symmetric monoidal (∞,1)-category of spectra
The little -disk operad or little -cubes operad (to distinguish from the framed little n-disk operad) is the topological operad/(∞,1)-operad whose -ary operations are parameterized by rectilinear disjoint embeddings of -dimensional cubes into another -dimensional cube.
When regarded as a topological operad, the topology on the space of all such embedding is such that a continuous path is given by continuously moving the images of these little cubes in the big cube around.
Many models for -operads in the literature are not in fact cofibrant in the model structure on operads, but are -cofibrant. By the therem at model structure on algebras over an operad, this is sufficient for their categories of algebras to present the correct -categories of E-∞ algebras.
Fix an integer . We let denote an open cube of dimension . We will say that a map is a rectilinear embedding if it is given by the formula for some real constants and , with .
More generally, if is a ﬁnite set, then we will say that a map is a rectilinear embedding if it is an open embedding whose restriction to each connected component of is rectilinear.
Let denote the collection of all rectitlinear embeddings from into . We will regard as a topological space (it can be identiﬁed with an open subset of .
The spaces constitute the -ary operations of a topological operad, which we will denote by and refer to as the little k-cubes operad.
This is Higher Algebra Definition 184.108.40.206.
We deﬁne a topological category as follows:
The objects of are the objects .
Given a pair of objects , a morphism from to in consists of the following data:
A morphism in .
For each a rectilinear embedding .
For every pair of objects , we regard as endowed with the topology induced by the presentation
(see 1.1.13 of Commutative Algebra)
is a groupoid object in .
Say an -algebra object is grouplike if it is grouplike as an -monoid. Say that an -algebra object in is grouplike if the restriction along is. Write
for the (∞,1)-category of grouplike -monoid objects.
The following result of (Lurie) makes precise for parameterized ∞-groupoids – for ∞-stacks – the general statement that -fold delooping provides a correspondence betwen n-categories that have trivial r-morphisms for and k-tuply monoidal n-categories.
Let , let be an ∞-stack (∞,1)-topos and let denote the full subcategory of the category of pointed objects, spanned by those pointed objects thar are -connected (i.e. their first ∞-stack homotopy groups) vanish. Then there is a canonical equivalence of (∞,1)-categories
This is EKAlg, theorem 1.3.6..
Specifically for , this refines to the classical theorem by (May).
This is EkAlg, theorem 1.3.16.
It has been long conjectured that it should be true that when suitably defined, there is a tensor product of -operads such that
This is discussed and realized in section 1.2. of (Lurie). The tensor product is defined in appendix B.7.
Explicit models of -operads include
A standard textbook reference is chapter 4 of
John Francis’ work on -actions on -categories is in
This influenced the revised version of
and is extended to include a discussion ot traces and centers in
(see also geometric ∞-function theory)
A detailed discussion of in the context of (∞,1)-operads is in
An elementary computation of the homology of the little -disk operad in terms of solar system calculus is in
For the relation to Poisson Operads see