nLab
A-n space

Contents

Definition

An A nA_n-space or A nA_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 nn variables.

  • An A 0A_0-space is a pointed space.
  • Same with an A 1A_1-space.
  • An A 2A_2-space is an H-space.
  • An A 3A_3-space is a homotopy associative H-space (but no coherence is required of the associator).
  • An A 4A_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.

Revised on January 24, 2013 20:00:08 by Urs Schreiber (82.113.99.233)