symmetric monoidal (∞,1)-category of spectra
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 would 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.
Does ‘unitoid’ generically mean a magma with identity? I can't verify this, but it would be convenient to have such a term. —Toby
Good question. Postnikov in his 1980-s course of homotopy theory, talks about internal unitoids in a fixed category before modifying the notion to H-version; however he does complaint that in the universal algebra the term is not standardized. – Zoran
Thanks, that's something to start with at least. —Toby
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 .
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 strructure
The terminology -space is a definition in a Chapter IV, Section 1 (dedicated to loop spaces) of
Some other papers in the 1950s include
E. H. Spanier, J. H, C, Whitehead, On fibre spaces in which the fibre is contractible, Comment. Math. Helv. 29, 1955, 1–8.
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 .