nLab stabilization




The stabilization of an (∞,1)-category CC with finite (∞,1)-limits is the free stable (∞,1)-category Stab(C)Stab(C) on CC. This is also called the (,1)(\infty,1)-category of spectrum objects of CC, because for the archetypical example where C=C = Top the stabilization is Stab(Top)SpecStab(Top) \simeq Spec the category of spectra.

There is a canonical forgetful (∞,1)-functor Ω :Stab(C)C\Omega^\infty : Stab(C) \to C that remembers of a spectrum object the underlying object of CC in degree 0. Under mild conditions, notably when CC is a presentable (∞,1)-category, this functor has a left adjoint Σ :CStab(C)\Sigma^\infty : C \to Stab(C) that freely stabilizes any given object of CC.

(Σ Ω ):Stab(C)Ω Σ C. (\Sigma^\infty \dashv \Omega^\infty) : Stab(C) \stackrel{\overset{\Sigma^\infty}{\leftarrow}}{\underset{\Omega^\infty}{\to}} C \,.

Going back and forth this way, i.e. applying the corresponding (∞,1)-monad Ω Σ \Omega^\infty \circ \Sigma^\infty yields the assignment

XΩ Σ X X \mapsto \Omega^\infty \Sigma^\infty X

that may be thought of as the stabilization of an object XX. Indeed, as the notation suggests, Ω Σ X\Omega^\infty \Sigma^\infty X may be thought of as the result as nn goes to infinity of the operation that forms from XX first the nn-fold suspension object Σ nX\Sigma^n X and then from that the nn-fold loop space object.


Abstract definition

Let CC be an (∞,1)-category with finite (∞,1)-limit and write C *:=C */C_* := C^{{*}/} for its (∞,1)-category of pointed objects, the undercategory of CC under the terminal object.

On C *C_* there is the loop space object (infinity,1)-functor Ω:C *C *\Omega : C_* \to C_*, that sends each object XX to the pullback of the point inclusion *X{*} \to X along itself. Recall that if a (,1)(\infty,1)-category is stable, the loop space object functor is an equivalence.

The stabilization Stab(C)Stab(C) of CC is the (∞,1)-limit (in the (∞,1)-category of (∞,1)-categories) of the tower of applications of the loop space functor

Stab(C)=lim(C *ΩC *ΩC *). Stab(C) = \underset{\leftarrow}{\lim} \left( \cdots \to C_* \stackrel{\Omega}{\to} C_* \stackrel{\Omega}{\to} C_* \right) \,.

This is (StabCat, proposition 8.14).

The canonical functor from Stab(C)Stab(C) to C *C_* and then further, via the functor that forgets the basepoint, to CC is therefore denoted

Ω :Stab(C)C. \Omega^\infty : Stab(C) \to C \,.

Construction in terms of spectrum objects

Concretely, for any CC with finite limits, Stab(C)Stab(C) may be constructed as the category of spectrum objects of C *C_*:

Stab(C)=Sp(C *). Stab(C) = Sp(C_*) \,.

This is definition 8.1, 8.2 in StabCat

Construction in terms of stable model categories

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.


  • If CC is an (,1)(\infty,1)-category with finite limits that is a

presentable (∞,1)-category, then the functor Ω :Stab(C)C\Omega^\infty : Stab(C) \to C

has a left adjoint

Σ :CStab(C). \Sigma^\infty : C \to Stab(C) \,.

Prop 15.4 (2) of StabCat.

  • stabilization is not in general functorial on all of (,1)Cat(\infty, 1)Cat. It’s failure of being functorial, and approximations to it, are studied in Goodwillie calculus.


  • For C=C = Top the stabilization is the category Spec of spectra. The functor Σ :Top *Spec\Sigma^\infty : Top_* \to Spec is that which forms suspension spectra.

  • For C=SetC=Set, the category of sets, the stabilization is trivial. An object in Stab(Set)Stab(Set) is a sequence of pointed sets (E 0,E 1,)(E_0, E_1, \ldots) together with isomorphisms Ω(E i+1)E i\Omega(E_{i+1}) \simeq E_i. But Ω(X)=*× X**\Omega(X) = \ast \times_X \ast \simeq \ast for every pointed set XX. So every object is isomorphic to (*,*,)(\ast, \ast, \ldots). The space of endomorphisms of this object is a limit of the endomorphism spaces Map Set *(*,*)*Map_{Set_*}(\ast, \ast) \simeq \ast, which is again contractible.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq ∞-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object


A general discussion in the context of (∞,1)-category theory is in

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

  • Marco Robalo, section 4 of Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, June 2012 (arxiv:1206.3645)

published as

with further remarks in

  • Marc Hoyois, section 6.1 of The six operations in equivariant motivic homotopy theory, Adv. Math. 305 (2017), 197-279 (arXiv:1509.02145)

Formalization of stabilization in dependent linear homotopy type theory (see also on the “exponential modalityhere):

Last revised on November 11, 2022 at 09:55:00. See the history of this page for a list of all contributions to it.