nLab
A-infinity-space

Definition

An $A_{\mathrm{\infty }}$-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

1. a binary product operation $\cdot :X\times X\to X$

2. a choice of associativity homotopy; ${\eta }_{x,y,z}:(x\cdot y)\cdot z\to x\cdot (y\cdot z)$;

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

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

In short one may say: an $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-space in the (∞,1)-category ∞Grpd/Top becomes an H-monoid in the homotopy Ho(Top). And lifting an H-monoid structure to an $A_{\mathrm{\infty }}$-space structure means lifting a monoid structure through the projection from the (∞,1)-category ∞Grpd/Top to Ho(Top).

Relation to $A_{\mathrm{\infty }}$-categories

The delooping of an $A_{\mathrm{\infty }}$-space is an A-∞ category/(∞,1)-category with a single object. (Beware that in standard literature ”$A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-spaces.

Related concepts

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ Γ-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

• model structure for dendroidal left fibrations

• H-space, A-n space

• A-infinity category

References

$A_{\mathrm{\infty }}$-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