For any kind of space (or possibly a directed space, viewed as some sort of category or higher dimensional analogue of one), its loop space objects canonically inherit a monoidal structure, coming from concatenation of loops.
If is essentially unique, then equipped with this monoidal structure remembers all of the structure of : we say call the delooping of the monoidal object .
What all these terms (“loops” , “delooping” etc.) mean in detail and how they are presented concretely depends on the given setup. We discuss some of these below in the section Examples.
There is an equivalence of (∞,1)-categories
(See groupoid object in an (infinity,1)-category for more details on this Quillen equivalence.)
The following result makes precise for parameterized ∞-groupoids – for ∞-stacks – the general statement that -fold delooping provides a correspondence between n-categories that have trivial r-morphisms for and k-tuply monoidal n-categories.
Write for the (∞,1)-category of groupal -algebra objects in .
Let , let be an (∞,1)-category of (∞,1)-sheaves and let denote the full subcategory of the category of pointed objects, spanned by those pointed objects thar are -connected (i.e. their first homotopy sheaves) vanish. Then there is a canonical equivalence of (∞,1)-categories
This is (Lurie, Higher Algebra, theorem 220.127.116.11).
This is EkAlg, theorem 1.3.16.
For we have a looping/delooping equivalence
If the ambient (∞,1)-topos has homotopy dimension 0 then every connected object admits a point . Still, the (∞,1)-category of pointed connected objects differs from that of unpointed connected objects (because in the latter the natural transformations may have nontrivial components on the point, while in the former case they may not).
See delooping hypothesis.
For any monoidal space, we may forget its monoidal structure and just remember the underlying space. The formation of loop space objects composed with this forgetful functor has a left adjoint which forms suspension objects.
|(∞,1)-operad||∞-algebra||grouplike version||in Top||generally|
|A-∞ operad||A-∞ algebra||∞-group||A-∞ space, e.g. loop space||loop space object|
|E-k operad||E-k algebra||k-monoidal ∞-group||iterated loop space||iterated loop space object|
|E-∞ operad||E-∞ algebra||abelian ∞-group||E-∞ space, if grouplike: infinite loop space Γ-space||infinite loop space object|
|connective spectrum||connective spectrum object|
Section 6.1.2 of
Section 5.1.3 of