A-infinity operad

The A A_\infty operad


An A A_\infty operad is an operad over some enriching category CC which is a (free) resolution of the standard associative operad enriched over CC (that is, the operad whose algebras are monoids).

Important examples, to be discussed below, include: * The topological operad of Stasheff associahedra. * The little 11-cubes operad. * The standard dg-A A_\infty operad. * The standard categorical A A_\infty operad.

An A A_\infty operad, like the standard associative operad, can be defined to be either a symmetric or a non-symmetric operad. On this page we assume the non-symmetric version. When regarded as a symmetric operad, an A A_\infty operad may also be called an E 1E_1 operad.

An algebra over an operad over an A A_\infty operad is called an A A_\infty-object or A-∞ algebra, where -object is often replaced with an appropriate noun; thus we have the notions of A A_\infty-space, A A_\infty-algebra, and so on. In general, A A_\infty-objects can be regarded as ‘monoids up to coherent homotopy.’ Likewise, a category over an A A_\infty operad is called an A A_\infty-category.

Some authors use the term ‘A A_\infty operad’ only for a particular chosen A A_\infty operad in their chosen ambient category, and thus use ‘A A_\infty-object’ and ‘A A_\infty-category’ for algebras and categories over this particular operad. The A A_\infty operads discussed below are common choices for this ‘standard’ A A_\infty operad.

The topological Stasheff associahedra operad


Let {K(n)}\{K(n)\} be the sequence of Stasheff associahedra. This is naturally equipped with the structure of a (non-symmetric) operad KK enriched over Top called the topological Stasheff associahedra operad or simply the Stasheff operad. Since each K(n)K(n) is contractible, KK is an A A_\infty operad.

The original article that defines associahedra, and in which the operad KK is used implicitly to define A A_\infty-topological spaces, is (Stasheff).

A textbook discussion (slightly modified) is in MarklShniderStasheff, section 1.6


Stasheff’s A A_\infty-operad is the relative Boardman-Vogt resolution W([0,1],I *Assoc)W([0,1], I_* \to Assoc) where I *I_* is the operad for pointed objects BergerMoerdijk.

The little 11-cubes operad

Let 𝒞 1(n)\mathcal{C}_1(n) denote the configuration space of nn disjoint intervals linearly embedded in [0,1][0,1]. Substitution gives the sequence {𝒞 1(n)}\{\mathcal{C}_1(n)\} an operad structure, called the little 1-cubes operad; it is again an A A_\infty operad. This is a special case of the little n-cubes operad 𝒞 n\mathcal{C}_n, which is in general an E nE_n operad.

The little nn-cubes operads (in their symmetric version) were among the first operads to be explicitly defined, in the book that first explicitly defined operads: The geometry of iterated loop spaces.

The standard dg-A A_\infty operad

The standard dg-A A_\infty operad is the dg-operad (that is, operad enriched in cochain complexes Ch (Vect)Ch^\bullet(Vect))

  • freely generated from one nn-ary operation f nf_n for each n1n \geq 1, taken to be in degree 2n2 - n;

  • with the differential of the nnth generator given by

    j+p+q=n 1<p<n(1) jp+qa p,j,n, - \sum_{j+p+q = n}^{1 \lt p \lt n} (-1)^{j p + q} a_{p,j,n} ,

    where a p,j,na_{p,j,n} is f pf_p attachched to the (j+1)(j+1)st input of f nf_{n}.

This can be shown to be a standard free resolution of the linear associative operad in the context of dg-operads; see Markl 94, proposition 3.3; therefore it is an A A_\infty operad.

It can also be shown to be isomorphic to the operad of top-dimensional (cellular) chains on the topological Stsheff associahedra operad. This is discussed on pages 26-27 of Markl 94

In the dg-context it is especially common to say ‘A A_\infty-algebra’ and ‘A A_\infty-category’ to mean specifically algebras and categories over this operad. The explicit description of this operad given above means that such A A_\infty-algebras and categories can be given a fairly direct description without explicit reference to operads.

In addition to

  • Martin Markl, Models for operads (arXiv)

another reference is section 1.18 of

  • Yu. Bespalov, V. Lyubashenko, O. Manzyuk, Pretriangulated A A_\infty-categories, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 (ps.gz)

A relation of the linear dg-A A_\infty operad to the Stasheff associahedra is in the proof of proposition 1.19 in Bespalov et al.

The standard categorical A A_\infty operad

Let OO be the operad in Set freely generated by a single binary operation and a single nullary operation. Thus, the elements of O(n)O(n) are ways to associate, and add units to, a product of nn things. Let B(n)B(n) be the indiscrete category on the set O(n)O(n); then BB is an A A_\infty operad in Cat. BB-algebras are precisely (non-strict, biased) monoidal categories, and BB-categories are precisely (biased) bicategories.

If instead of OO we use the SetSet-operad freely generated by a single nn-ary operation for every nn, we obtain a CatCat-operad whose algebras and categories are unbiased monoidal categories and bicategories.


Revised on November 3, 2015 14:04:53 by Urs Schreiber (