The stabilization of an (∞,1)-category with finite (∞,1)-limits is the free stable (∞,1)-category on . This is also called the -category of spectrum objects of , because for the archetypical example where Top the stabilization is the category of spectra.
There is a canonical forgetful (∞,1)-functor that remembers of a spectrum object the underlying object of in degree 0. Under mild conditions, notably when is a presentable (∞,1)-category, this functor has a left adjoint that freely stabilizes any given object of .
Going back and forth this way, i.e. applying the corresponding (∞,1)-monad yields the assignment
that may be thought of as the stabilization of an object . Indeed, as the notation suggests, may be thought of as the result as goes to infinity of the operation that forms from first the -fold suspension object and then from that the -fold loop space object.
On there is the loop space object (infinity,1)-functor , that sends each object to the pullback of the point inclusion along itself. Recall that if a -category is stable, the loop space object functor is an equivalence.
This is (StabCat, proposition 8.14).
The canonical functor from to and then further, via the functor that forgets the basepoint, to is therefore denoted
Concretely, for any with finite limits, may be constructed as the category of spectrum objects of :
This is definition 8.1, 8.2 in StabCat
Given a presentation of an (∞,1)-category by a model category, there is a notion of stabilization of this model category to a stable model category. That this in turn presents the abstractly defined stabilization of the corresponding (∞,1)-category is due to (Robalo 12, prop. 4.14).
Prop 15.4 (2) of StabCat.
stabilization is not in general functorial. It’s failure of being functorial, and approximations to it, are studied in Goodwillie calculus.
|(∞,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|
A general discussion in the context of (∞,1)-category theory is in section 1.4 of