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.
Let be an (∞,1)-category with finite (∞,1)-limit and write for its (∞,1)-category of pointed objects, the undercategory of under the terminal 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.
The stabilization of is the (∞,1)-limit (in the (∞,1)-category of (∞,1)-categories) of the tower of applications of the loop space functor
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).
For classical discussion see also spectrification and the Bousfield-Friedlander model structure.
Prop 15.4 (2) of StabCat.
In the case of ordinary homotopy types (at least), the stabilization adjunction
is a comonadic adjunction on simply connected homotopy types, meaning that simply-connected classical (but pointed) homotopy types are identified with the -modales among stable homotopy types.
Here is the (pointed) exponential modality in linear homotopy type theory, see there for more.
For Top the stabilization is the category Spec of spectra. The functor is that which forms suspension spectra.
For , the category of sets, the stabilization is trivial. An object in is a sequence of pointed sets together with isomorphisms . But for every pointed set . So every object is isomorphic to . The space of endomorphisms of this object is a limit of the endomorphism spaces , which is again contractible.
A general discussion in the context of (∞,1)-category theory is in
Jacob Lurie, section 1.4 of: Higher Algebra
Jacob Lurie, section 1 of: Spectral Schemes
Discussion of stabilization as inversion of smashing with a suspension objects, and the relation between stabilization of (∞,1)-categories (to stable (∞,1)-categories) and of model categories (to stable model categories) in
published as
with further remarks in
Formalization of stabilization in dependent linear homotopy type theory (see also on the “exponential modality” here):
On comonadicity of the “exponential modality” :
Jacobson R. Blomquist, John E. Harper, Thm. 1.8 in: Suspension spectra and higher stabilization [arXiv:1612.08623]
Kathryn Hess, Magdalena Kedziorek, Thm. 3.11 in: The homotopy theory of coalgebras over simplicial comonads, Homology, Homotopy and Applications 21 1 (2019) [arXiv:1707.07104, doi:10.4310/HHA.2019.v21.n1.a11]
Last revised on August 25, 2023 at 19:01:55. See the history of this page for a list of all contributions to it.