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\begin{document}

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\section*{stabilization}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{abstract_definition}{Abstract definition}\dotfill \pageref*{abstract_definition} \linebreak
\noindent\hyperlink{construction_in_terms_of_spectrum_objects}{Construction in terms of spectrum objects}\dotfill \pageref*{construction_in_terms_of_spectrum_objects} \linebreak
\noindent\hyperlink{ConstructionInTermsOfStableModelCategories}{Construction in terms of stable model categories}\dotfill \pageref*{ConstructionInTermsOfStableModelCategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{stabilization} of an [[(∞,1)-category]] $C$ with [[finite (∞,1)-limits]] is the [[free construction|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 [[spectrum|spectra]].

There is a canonical [[forgetful functor|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 \emph{freely stabilizes} any given object of $C$.

\begin{displaymath}
(\Sigma^\infty \dashv \Omega^\infty)
  :
  Stab(C)
   \stackrel{\overset{\Sigma^\infty}{\leftarrow}}{\underset{\Omega^\infty}{\to}}
  C
  \,.
\end{displaymath}
Going back and forth this way, i.e. applying the corresponding [[(∞,1)-monad]] $\Omega^\infty \circ \Sigma^\infty$ yields the assignment

\begin{displaymath}
X \mapsto \Omega^\infty \Sigma^\infty X
\end{displaymath}
that may be thought of as the \textbf{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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{abstract_definition}{}\subsubsection*{{Abstract definition}}\label{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 \emph{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

\begin{displaymath}
Stab(C)
  =
  \underset{\leftarrow}{\lim}
  \left(
    \cdots \to C_* \stackrel{\Omega}{\to} C_* \stackrel{\Omega}{\to} C_*
  \right)
  \,.
\end{displaymath}
This is (\href{http://arxiv.org/abs/math/0608228}{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

\begin{displaymath}
\Omega^\infty : Stab(C) \to C
  \,.
\end{displaymath}
\hypertarget{construction_in_terms_of_spectrum_objects}{}\subsubsection*{{Construction in terms of spectrum objects}}\label{construction_in_terms_of_spectrum_objects}

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

\begin{displaymath}
Stab(C) = Sp(C_*)
  \,.
\end{displaymath}
This is definition 8.1, 8.2 in \href{http://arxiv.org/abs/math/0608228}{StabCat}

\hypertarget{ConstructionInTermsOfStableModelCategories}{}\subsubsection*{{Construction in terms of stable model categories}}\label{ConstructionInTermsOfStableModelCategories}

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 (\hyperlink{Robalo12}{Robalo 12, prop. 4.14}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item 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]]

\begin{displaymath}
\Sigma^\infty : C \to Stab(C)
  \,.
\end{displaymath}
Prop 15.4 (2) of \href{http://arxiv.org/abs/math/0608228}{StabCat}.


\item stabilization is \emph{not} in general functorial. It's failure of being functorial, and approximations to it, are studied in [[Goodwillie calculus]].



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item For $C =$ [[Top]] the stabilization is the category [[Spec]] of [[spectra]]. The functor $\Sigma^\infty : Top_* \to Spec$ is that which forms [[suspension spectra]].

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include k-monoidal table]]

\hypertarget{References}{}\subsection*{{References}}\label{References}

A general discussion in the context of [[(∞,1)-category theory]] is in

\begin{itemize}%
\item [[Jacob Lurie]], section 1.4 of \emph{[[Higher Algebra]]}


\item [[Jacob Lurie]], section 1 of \emph{[[Spectral Schemes]]}



\end{itemize}
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

\begin{itemize}%
\item [[Marco Robalo]], section 4 of \emph{Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes}, June 2012 (\href{http://arxiv.org/abs/1206.3645}{arxiv:1206.3645})

published as


\item [[Marco Robalo]], section 2 of \emph{K-theory and the bridge from motives to noncommutative motives}, Advances in Mathematics Volume 269, 10 January 2015 (\href{https://doi.org/10.1016/j.aim.2014.10.011}{doi:10.1016/j.aim.2014.10.011})



\end{itemize}
with further remarks in

\begin{itemize}%
\item [[Marc Hoyois]], section 6.1 of \emph{The six operations in equivariant motivic homotopy theory}, Adv. Math. 305 (2017), 197-279 (\href{https://arxiv.org/abs/1509.02145}{arXiv:1509.02145})

\end{itemize}
[[!redirects stabilizations]] [[!redirects stabilization of an (∞,1)-category]] [[!redirects stabilization of an (infinity,1)-category]] [[!redirects stabilization of an ∞-category]] [[!redirects stabilization of an infinity-category]] [[!redirects stabilization of (∞,1)-categories]] [[!redirects stabilization of (infinity,1)-categories]] [[!redirects stabilization of ∞-categories]] [[!redirects stabilization of infinity-categories]]



\end{document}