nLab A-n space

Definition

An $A_n$-space or $A_n$-algebra in spaces is a space (in the sense of an homotopy type/$\infty$-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 $A_\infty$-space has all coherent higher associativity homotopies.

Related concepts

• $A_\infty$-space

• E-n algebra

• A2-groupoid

• A3-groupoid?

• k-tuply associative n-groupoid

References

The notion is originally due to:

• Jim Stasheff, Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc. 108 2 (1963) 275-292 [doi:10.2307/1993608]

• Jim Stasheff, Homotopy associativity of H-spaces II 108 2 (1963) 293-312 [doi:10.2307/1993609, doi:10.1090/S0002-9947-1963-0158400-5]

Early review:

• John Adams, §2.2 of: Infinite loop spaces, Hermann Weyl lectures at IAS 1974, Annals of Mathematics Studies 90, Princeton University Press (1978) [ISBN:9780691082066, doi:10.1515/9781400821259]