symmetric monoidal (∞,1)-category of spectra
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
An $H$-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 $Ho(Top)_*$ of pointed topological spaces, which has a unit up to homotopy.
Similarly:
An $H$-monoid? is a monoid object in Ho(Top), hence an $H$-space is an $H$-monoid if the product of the magma is associative up to homotopy.
An $H$-group is a group object in Ho(Top), so an $H$-monoid is an $H$-group if it also has inverses up to homotopy.
An $H$-ring is a ring object in (pointed) Ho(Top).
To continue in this pattern, one could say that an H-category is a category internal to Ho(Top).
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 $H$-monoid equipped with such higher and coherent homotopies is instead called a strongly homotopy associative space or $A_\infty$-space for short. If it has only higher homotopies up to level $n$, it is called an $A_n$-space.
A better name for an $H$-space might be be $H$-unitoid, but it is rarely used. The $H$ 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 $H$-space structure.
There are dual notions of $H$-counitoid (or $H'$-space, or co-H-space), $H$-comonoid (or $H'$-monoid) and $H$-cogroup (or $H'$-group) having co-operations with the usual identities up to homotopy.
The main example of an $H$-group is the loop space $\Omega X$ of a space $X$, which however is naturally even an A-infinity space.
The main example of an H-cogroup in $Top_*$ is the suspension $S X= S^1\wedge X$ of a pointed topological space $X$.
The only n-spheres $S^n$ which have $H$-space structure are those for $n = 0,1,3,7$, 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 $S^7$. 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 $p$ Steenrod operations for any odd $p$, see Lemma 2, Adams61. The 7-sphere is also not an $H$-group.
If $K$ is an $H$-group then for any topological space $X$, the set of homotopy classes $[X,K]$ has a natural group structure in the strict sense; analogously if $K'$ is an $H$-cogroup then $[K',X]$ has a group structure. If there is more than one $H$-group structure on a space, then the induced group structures on the set of homotopy classes coincide.
If an $H$-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.
The classifying space $B U \times \mathbb{Z}$ for (complex) topological K-theory is an H-ring space (p. 205 (213 of 251) in A Concise Course in Algebraic Topology.)
See at Dwyer-Wilkerson H-space
Let $A$ be a connected H-space. Then for every $a:A$, the maps $\mu(a,-),\mu(-,a):A \to A$ are homotopy equivalences.
Every connected H-space is nilpotent (see there).
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 $H$-space. Conversely, refining an H-space to a genuine $A_\infty$-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.
The operations on an H-space $X$ equip its homology with the structure of ring. At least for ordinary homology this is known as the Pontrjagin ring $H_*(X)$ of $X$.
In homotopy type theory, an H-Space consists of
A homomorphism of H-spaces between two H-spaces $A$ and $B$ consists of
A function $\phi:A \to B$ such that
A function
such that the left unitor is preserved:
such that the right unitor is preserved:
We define $\mu : S^1 \to S^1 \to S^1$ by circle induction:
where $h : \prod_{x : S^1} x = x$ is defined in Lemma 6.4.2 of the HoTT book. We now need to show that $\mu(x,e)=\mu(e,x)=x$ for every $x : S^1$. Showing $\mu(e,x)=x$ is quite simple, the other way requires some more manipulation. Both of which are done in the book.
Every loop space is naturally a H-space with path concatenation as the operation. In fact every loop space is a group.
The type of maps $A \to A$ has the structure of a H-space, with basepoint $id_A$, operation function composition.
An A3-space $A$ is an H-space with an equality $\mu(\mu(a, b),c)=\mu(a,\mu(b,c))$ for every $a:A$, $b:A$, $c:A$ representing associativity up to homotopy.
A unital magma is a 0-truncated H-space.
Introduction and survey:
The terminology “$H$-space” is a definition in a Chapter IV, Section 1 (dedicated to loop spaces) of
More general notion of internal H-objects:
Beno Eckmann, Peter Hilton, Group-like structures in general categories I multiplications and comultiplications, Math. Ann. 145, 227–255 (1962) (doi:10.1007/BF01451367)
Beno Eckmann, Peter Hilton, Group-like structures in general categories III primitive categories, Math. Ann. 150 165–187 (1963) (doi:10.1007/BF01470843)
Original articles:
Edwin Spanier, Henry Whitehead, On fibre spaces in which the fibre is contractible, Comment. Math. Helv. 29, 1955, 1–8.
Arthur H. Copeland, On $H$-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 $H$-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 .
Wikipedia's definition (at time of this writing, and phrased in the language of homotopy theory) is rather a unitoid object in the $(\infty,1)$-category Top.
For H-spaces in homotopy type theory:
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, Egbert Rijke, Central H-spaces and banded types [arXiv:2301.02636]
See also
John Adams, Finite $H$-Spaces and Lie groups, Journal of Pure and Applied Algebra 19 (1980) 1-8 (pdf)
John Adams, The sphere, considered as an $H$-space mod $p$, Quart. J. Math. Oxford. Ser. (2) vol 12 52-60 (pdf)
Last revised on January 20, 2023 at 17:36:33. See the history of this page for a list of all contributions to it.