An H-space produces a commutative monoid in the homotopy category of based CW-complexes, but so does an E∞-space. In my approach to higher mathematics, I intend to use one of these or perhaps both, for instance using the Dold-Thom theorem so as to remain in the homotopy category at times, using structure on spaces.
One way these will arise in what I'd like to do is via the cup product structure [X,A] ⭢ [X,A] ∧ [X,A], which seems to be an example of both of these.
Here are my beginner questions:
- What is a way of understanding the difference between these?
- Is there a functor from one of these categories to the other?
- Which of these structures is more common, typical, or recommended for particular purposes, if this question ever comes up?
There are some slightly different notions of $H$-spaces around, but I will assume you mean it is a CW-complex equipped with a multiplication map that is unital, associative and commutative up to homotopy. This is almost the same as a commutative monoid object in the homotopy category of CW-complexes, so in case you meant that, the discussion won't fundamentally change.
The difference between an $H$-space and an $\mathbb{E}_\infty$-space is related to the difference between the homotopy $1$-category $\mathsf{hTop}$ of spaces (equivalent to the homotopy $1$-category $\mathsf{hCW}$ of CW-complexes), and between the $\infty$-category $\mathsf{Spc}$ of spaces. In an appropriate $\infty$-categorical sense, an $\mathbb{E}_\infty$-space is a commutative monoid object in $\mathsf{Spc}$, while indeed an $H$-space is essentially a commutative monoid object in $\mathsf{hCW}$.
As said in the first paragraph, with an $H$-space I will mean a space with a structure of a commutative monoid up to homotopy. However, when describing an $H$-space, I am not required to specify which homotopies I use, and the homotopies do not need to satisfy any coherence conditions. To illustrate why we might want specified homotopies and coherence relations, let us look at a more classical setting, say groups. If two groups $G$ and $H$ are isomorphic, then informally we say that $G$ and $H$ share all their group theoretical properties: $G$ satisfies a certain property iff $H$ satisfies it. But suppose we are able to perform a certain construction on $G$, or have a certain relation between particular elements in $G$. This corresponds to an analogous construction or relation in $H$, but to figure out which one exactly, we need to know exactly which isomorphism we have used between $G$ and $H$. We need a specific isomorphism in order to actually translate group theory related to $G$ to group theory related to $H$. Just knowing an isomorphism exists can be insufficient in practice.
As for coherence data: let us look at the tensor product of abelian groups. Given two abelian groups $A$ and $B$, there exists an isomorphism $A\otimes B\cong B\otimes A$. Suppose with randomly choose for every $A$ and $B$ such an isomorphism $\varphi_{A,B}$. These allow us to conclude that $A\otimes B\otimes C\cong C\otimes B\otimes A$ as well. However, which isomorphism is this? There are actually multiple ways to get such an isomorphism: either we write $$ A\otimes B\otimes C = (A\otimes B) \otimes C \cong C \otimes (A\otimes B) \cong C\otimes (B\otimes A) = C \otimes B\otimes A $$ or we write $$ A\otimes B\otimes C = A\otimes (B\otimes C)\cong (B\otimes C) \otimes A \cong (C\otimes B)\otimes A = C\otimes B\otimes A, $$ and there some more options. We don't know if these isomorphisms are actually equal to one another, and as such, we don't have a preferred way to compare $A\otimes B\otimes C$ with $C\otimes B\otimes A$. This is a problem, as we just saw we need specified isomorphisms to do our work, and we cannot guarantee there is a uniform way to get such specified isomorphisms. Note that the problem does not disappear if we assume the isomorphisms $\varphi_{A,B}$ are natural in $A$ and $B$: we still have the problem that the two isomorphisms I wrote above may be different.
Requiring that all these potentially different isomorphisms between $A\otimes B\otimes C$ and $C\otimes B\otimes A$ are actually all equal is a coherence condition. We require our chosen isomorphisms witnessing commutativity of $\otimes$ to be coherent in a way. In mathematical practise, the commutativity isomorphisms that we choose come from the universal property of the tensor product, and are indeed coherent in this sense, which explains why in practice we don't have to worry and can pretend we can permute multiple objects in a tensor product and still have ''the same'' tensor product.
Given three elements $a,b,c$ in an $H$-space $X$, it is likewise unclear what homotopy you use when writing $abc\simeq cba$, even if you choose a specified homotopy witnessing commutativity of multiplication of any two elements. There is a similar problem with associativity: given four elements $a,b,c,d\in X$, there may be multiple homotopies $((ab)c)d)\simeq a(b(cd))$. And even unitality is problematic: if $e\in X$ serves as a unit element, then $(ab)e\simeq ab$ and $(ab)e\simeq a(be)\simeq a(b)=ab$ may be different homotopies. The situation gets even worse once we ask about the commutativity, associativity and unitality of multiplication of $n$ elements, for $n$ large.
