nLab stabilization

Idea

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

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

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

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

$$X \mapsto \Omega^\infty \Sigma^\infty X$$

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

Definition

Abstract definition

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

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

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

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

This is (StabCat, proposition 8.14).

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

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

Construction in terms of spectrum objects

Let $\mathscr{S}^{fin} \subseteq \mathscr{S}$ be the full subcategory of spaces generated under finite colimits by the point (i.e. the finite CW-complexes, for concreteness) and let $\mathscr{S}^{fin}_*$ be the category of pointed objects in $\mathscr{S}^{fin}$. For any category $C$ with finite limits, $Stab(C)$ may be constructed as the category of spectrum objects of $C$, that is reduced excisive functors from $\mathscr{S}^{fin}_*$ to $C$:

$$Stab(C) = \mathrm{Exc}_*(\mathscr{S}^{\mathrm{fin}}_*, C) \,.$$

This is Higher Algebra, Definition 1.4.2.8. One sees, in particular, that $Stab(C) \cong Stab(C^{*/})$: stabilizing $C$ is the same as stabilizing the category of pointed objects in $C$.

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.

Properties

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

Prop 15.4 (2) of StabCat.

• stabilization extends to a functor $Cat^{lex}_{\infty} \to Cat^{ex}_{\infty}$., i.e. from the (large) category of $\infty$-categories with finite limits and left exact functors between them to the (large) category of stable $\infty$-categories and exact functors. The stabilization of a left exact functor $F\colon C\to D$ is the functor $\mathrm{Exc}_*(\mathscr{S}^{\mathrm{fin}}_*, C) \to \mathrm{Exc}_*(\mathscr{S}^{\mathrm{fin}}_*, D)$ given by post-composition with $F$. This requires crucially $F$ to be left exact in order to be well-defined! However, it turns out that stabilization can also be made functorial in another situation: that of reduced functors $F\colon C \to D$ between differentiable $\infty$-categories that preserve sequential colimits. The image of $F$ under the stabilization functor is called in this case the linearization ((infinity,2)-Categories and the Goodwillie Calculus, Definition 5.1.5) or the derivative (Higher Algebra, Definition 6.2.1.1) of $F$. (While these objects may be defined for a more general functor $F$, these hypotheses are needed in order to have functoriality, i.e. in order for the chain rule (Higher Algebra, Corollary 6.2.1.24) to hold). The derivative of $F$ is strictly linked with the first approximation of $F$, as defined in Goodwillie calculus.

Proposition

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

$$\Sigma^\infty \dashv \Omega^\infty \;\colon\; Grpd_\infty^{\ast/} \rightleftarrows Spectra$$

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

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

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

Examples

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

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

Related concepts

	(∞,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 $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

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

An exposition of Goodwillie calculus and its role in making stabilization functorial can be found in Higher Algebra, Section 6, and in

• Jacob Lurie, section 5 of: (infinity,2)-Categories and the Goodwillie Calculus

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

• Marco Robalo, section 2 of K-theory and the bridge from motives to noncommutative motives, Advances in Mathematics Volume 269, 10 January 2015 (doi:10.1016/j.aim.2014.10.011)

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 modality” here):

• Mitchell Riley, §2.1.2 in: A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory, PhD Thesis (2022) [doi:10.14418/wes01.3.139]

On comonadicity of the “exponential modality” $\Sigma^\infty \Omega^\infty$:

• 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]