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 126.96.36.199.
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.
Proofs independent of higher order categories can be extracted from the literature. See this MO answer by Tyler Lawson for details.
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