symmetric monoidal (∞,1)-category of spectra
An -space (“H” for Hopf, as in Hopf construction) is a magma in the homotopy category of topological spaces Ho(Top), or in the homotopy category category of pointed spaces, which has a unit up to homotopy.
Notice that here the homotopies for units, associativity etc. are only required to exist for an H-space, not required to be equipped with higher coherent homotopies. An -monoid equipped with such higher and coherent homotopies is instead called a strongly homotopy associative space or -space for short. If it has only higher homotopies up to level , it is called an -space.
A better name for an -space might be be -unitoid, but it is rarely used. The stands for Heinz Hopf, and reflects the sad fact that the natural name ‘homotopy group’ was already occupied; Hopf and A. Borel found necessary algebraic conditions for a space to admit an -space structure.
The only n-spheres which have -space structure are those for , and their H-space structure is given by identifying them as the unit spheres in one of the four normed division algebras (real numbers, complex numbers, quaternions, octonions) and taking the product to be that induced by the algebra product.
An example of an H-space that does not lift to an A-infinity space is the 7-sphere . It can’t be delooped because its delooping would have cohomology group a polynomial ring on a generator in degree 8, and this is impossible by mod Steenrod operations for any odd , see Lemma 2, Adams61. The 7-sphere is also not an -group.
If is an -group then for any topological space , the set of homotopy classes has a natural group structure in the strict sense; analogously if is an -cogroup then has a group structure. If there is more than one -group structure on a space, then the induced group structures on the set of homotopy classes coincide.
For more details see at loop space.
Given an A-∞ space in the (∞,1)-category ∞Grpd/Top, its image in the corresponding homotopy category of an (∞,1)-category Ho(Top) in an -space. Conversely, refining an H-space to a genuine -space means lifting the structure of a monoid object in an (∞,1)-category from the homotopy category Ho(Top) to the genuine (∞,1)-category ∞Grpd/Top.
Further discussion of this is also at loop space – Homotopy associative structure
Introduction and survey includes
The terminology -space is a definition in a Chapter IV, Section 1 (dedicated to loop spaces) of
Some other papers in the 1950s include
Arthur H. Copeland, On -spaces with two nontrivial homotopy groups, Proc. AMS 8, 1957, 109–129.
F. I. Karpelevič, A. L. Oniščik, Algebra of homologies of space of paths, Dokl. Akad. Nauk. SSSR, (N.S.), 106 (1956), 967–969. MR0081478
The theory of -spaces was widely established in the 1950s and studied by Serre, Postnikov, Spanier, Whitehead, Dold, Eckmann and Hilton and many others.
The deloopable case has more coherent structure which has been discovered few years later in J. Stasheff’s thesis and published in
For a historical account see
The description in terms of groupoid object in an (∞,1)-category is due to
see last remark of section 6.1.2 .