Every based loop space has the homotopy type of a topological group. I would like to understand this fact, and this is what this question is about : why is it true, and how does one prove it?
I think i have a proof of this fact (which I'll post below), but I would like to get a more illuminating explanation, actually, if possible, some moral justification for this fact, since the result seems unlikely.
Here is a very naive explanation with all technicalities skipped.
Let $X$ be a simplicial set (this is a type of discretisation of a CW complex which is a generalization of a simplicial complex). In his paper Kan constructs out of $X$ another simplicial set, which he denotes by $GX$. The set of $i$-simplices of $GX$ is the free non-abelian group with generators the $i$-simplices of $X$ (with a small part of them cut out, but that is not that important here). The obtained simplicial set $GX$ is thus a simplicial group, meaning that for every $i \geq 0$ the set of $i$-simplices of $GX$ is a group and in addition, the face and degeneracy operators are homomorphisms (faces are defined in a similar way as for a simplicial complex). If one is familiar with simplicial sets, it is not hard to see that the geometric realization of a simplicial group is a topological group (the geometric realization of a simplicial set is defined similarly to the geometric realization of a simplicial complex and is a CW complex).
The crucial thing in that part of Kan's paper is to prove that $GX$ is a model of a loop space for $X$. This goes as follows:
Step 1: Out of $X$ and $GX$ Kan constructs another simplicial set $GX \times_\tau X$ ($\tau$ is a certain explicit map of simplicial sets which comes naturally together with the construction of $GX$) Step 2: Prove that the projection $GX \hookrightarrow GX \times_\tau X \xrightarrow{p} X$ is a Kan fibration (the equivalent of a Serre fibration but in the category of simplicial sets ; it has identical properties) Step 3: Prove that the simplicial set $GX \times_\tau X$ is contractible (this part is rather technical)
So, the geometric realizations of $GX \times_\tau X$ and $GX$ give a path-space and loop space of $X$, thus they are homotopy equivalent to the standard path space and loop space of $X$, and moreover, the geometric realization of $GX$ is a topological group.
There are a couple of other nice places where you could read the details such as:
Peter May - Simplicial objects in algebraic topology - a more combinatorial approach Goerss, Paul G., Jardine, John - Simplicial homotopy theory - a more category theoretical approach
I hope that was useful.
I know that every based loop space is homotopy equivalent to a strictly associative monoid : the space of based Moore loops. The full result should follow from the theory of simplicial sets, where one can construct a functor $\mathbb G$ from reduced simplicial sets to simplicial groups that satisfies, for any reduced simplicial set $X=X_{\bullet}$,$$|\mathbb{G}X|\simeq\Omega|X|$$ (just in case : a simplicial set $X_{\bullet}$ is reduced if $X_ 0=\mathrm{pt}$). Because geometric realization (in a convenient category of spaces) admits natural homeomorphisms $|X\times Y|\simeq|X|\times|Y|$, $|\mathbb{G}X|$ is always a topological group being a group object in the category of simplicial sets. Furthermore, there most certainly is a "reduced singular simplices functor" $S_{\bullet}^{r}:\mathbf{Top}_*\to sSet^r$ from path connected pointed spaces to reduced simplicial sets, that takes a path connected space $\mathcal X$ to the reduced simplicial set of all singular $n$-simplices $\Delta^n\to \mathcal X$ that map the vertices of $\Delta^n$ to the base point of $\mathcal X$, and that satisfies similar properties as the standard singular simplex functor. Then we would have, for any connected based space $(\mathcal X,x_0)$, setting $X=S^r(\mathcal X)$, $$\Omega\mathcal X\simeq\Omega|X|\simeq\mathcal |\mathbb{G}X|$$ This seems to work, but I would like to
The result is supposed to be contained both in Milnor's paper Construction of universal bundles I and in Kan's paper A combinatorial definition of homotopy groups, but a (very) cursory look didn't reveal much.