You can think about an $\mathbb{E}_\infty$-space $Y$ as a space $Y$ with a multiplication operation which is associative, unital and commutative up to specified homotopies, such that those homotopies satisfy all the coherence relations that makes that all different choices of homotopies witnessing associativity, unitality and commutativity of multiplication of $n$ elements are actually equal to one another. (This is slightly imprecise: any two homotopies witnessing that $abc\simeq cba$ are not so much equal, but rather themselves homotopic in a specified sense which satisfies coherence conditions with the higher commutativity homotopies; these homotopies witnessing all the coherence relations hold are themselves also part of the data of an $\mathbb{E}_\infty$-space and satisfy further coherence relations!) It gets really messy to spell out these coherence relations out explicitly, but luckily the machinery of operads and higher category theory manages to capture them without writing all relations out algebraically.
In an $\mathbb{E}_\infty$-space, you can actually do some nontrivial algebraic work without running into the problem that you don't know how to identify certain terms with each other anymore. You will find that the algebra you can do in $H$-space is mostly confined to considerations in the homotopy category $\mathrm{hCW}$. As a slogan, given a homotopical theory like spaces, chain complexes, and the like, you don't want to do algebra and category theory inside the associated homotopy $1$-category, but want to do algebra and category theory inside the associated $\infty$-category. (The same applies to homotopy (co)limits: you don't want to take them in the homotopy category, but want to consider limits in the $\infty$-category.)
So as a one-sentence summary: an $H$-space is a space with a commutative monoid structure up to unspecified homotopy, in which you don't require any relations between those homotopies; an $\mathbb{E}_\infty$-space is a space with a commutative monoid structure up to specified, coherent homotopy. This hopefully answers your first question.
For your second question: due to their higher homotopical nature, there is not really a convenient $1$-category of $\mathbb{E}_\infty$-spaces, but there is an $\infty$-category of them, which I will write as $\mathrm{CMon}(\mathsf{Spc})$. There is a localization functor $\mathsf{Spc}\to\mathsf{hCW}$ that basically acts as the identity on objects and sends a morphism to the homotopy class of that morphism. It induces a functor $\mathrm{CMon}(\mathsf{Spc})\to\mathrm{CMon}(\mathsf{hCW})$, in which the codomain is the $1$-category of commutative monoid objects in $\mathsf{hCW}$. This functor basically takes an $\mathbb{E}_\infty$-space $Y$ and forgets which specified homotopies we used to witness unitality, associativity, commutativity and the higher coherence relations, and only remembers that unitality, associativity and commutativity holds up to (noncoherent) homotopy, thus giving you (essentially) an $H$-space. There is no interesting functor going from $\mathrm{CMon}(\mathsf{hCW})$ to $\mathrm{CMon}(\mathsf{Spc})$ (you have things like a constant functor, but that is not telling you anything interesting about how $H$-spaces relate to $\mathbb{E}_\infty$-spaces). The reason is that not every $H$-space can be enhanced to an $\mathbb{E}_\infty$-space, so there is generally no interesting way to associate to $H$-spaces a corresponding $\mathbb{E}_\infty$-space.
For your third question: I definitely think that $\mathbb{E}_\infty$-spaces are the more natural and useful notion. For more advanced purposes like obtaining multiplicative structures on cohomology theories, you need $\mathbb{E}_\infty$-spaces (or $\mathbb{E}_\infty$-groups and the like), whereas an $H$-space just does not suffice. Moreover, I'd say that all $H$-spaces we encounter in practice can be enhanced to $\mathbb{E}_\infty$-spaces. Thirdly, as said before, in categorical and homotopical mathematics it is generally much better practice to remember how things are isomorphic or homotopic and to require coherence relations on this data (the development of higher category theory was heavily motivated by this desire). For instance, you can develop a fully homotopical analogue of commutative algebra with $\mathbb{E}_\infty$-ring spectra, including a homotopical version of algebraic geometry. None of that is possible with $H$-rings, because of a lack of structure.
$H$-spaces are however much simpler to define (with this I mean it is both easier to define the concept, and to provably give actual examples), and suffice for instance to prove some things about abelianness of higher homotopy groups. This certainly gives them some merit, but I'd say they only serve as a lower shadow of a higher concept.
(I am willing to change this opinion if someone can give me an interesting example of an $H$-space that is not an $\mathbb{E}_\infty$-space or a piece of mathematics that uses $H$-spaces and could not be enhanced to be done by $\mathbb{E}_\infty$-spaces.)