Higher algebra

Homotopy theory



An HH-space (“H” for Hopf, as in Hopf construction) is a magma internal to the classical homotopy category of topological spaces Ho(Top), or in the homotopy category category Ho(Top) *Ho(Top)_* of pointed topological spaces, which has a unit up to homotopy.


An HH-monoid is a monoid object in Ho(Top), hence an HH-space is an HH-monoid if the product of the magma is associative up to homotopy.

An HH-group is a group object in Ho(Top), so an HH-monoid is an HH-group if has also a inverses up to homotopy.

An HH-ring is a ring object in (pointed) Ho(Top).

To continue in this pattern, one could say that an H-category is an Ho(Top)-enriched category.

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 HH-monoid equipped with such higher and coherent homotopies is instead called a strongly homotopy associative space or A A_\infty-space for short. If it has only higher homotopies up to level nn, it is called an A nA_n-space.

A better name for an HH-space might be be HH-unitoid, but it is rarely used. The HH 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 HH-space structure.

There are dual notions of HH-counitoid (or HH'-space, or co-H-space), HH-comonoid (or HH'-monoid) and HH-cogroup (or HH'-group) having co-operations with the usual identities up to homotopy.


The main example of an HH-group is the loop space ΩX\Omega X of a space XX, which however is naturally even an A-infinity space.

(The main example of an H-cogroup in Top *Top_* is the suspension SX=S 1XS X= S^1\wedge X of a pointed topological space XX. )

The only n-spheres S nS^n which have HH-space structure are those for n=0,1,3,7n = 0,1,3,7, 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 S 7S^7. 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 pp Steenrod operations for any odd pp, see Lemma 2, Adams61. The 7-sphere is also not an HH-group.

If KK is an HH-group then for any topological space XX, the set of homotopy classes [X,K][X,K] has a natural group structure in the strict sense; analogously if KK' is an HH-cogroup then [K,X][K',X] has a group structure. If there is more than one HH-group structure on a space, then the induced group structures on the set of homotopy classes coincide.

If an HH-space is equivalent to a deloopable one, then it is a group object in the (∞,1)-category Top. In other words, in that case, the associativity and other axioms hold up to coherent homotopy.

For more details see at loop space.

The classifying space BU×B U \times \mathbb{Z} for (complex) topological K-theory is an H-ring space (p. 205 (213 of 251) in A Concise Course in Algebraic Topology.)


Relation to A A_\infty-spaces

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 HH-space. Conversely, refining an H-space to a genuine A A_\infty-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

  • Martin Arkowitz, H-Spaces and Co-H-Spaces, chapter 2 in Introduction to homotopy theory, Springer 2011 (pdf)

The terminology HH-space is a definition in a Chapter IV, Section 1 (dedicated to loop spaces) of

  • Jean-Pierre Serre, Homologie singulière des espaces fibrés. Applications., Ann. Math. 54 (1951), 425–505.

Some other papers in the 1950s include

  • Edwin Spanier, Henry Whitehead, On fibre spaces in which the fibre is contractible, Comment. Math. Helv. 29, 1955, 1–8.

  • Arthur H. Copeland, On HH-spaces with two nontrivial homotopy groups, Proc. AMS 8, 1957, 109–129.

  • M. Sugawara:

    • HH-spaces and spaces of loops, Math. J. Okazama Univ, 5, 1955, 5–11;
    • A condition that a space is an HH-space, Math. J. Okayama Univ. 6, 1957, 109–129;
    • A condition that a space is group-like, Math. J. Okayama Univ. 7, 1957, 123–149.
  • 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 HH-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

  • Jim Stasheff, Homotopy associative HH-spaces I, II, Trans. Amer. Math. Soc. 108, 1963, 275-312

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 .

Wikipedia's definition (at time of this writing, and phrased in the language of homotopy theory) is rather a unitoid object in the (,1)(\infty,1)-category Top.

See also

  • John Adams, Finite HH-Spaces and Lie groups, Journal of Pure and Applied Algebra 19 (1980) 1-8 (pdf)

  • John Adams, The sphere, considered as an HH-space mod pp, Quart. J. Math. Oxford. Ser. (2) vol 12 52-60 (pdf)

Revised on June 15, 2017 09:51:13 by Urs Schreiber (