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\begin{document}

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\section*{spectrification}

\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{existence}{Existence}\dotfill \pageref*{existence} \linebreak
\noindent\hyperlink{for_sequential_spectra}{For sequential spectra}\dotfill \pageref*{for_sequential_spectra} \linebreak
\noindent\hyperlink{for_excisive_functors}{For excisive functors}\dotfill \pageref*{for_excisive_functors} \linebreak
\noindent\hyperlink{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{for_sequential_spectra_2}{For sequential spectra}\dotfill \pageref*{for_sequential_spectra_2} \linebreak
\noindent\hyperlink{for_coordinatefree_spectra}{For coordinate-free spectra}\dotfill \pageref*{for_coordinatefree_spectra} \linebreak
\noindent\hyperlink{for_symmetric_spectra}{For symmetric spectra}\dotfill \pageref*{for_symmetric_spectra} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The [[full subcategory|inclusion]] of actual [[spectra]] (e.g. [[sequential spectrum|sequential]] [[Omega-spectra]] or [[excisive functors]] on $(\infty Grpd_{fin}^{\ast/})^{op}$) into [[pre-spectra]] (e.g. [[sequential spectrum|sequential]] pre-spectra or all functors on on $(\infty Grpd_{fin}^{\ast/})^{op}$) has a ([[adjoint (infinity,1)-functor|infinity]]-)[[left adjoint]], ``spectrification''.

\hypertarget{existence}{}\subsection*{{Existence}}\label{existence}

\hypertarget{for_sequential_spectra}{}\subsubsection*{{For sequential spectra}}\label{for_sequential_spectra}

An original account is (\hyperlink{Lewis86}{Lewis 86}). The following is the approach due to (\hyperlink{Joyal08}{Joyal 08}), in the generality of [[parameterized spectra]] (which happens to make the analysis easier instead of harder).

\begin{defn}
\label{}\hypertarget{}{}
Let $seq$ be the [[diagram]] category as follows:

\begin{displaymath}
seq 
  \coloneqq
  \left\{
    \itexarray{
    && \vdots && \vdots
    \\
    && \downarrow && 
    \\
    \cdots &\to& x_{n-1} &\stackrel{p_{n-1}}{\longrightarrow}& \ast
    \\
    &&{}^{\mathllap{p_{n-1}}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_n}} & \searrow^{\mathrm{id}}
    \\
    &&\ast &\underset{i_n}{\longrightarrow}& x_n &\stackrel{p_n}{\longrightarrow}& \ast
    \\
    && &{}_{\mathllap{id}}\searrow& {}^{\mathllap{p_n}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_{n+1}}}
    \\
    && && \ast &\stackrel{i_{n+1}}{\longrightarrow}& x_{n+1} &\to& \cdots
    \\
     && && && \downarrow
     \\
     && && && \vdots
    }
  \right\}_{n \in \mathbb{N}}
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Joyal08}{Joyal 08, section 35.5})

Let $\mathbf{H}$ be an [[(∞,1)-topos]], for instance $\mathbf{H} =$ [[∞Grpd]] for purposes of traditional [[stable homotopy theory]].

\begin{remark}
\label{}\hypertarget{}{}
An [[(∞,1)-functor]]

\begin{displaymath}
X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}
\end{displaymath}
is equivalently

\begin{enumerate}%
\item a choice of [[object]] $B \in \mathbf{H}$ (the image of $\ast \in seq$);


\item a sequence of objects $\{X_n\} \in \mathbf{H}_{/B}$ in the [[slice (∞,1)-topos]] over $B$;


\item a sequence of [[morphisms]] $X_n \longrightarrow \Omega_B X_{n+1}$ from $X_n$ into the [[loop space object]] of $X_{n+1}$ in the slice.



\end{enumerate}
This is a [[prespectrum object]] in the [[slice (∞,1)-topos]] $\mathbf{H}_{/B}$.

A [[natural transformation]] $f \;\colon \;X_\bullet \to Y_\bullet$ between two such functors with components

\begin{displaymath}
\left\{
  \itexarray{
    X_n &\stackrel{f_n}{\longrightarrow}& Y_n
    \\
    \downarrow^{\mathrlap{p_n^X}} && \downarrow^{\mathrlap{p_n^Y}}
    \\
    B_1 &\stackrel{f_b}{\longrightarrow}& B_2
  }
  \right\}
\end{displaymath}
is equivalently a morphism of base objects $f_b \;\colon\; B_1 \longrightarrow B_2$ in $\mathbf{H}$ together with morphisms $X_n \longrightarrow f_b^\ast Y_n$ into the [[(∞,1)-pullback]] of the components of $Y_\bullet$ along $f_b$.

Therefore the [[(∞,1)-presheaf (∞,1)-topos]]

\begin{displaymath}
\mathbf{H}^{seq}
  \coloneqq
  Func(seq, \mathbf{H})
\end{displaymath}
is the [[codomain fibration]] of $\mathbf{H}$ with ``fiberwise pre-stabilization''.

A genuine [[spectrum object]] is a [[prespectrum object]] for which all the structure maps $X_n \stackrel{\simeq}{\longrightarrow} \Omega_B X_{n+1}$ are [[equivalence in an (∞,1)-category|equivalences]]. The [[full sub-(∞,1)-category]]

\begin{displaymath}
T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}
\end{displaymath}
on the genuine [[spectrum objects]] is therefore the ``fiberwise [[stabilization]]'' of the self-indexing, hence the tangent $(\infty,1)$-category.

