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.



Properties of loop space functor from homotopy types to group objects in homotopy types
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](https://en.wikipedia.org/wiki/Classifying_space). The things that have classifying spaces are topological groups or, more generally, [$E\_1$](http://ncatlab.org/nlab/show/little+cubes+operad) / [$A\_{\infty}$](http://ncatlab.org/nlab/show/A-infinity-space) 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. 


