Bⁿ and coherence

Question

I understand that an internal abelian group in CW-complexes (while it is not the most extensive structure for which this can be done) produces a classifying space which itself has the structure of an internal abelian group in the derived category:

$$BA \times BA \rightarrow BA$$

It would be nice to hear about some kind of product preservation property of the classifying space functor

$$B : CW \rightarrow Ho(CW)$$

This produces in fact an internal abelian group, and it is the way I prefer to form Eilenberg-Maclane spaces. The main feature I am examining is a B-functor which can be iterated like so:

$$B^{n} := B \circ \cdots \circ B$$

This specifically entails abelian groups of some kind, at least in the homotopy category.

There are many subtleties here that I wanted some help with:

1. (Coherence concerning the target of B) it is said that not all H-spaces are E-infinity spaces. I am more interested in a B construction which can be iterated than having the most natural notion of coherence, but with that said, what is the most natural structure to say that BA has for an internal abelian group in CW?

2. (Coherence concerning the domain of B) B can be constructed for an internal group in CW, but also for a more extensive kind of object. What is the most natural higher structure that the domain of B would have?

With this said, in the context of Eilenberg-Maclane spaces, I am far more interested in a functorial B which can be iterated, as is suggested by $B^{n}$, than in using the most extensive or natural higher or operadic notion of coherence (though I guess I could get both). My last question is what the most typical domain and codomain are for B in a context in which B can be iterated, i.e. we can for the composite

My question is what the most natural definition of B is which can specifically be iterated, if not abelian groups in the homotopy category. I have an impression that people prefer operadic structures and coherence of various kinds, but I have also heard that it is the case that there are H-spaces (internal commutative monoids in the homotopy category) which do not arise from E-infinity spaces.

Answer 1

One natural setting where the delooping operation $\def\B{{\sf B}}\B$ can be defined very quickly is Γ-spaces, described by Segal in Categories and cohomology theories. A Γ-space is simply a functor $\def\FinSet{{\sf FinSet}}\def\Spc{{\sf Spc}}\FinSet_*→\Spc$. The delooping operation simply sends a Γ-space $A$ to the Γ-space $\B A$, where $\B A(S)$ is the realization of the simplicial space $T↦A(S⨯T)$. Additional references on Γ-spaces can be found in the nLab article Γ-space.

The model category of Γ-spaces is Quillen equivalent to the other models of homotopy coherent commutative monoids and/or infinite loop spaces, e.g., $\def\Ei{{\sf E}_{\sf\infty}}\Ei$-spaces (group-like or not), defined using some choice of an $\Ei$-operad.