symmetric monoidal (∞,1)-category of spectra
This are notes on
Using the definition of the notion (∞,1)-operad in terms of a vertical categorification of the notion of category of operators, the article discusses the - or little cubes operads and its En-algebras.
A major application in the second part of the article is the study of topological chiral homology.
(see 1.1.13 of Commutative Algebra)
is a groupoid object in .
Say an -algebra object is grouplike if it is grouplike as an Assoc-monoid. Say that an -algebra object in is grouplike is the restriction along is. Write
for the (∞,1)-category of grouplike -monoid objects.
The following result 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 reduces to the classical theorem by Peter May
This is EkAlg, theorem 1.3.16.
Lurie’s proof of the equivalence of -connected objects with grouplike -objects is entirely at the level of (∞,1)-categories. One would hope that in addition there is a model for this equivalence at the level of model categories.
There is a model category structure on the category of pointed topological spaces, such that the cofibrant objects are -connected CW-complexes, described in
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. The tensor product is defined in appendix B.7.
Section 2.5 gives a proof of a generalization of the Deligne conjecture.