# nLab A-n space

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

An $A_n$-space or $A_n$-algebra in spaces is a space (in the sense of an infinity-groupoid, usually presented by a topological space or a simplicial set) with a multiplication that is associative up to higher homotopies involving up to $n$ variables.

• An $A_0$-space is a pointed space.
• Same with an $A_1$-space.
• An $A_2$-space is an H-space.
• An $A_3$-space is a homotopy associative H-space (but no coherence is required of the associator).
• An $A_4$-space has an associativity homotopy that satisfies the pentagon identity up to homotopy, but no further coherence.
• An A-infinity space has all coherent higher associativity homotopies.

Last revised on January 24, 2013 at 20:00:08. See the history of this page for a list of all contributions to it.