctor, and when we restrict attention to the group of components of the loop-space, this is the fundamental group. I am wondering about when we retain all of this information. What properties does the resulting functor to the category of group objects in the homotopy category have (where morphisms are morphisms of homotopy types which respect the group structure)?

Is it full? Is it faithful? Is it essentially surjective (this is a very interesting question on its own for me -- does every homotopy type group object arise from a loop space)? Is it a right adjoint (my intuition tells me yes)?

More context: there is an obvious composable pair of right adjoints $\text{Set}\rightarrow\text{Grpd}\rightarrow\text{Cat}$ whose left adjoints turn a category into a groupoid by freely inverting all arrows (quotients become necessary in the case of idempotent arrows; I think of this as the categorification of the Grothendieck group operation) and turn a groupoid into its set of components. I would like to find an analogous sequence of functors when $\text{Set}$, $\text{Grpd}$, and $\text{Cat}$ are replaced by $\text{HoSSet}$, $\text{HoSSet-Grpd}$, and $\text{HoSSet-Cat}$ respectively.

One more point: the functor sending a groupoid to its components induces a functor $\text{Grpd-Cat}\rightarrow\text{Set-Cat}\cong\text{Cat}$. When we use its closed structure to think of $\text{Grpd}$ as an object of $\text{Grpd-Cat}$, we see that this functor sends the $\text{Grpd}$-category $\text{Grpd}$ to a category equivalent to the category of sets (forgive me for playing fast and loose with size issues; insert "small" where appropriate). I wonder if there is a functor which does something similar in the case of $\text{HoSSet}$-categories.

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  • $\begingroup$ I've just (rather serendipitously) stumbled upon the fact that the appropriate "classifying space" functor is probably the adjoint I am seeking. However, I don't know very much about this functor. $\endgroup$ – Kyle Ferendo Jul 14 '15 at 23:49
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    $\begingroup$ A phrase that might help in your search is "Kan loop group". $\endgroup$ – Tyler Lawson Jul 15 '15 at 2:38
  • $\begingroup$ @TylerLawson, yes, thank you. When we take simplicial sets as our $(\infty,0)$-categories and simplicially enriched categories as a our $(\infty,1)$-categories, this construction seems to give the correct forgetful functor. I had been wondering if one could discard the apparently superficial information of simplicial sets and take homotopy types themselves as infinity groupoids. As Qiaochu points out, even if this is possible, it is not in so straightforward a manner as I'd been attempting. $\endgroup$ – Kyle Ferendo Jul 16 '15 at 0:13
  • $\begingroup$ I'm not an expert in any of this material, but it seems to me that it might be helpful to learn about classifying spaces before jumping to $(\infty,1)$-categories. $\endgroup$ – Tom Church Jul 17 '15 at 2:05
  • $\begingroup$ Yes, as I and others have already noted, it is a useful topic for me to familiarize myself with. However, it is not necessary for understanding $(\infty,1)$-categories; it is not relevant to any of the classical expositions I am familiar with on quasicategories, Segal categories, complete Segal spaces, or simplicial categories (or model categories). If you know of its importance in some facet that hasn't been discussed in this thread, however, I would be grateful for a link to more exposition. $\endgroup$ – Kyle Ferendo Jul 18 '15 at 15:30
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Group objects in the homotopy category are the wrong thing to look at, and the reason is precisely that this is not enough structure to construct a classifying space. The things that have classifying spaces are topological groups or, more generally, $E_1$ / $A_{\infty}$ spaces. These are higher categorical and cannot be defined solely in terms of the homotopy category.

With this modification, the loop space functor establishes an equivalence of higher categories between pointed connected spaces (by which I mean weak homotopy types) and grouplike $E_1$ spaces, with the inverse given by taking classifying spaces. In particular, every grouplike $E_1$ space is a loop space; this is the recognition principle for loop spaces.

Regarding why group objects in the homotopy category are the wrong thing to look at, I don't have a counterexample off the top of my head on the level of objects, but here's a counterexample on the level of morphisms. Let $G$ be a group and let $A$ be an abelian group. Homotopy classes of pointed maps from $BG$ (the Eilenberg-MacLane space $K(G, 1)$) to $B^2 A$ (the Eilenberg-MacLane space $K(A, 2)$) are classified by second group cohomology $H^2(G, A)$. After taking loop spaces, whatever homotopy classes of maps from the group object $G$ to the group object $BA$ is, it's a subset of homotopy classes of maps from $G$ to $BA$. But since $G$ is a discrete space and $BA$ is connected, there is a unique such homotopy class.

The problem is that the data of a map of $E_1$ / $A_{\infty}$ spaces from $G$ to $BA$ involves more information than the data of a map between their underlying spaces: that is, "preserving $E_1$ / $A_{\infty}$ structure" is itself a structure, not a property.

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    $\begingroup$ This is a fantastic answer, and I am fascinated to learn that "preserving $E_1$ structure" is a structure; I had certainly been thinking too much in terms of plain old sets and groups. I'll begin researching $E_1$ spaces now. My adviser presented a good, intuitive argument for why the approach I described above might be wrong: in terms of (strictly) monoidal (fibrant-cofibrant) topological spaces, only a subspace of the mapping space between them will correspond to strictly monoidal maps, so there may be fewer monoidal homotopies. Thanks so much for the clarification and research suggestions! $\endgroup$ – Kyle Ferendo Jul 15 '15 at 3:51

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