# nLab stabilization

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## 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

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

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

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).

## Properties

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

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

has a left adjoint

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

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 $\Sigma^\infty : Top_* \to Spec$ is that which forms suspension spectra.
(∞,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

• 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)