Higher algebra

Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts




An A A_\infty-space is a homotopy type XX that is equipped with the structure of a monoid up to coherent higher homotopy:

that means it is equipped with

  1. a binary product operation :X×XX\cdot \colon X \times X \to X

  2. a choice of associativity homotopy; η x,y,z:(xy)zx(yz)\eta_{x,y,z} : (x\cdot y) \cdot z \to x \cdot (y \cdot z);

  3. a choice of pentagon law? homotopy between five such η\etas;

  4. and so ever on, as controlled by the associahedra.

In short one may say: an A A_\infty-space is an A-∞ algebra/monoid in an (∞,1)-category in the (∞,1)-category ∞Grpd/Top. See there for more details.


Relation to H-monoids

If in the definition of an A A_\infty-space one discards all the higher homotopies and retains only the existence of an associativity-homotopy, then one has the notion of H-monoid. Put another way, An A A_\infty-space in the (∞,1)-category ∞Grpd/Top becomes an H-monoid in the homotopy Ho(Top). And lifting an H-monoid structure to an A A_\infty-space structure means lifting a monoid structure through the projection from the (∞,1)-category ∞Grpd/Top to Ho(Top).

Relation to A A_\infty-categories

The delooping of an A A_\infty-space is an A-∞ category/(∞,1)-category with a single object. (Beware that in standard literature “A A_\infty-category” is often, but not necessarily, reserved for a stable (∞,1)-category).

There is an equivalence of (∞,1)-categories between pointed connected A-∞ categories/(∞,1)-categories and A A_\infty-spaces.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq ∞-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object


A A_\infty-spaces were introduced by Jim Stasheff as a refinement of an H-group taking into account higher coherences.

  • Jim Stasheff, Homotopy associative H-spaces I, II, Trans. Amer. Math. Soc. 108 (1963), 275–312 MR158400

Last revised on December 21, 2014 at 23:37:28. See the history of this page for a list of all contributions to it.