# nLab A-infinity-space

### Context

#### Higher algebra

higher algebra

universal algebra

# 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

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

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

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 ${A}_{\infty }$-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.

E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
${A}_{\infty }$-spaces were introduced by Jim Stasheff as a refinement of an H-group taking into account higher coherences.