nLab
stabilization

Idea

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

There is a canonical forgetful (∞,1)-functor ${\Omega }^{\mathrm{\infty }}:\mathrm{Stab}(C)\to C$ 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 ${\Sigma }^{\mathrm{\infty }}:C\to \mathrm{Stab}(C)$ that freely stabilizes any given object of $C$.

$$({\Sigma }^{\mathrm{\infty }}⊢{\Omega }^{\mathrm{\infty }}):\mathrm{Stab}(C)\stackrel{\stackrel{{\Sigma }^{\mathrm{\infty }}}{\leftarrow }}{\underset{{\Omega }^{\mathrm{\infty }}}{\to }}C.$$(\Sigma^\infty \vdash \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 ${\Omega }^{\mathrm{\infty }}\circ {\Sigma }^{\mathrm{\infty }}$ yields the assignment

$$X\mapsto {\Omega }^{\mathrm{\infty }}{\Sigma }^{\mathrm{\infty }}X$$X \mapsto \Omega^\infty \Sigma^\infty X

that may be thought of as the stabilization of an object $X$. Indeed, as the notation suggests, ${\Omega }^{\mathrm{\infty }}{\Sigma }^{\mathrm{\infty }}X$ 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 limits and write $C_{*}:=C^{*/}$ for its $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-category is stable, the loop object functor is an equivalence.

The stabilization $\mathrm{Stab}(C)$ of $C$ is the limit (in the (infinity,1)-category of (infinity,1)-categories) of the tower of applications of the loop space functor

$$\mathrm{Stab}(C)=\lim (\cdots \to C_{*}\stackrel{\Omega }{\to }{C}_{*}\stackrel{\Omega }{\to }{C}_{*}).$$Stab(C) = \lim( \cdots \to C_* \stackrel{\Omega}{\to} C_* \stackrel{\Omega}{\to} C_* ) \,.

This is proposition 8.14 in StabCat.

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

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

Construction in terms of spectrum objects

Concretely, for any $C$ with finite limits, $\mathrm{Stab}(C)$ may be constructed as the category of spectrum objects of $C_{*}$:

$$\mathrm{Stab}(C)=\mathrm{Sp}(C_{*}).$$Stab(C) = Sp(C_*) \,.

This is definition 8.1, 8.2 in StabCat

Properties

• If $C$ is an $(\mathrm{\infty },1)$-category with finite limits that is a presentable (∞,1)-category, then the functor ${\Omega }^{\mathrm{\infty }}:\mathrm{Stab}(C)\to C$ has a left adjoint

  $${\Sigma }^{\mathrm{\infty }}:C\to \mathrm{Stab}(C).$$\Sigma^\infty : C \to Stab(C) \,.

  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.

Examples

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

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 $\simeq$ Γ-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

Section 1.4 of

• Jacob Lurie, Higher Algebra