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.


  • For CC a presentable (,1)(\infty,1)-category with finite limits, the functor Ω :Stab(C)C\Omega^\infty \colon Stab(C) \to C has a left adjoint Σ :CStab(C)\Sigma^\infty : C \to Stab(C) (forming suspension spectra).

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.


In the case of ordinary homotopy types (at least), the stabilization adjunction

Σ Ω :Grpd */Spectra \Sigma^\infty \dashv \Omega^\infty \;\colon\; Grpd_\infty^{\ast/} \rightleftarrows Spectra

is a comonadic adjunction on simply connected homotopy types, meaning that simply-connected classical (but pointed) homotopy types are identified with the Σ Ω \Sigma^\infty \Omega^\infty-modales among stable homotopy types.

[Blomquist & Harper 2016 Thm. 1.8; Hess & Kedziorek 2017 Thm. 3.11]

Here Σ Ω \Sigma^\infty \Omega^\infty is the (pointed) exponential modality in linear homotopy type theory, see there for more.


  • 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):

On comonadicity of the “exponential modalityΣ Ω \Sigma^\infty \Omega^\infty:

Last revised on August 25, 2023 at 19:01:55. See the history of this page for a list of all contributions to it.