nLab
A-infinity-space
Context
Higher algebra
Homotopy theory
homotopy theory

Introductions
Background
Variations
Definitions
Paths and cylinders
Homotopy groups
Theorems
Contents
Definition
An $A_\infty$ -space is a homotopy type $X$ that is equipped with the structure of a monoid up to coherent higher homotopy :

that means it is equipped with

a binary product operation $\cdot \colon X \times X \to X$

a choice of associativity homotopy ; $\eta_{x,y,z} : (x\cdot y) \cdot z \to x \cdot (y \cdot z)$ ;

a choice of pentagon law? homotopy between five such $\eta$ s;

and so ever on, as controlled by the associahedra .

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

Properties
Relation to H-monoids
If in the definition of an $A_\infty$ -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 $A_\infty$ -space in the (∞,1)-category ∞Grpd /Top becomes an H-monoid in the homotopy Ho(Top) . And lifting an H-monoid structure to an $A_\infty$ -space structure means lifting a monoid structure through the projection from the (∞,1)-category ∞Grpd /Top to Ho(Top) .

Relation to $A_\infty$ -categories
The delooping of an $A_\infty$ -space is an A-∞ category /(∞,1)-category with a single object. (Beware that in standard literature “$A_\infty$ -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 $A_\infty$ -spaces.

References
$A_\infty$ -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