nLab
A-infinity operad

Idea

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

Important examples, to be discussed below, include:

• The topological operad of Stasheff associahedra.
• The little $1$-cubes operad.
• The standard dg-$A_{\mathrm{\infty }}$ operad.
• The standard categorical $A_{\mathrm{\infty }}$ operad.

An $A_{\mathrm{\infty }}$ 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 $A_{\mathrm{\infty }}$ operad may also be called an $E_1$ operad.

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

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

The topological Stasheff associahedra operad

Definition

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

The original article that defines associahedra, and in which the operad $K$ is used implicitly to define $A_{\mathrm{\infty }}$-topological spaces, is (Stasheff).

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

Properties

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

The little $1$-cubes operad

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

The little $n$-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-$A_{\mathrm{\infty }}$ operad

The standard dg-$A_{\mathrm{\infty }}$ operad is the dg-operad (that is, operad enriched in cochain complexes ${\mathrm{Ch}}^{•}(\mathrm{Vect})$)

• freely generated from one $n$-ary operation $f_n$ for each $n\ge 1$, taken to be in degree $2-n$;

• with the differential of the $n$th generator given by

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

  where $a_{p,j,n}$ is $f_p$ attachched to the $(j+1)$st input of $f_n$.

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 $A_{\mathrm{\infty }}$ 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 ‘$A_{\mathrm{\infty }}$-algebra’ and ‘$A_{\mathrm{\infty }}$-category’ to mean specifically algebras and categories over this operad. The explicit description of this operad given above means that such $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-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-$A_{\mathrm{\infty }}$ operad to the Stasheff associahedra is in the proof of proposition 1.19 in Bespalov et al.

The standard categorical $A_{\mathrm{\infty }}$ operad

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

If instead of $O$ we use the $\mathrm{Set}$-operad freely generated by a single $n$-ary operation for every $n$, we obtain a $\mathrm{Cat}$-operad whose algebras and categories are unbiased monoidal categories and bicategories.

Related concepts

• $A_{\mathrm{\infty }}$-operad

• E-∞ operad

• L-∞ operad

References

• Jim Stasheff, Homotopy associative H-spaces I, II, Trans. Amer. Math. Soc. 108 (1963), 275-312

• Peter May, The Geometry of Iterated Loop Spaces

• Martin Markl, Steve Shnider, Jim Stasheff, Operads in Algebra, Topology and Physics (web)

• Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras (arXiv:math/0512576)