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.
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 $A_\infty$ 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
Stasheff’s $A_\infty$-operad is the relative Boardman-Vogt resolution $W([0,1], I_* \to Assoc)$ where $I_*$ is the operad for pointed objects BergerMoerdijk.
Let $\mathcal{C}_1(n)$ denote the configuration space of $n$ disjoint intervals linearly embedded in $[0,1]$. Substitution gives the sequence $\{\mathcal{C}_1(n)\}$ 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 $\mathcal{C}_n$, 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 is the dg-operad (that is, operad enriched in cochain complexes $Ch^\bullet(Vect)$)
freely generated from one $n$-ary operation $f_n$ for each $n \geq 1$, taken to be in degree $2 - n$;
with the differential of the $n$th generator given by
where $a_{p,j,n}$ is $f_p$ attached 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 $A_\infty$ 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.
In addition to
another reference is section 1.18 of
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.
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 $A_\infty$ 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 $Set$-operad freely generated by a single $n$-ary operation for every $n$, we obtain a $Cat$-operad whose algebras and categories are unbiased monoidal categories and bicategories.
$A_\infty$-operad
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)