The ${A}_{\infty }$ operad

Idea

An ${A}_{\infty }$ 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-${A}_{\infty }$ operad.
• The standard categorical ${A}_{\infty }$ operad.

An ${A}_{\infty }$ 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 ${A}_{\infty }$ operad may also be called an ${E}_{1}$ operad.

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

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

Definition

Let $\left\{K\left(n\right)\right\}$ 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 $K\left(n\right)$ is contractible, $K$ is an ${A}_{\infty }$ operad.

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

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

Properties

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

The little $1$-cubes operad

Let ${𝒞}_{1}\left(n\right)$ denote the configuration space of $n$ disjoint intervals linearly embedded in $\left[0,1\right]$. Substitution gives the sequence $\left\{{𝒞}_{1}\left(n\right)\right\}$ an operad structure, called the little 1-cubes operad; it is again an ${A}_{\infty }$ 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-${A}_{\infty }$ operad

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

• 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- \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 $\left(j+1\right)$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 ${A}_{\infty }$ 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 ‘${A}_{\infty }$-algebra’ and ‘${A}_{\infty }$-category’ to mean specifically algebras and categories over this operad. The explicit description of this operad given above means that such ${A}_{\infty }$-algebras and categories can be given a fairly direct description without explicit reference to operads.

• Martin Markl, Models for operads (arXiv)

another reference is section 1.18 of

• Yu. Bespalov, V. Lyubashenko, O. Manzyuk, Pretriangulated ${A}_{\infty }$-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-${A}_{\infty }$ operad to the Stasheff associahedra is in the proof of proposition 1.19 in Bespalov et al.

The standard categorical ${A}_{\infty }$ operad

Let $O$ be the operad in Set freely generated by a single binary operation and a single nullary operation. Thus, the elements of $O\left(n\right)$ are ways to associate, and add units to, a product of $n$ things. Let $B\left(n\right)$ be the indiscrete category on the set $O\left(n\right)$; then $B$ is an ${A}_{\infty }$ 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.

References

Revised on November 11, 2010 21:29:58 by Urs Schreiber (87.212.203.135)