Important examples, to be discussed below, include:
The topological operad of Stasheff associahedra.
The little -cubes operad.
The standard dg- operad.
The standard categorical operad.
An 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 operad may also be called an operad.
An algebra over an operad over an operad is called an -object or A-∞ algebra, where -object is often replaced with an appropriate noun; thus we have the notions of -space, -algebra, and so on. In general, -objects can be regarded as ‘monoids up to coherent homotopy.’ Likewise, a category over an operad is called an -category.
Some authors use the term ‘ operad’ only for a particular chosen operad in their chosen ambient category, and thus use ‘-object’ and ‘-category’ for algebras and categories over this particular operad. The operads discussed below are common choices for this ‘standard’ operad.
Let be the sequence of Stasheff associahedra. This is naturally equipped with the structure of a (non-symmetric) operad enriched over Top called the topological Stasheff associahedra operad or simply the Stasheff operad. Since each is contractible, is an operad.
A textbook discussion (slightly modified) is in MarklShniderStasheff, section 1.6
Let denote the configuration space of disjoint intervals linearly embedded in . Substitution gives the sequence an operad structure, called the little 1-cubes operad; it is again an operad. This is a special case of the little n-cubes operad , which is in general an operad.
The little -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- operad is the dg-operad (that is, operad enriched in cochain complexes )
freely generated from one -ary operation for each , taken to be in degree ;
with the differential of the th generator given by
where is attached to the st input of .
In the dg-context it is especially common to say ‘-algebra’ and ‘-category’ to mean specifically algebras and categories over this operad. The explicit description of this operad given above means that such -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- operad to the Stasheff associahedra is in the proof of proposition 1.19 in Bespalov et al.
Let be the operad in Set freely generated by a single binary operation and a single nullary operation. Thus, the elements of are ways to associate, and add units to, a product of things. Let be the indiscrete category on the set ; then is an operad in Cat. -algebras are precisely (non-strict, biased) monoidal categories, and -categories are precisely (biased) bicategories.
If instead of we use the -operad freely generated by a single -ary operation for every , we obtain a -operad whose algebras and categories are unbiased monoidal categories and bicategories.
Jim Stasheff, Homotopy associative H-spaces I, II, Trans. Amer. Math. Soc. 108 (1963), 275-312
Peter May, The Geometry of Iterated Loop Spaces