# nLab stabilization

### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

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

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

$\left({\Sigma }^{\infty }⊢{\Omega }^{\infty }\right):\mathrm{Stab}\left(C\right)\stackrel{\stackrel{{\Sigma }^{\infty }}{←}}{\underset{{\Omega }^{\infty }}{\to }}C\phantom{\rule{thinmathspace}{0ex}}.$(\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 ${\Omega }^{\infty }\circ {\Sigma }^{\infty }$ yields the assignment

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

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

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

$\mathrm{Stab}\left(C\right)=\mathrm{lim}\left(\cdots \to {C}_{*}\stackrel{\Omega }{\to }{C}_{*}\stackrel{\Omega }{\to }{C}_{*}\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $\mathrm{Stab}\left(C\right)$ to ${C}_{*}$ and then further, via the functor that forgets the basepoint, to $C$ is therefore denoted

${\Omega }^{\infty }:\mathrm{Stab}\left(C\right)\to C\phantom{\rule{thinmathspace}{0ex}}.$\Omega^\infty : Stab(C) \to C \,.

### Construction in terms of spectrum objects

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

$\mathrm{Stab}\left(C\right)=\mathrm{Sp}\left({C}_{*}\right)\phantom{\rule{thinmathspace}{0ex}}.$Stab(C) = Sp(C_*) \,.

This is definition 8.1, 8.2 in StabCat

## Properties

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

${\Sigma }^{\infty }:C\to \mathrm{Stab}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\Sigma }^{\infty }:{\mathrm{Top}}_{*}\to \mathrm{Spec}$ is that which forms suspension spectra.
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object