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 $C$.
Going back and forth this way, i.e. applying the corresponding (∞,1)-monad $\Omega^\infty \circ \Sigma^\infty$ yields the assignment
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.
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
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
Concretely, for any $C$ with finite limits, $Stab(C)$ may be constructed as the category of spectrum objects of $C_*$:
This is definition 8.1, 8.2 in StabCat
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.
presentable (∞,1)-category, then the functor $\Omega^\infty : Stab(C) \to C$
has a left adjoint
Prop 15.4 (2) of StabCat.
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.
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
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
published as
with further remarks in
Formalization of stabilization in dependent linear homotopy type theory (see also on the “exponential modality” here):
Last revised on November 11, 2022 at 09:55:00. See the history of this page for a list of all contributions to it.