\end{remark}
\begin{lemma}
\label{SpectrificationLemma}\hypertarget{SpectrificationLemma}{}
\textbf{(spectrification is left exact reflective)}

The inclusion of [[spectrum objects]] into $\mathbf{H}^{seq}$ is [[left exact (infinity,1)-functor|left]] [[reflective sub-(infinity,1)-category|reflective]], hence it has a [[left adjoint]] [[(∞,1)-functor]] $L$ -- [[spectrification]] -- which preserves [[finite (∞,1)-limits]].

\begin{displaymath}
T \mathbf{H}
  \stackrel{\overset{L_{lex}}{\leftarrow}}{\hookrightarrow}
  \mathbf{H}^{seq}
  \,.
\end{displaymath}
\end{lemma}
(\hyperlink{Joyal08}{Joyal 08, section 35.1})

\begin{proof}
Forming degreewise [[loop space objects]] constitutes an [[(∞,1)-functor]] $\Omega \colon \mathbf{H}^{seq} \longrightarrow \mathbf{H}^{seq}$ and by definition of $seq$ this comes with a [[natural transformation]] out of the identity

\begin{displaymath}
\theta \;\colon\; id \longrightarrow \Omega
  \,.
\end{displaymath}
This in turn is compatible with $\Omega$ in that

\begin{displaymath}
\theta \circ \Omega \simeq \Omega \circ \theta
  \;\colon\;
  \rho \longrightarrow \rho \circ \rho = \rho^2
  \,.
\end{displaymath}
Consider then a sufficiently deep [[transfinite composition]] $\rho^{tf}$. By the [[small object argument]] available in the [[presentable (∞,1)-category]] $\mathbf{H}$ this stabilizes, and hence provides a [[reflective sub-(infinity,1)-category|reflection]] $L \;\colon\; \mathbf{H}^{seq} \longrightarrow T \mathbf{H}$.

Since [[transfinite composition]] is a [[filtered (∞,1)-colimit]] and since in an [[(∞,1)-topos]] these commute with [[finite (∞,1)-limits]], it follows that spectrum objects are an [[exact (∞,1)-functor|left exact]] [[reflective sub-(∞,1)-category]].

\end{proof}
See also at \emph{[[tangent (∞,1)-topos]]}.

\hypertarget{for_excisive_functors}{}\subsubsection*{{For excisive functors}}\label{for_excisive_functors}

See at \emph{\href{n-excisive+functor#nExcisiveApproximation}{n-excisive functor -- n-Excisive approximation and reflection}}

\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

\hypertarget{for_sequential_spectra_2}{}\subsubsection*{{For sequential spectra}}\label{for_sequential_spectra_2}

For $E$ a [[sequential spectrum|sequential]] [[CW-spectrum|CW-]][[pre-spectrum]], its spectrification to an [[Omega-spectrum]] may be constructed

\begin{displaymath}
(L E)_n = \underset{\longrightarrow}{\lim}_k \Omega^k E_{n+k}
  \,.
\end{displaymath}
(\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, p. 3}, \hyperlink{Weibel94}{Weibel 94, 10.9.6 and topology exercise 10.9.2})

In the special case that $E = \Sigma^\infty X$ a [[suspension spectrum]], i.e. with $E_n = \Sigma^n X$, then $(L E)_0$ is the [[free infinite loop space]] construction.

If the pre-spectrum $E$ is not a [[CW-spectrum]] then the construction of the spectrification is more involved (\hyperlink{Lewis86}{Lewis 86}).

For $E$ a [[sequential spectrum|sequential]] [[prespectrum]] in [[pointed object|pointed]] [[simplicial sets]] the spectrification may be constructed by (\hyperlink{BousfieldFriedlander78}{Bousfield-Friedlander 78, section 2.3})

\begin{displaymath}
(L E)_n = \underset{\longrightarrow}{\lim}_k Sing {\Omega^k {\vert E_{n+k} \vert}}
\end{displaymath}
(i.e. by the previous formula combined with [[geometric realization]]/[[Kan fibrant replacement]]).

Generally, for sequential spectra in any [[proper model category|proper]] [[pointed category|pointed]] [[simplicial model category]] which admits a [[small object argument]], spectrification is discussed in (\hyperlink{Schwede97}{Schwede 97, section 2.1}).

These spectrification functors on sequential prespectra satisfy the conditions of the [[Bousfield-Friedlander theorem]], and hence the [[left Bousfield localization]] of pre-spectra with degree-wise fibrations weak equivalences at the morphisms of prespectra that become weak equivalences under spectrification exists. This is the stable \emph{[[Bousfield-Friedlander model structure]]}.

\hypertarget{for_coordinatefree_spectra}{}\subsubsection*{{For coordinate-free spectra}}\label{for_coordinatefree_spectra}

Similarly for a [[coordinate-free spectrum]] $E$, if all the structure maps are inclusions

\begin{displaymath}
E_V \hookrightarrow \Omega^{W-V}E_W
\end{displaymath}
then the spectrification is

\begin{displaymath}
(L E)_V = \underset{V\subset W}{\lim} \Omega^{W-V}E_V
  \,-
\end{displaymath}
(\hyperlink{ElmendorfKrizMay95}{Elmendorf-Kriz-May 95, p. 7})

\hypertarget{for_symmetric_spectra}{}\subsubsection*{{For symmetric spectra}}\label{for_symmetric_spectra}

For [[symmetric spectra]], see (\hyperlink{Schwede12}{Schwede 12, prop. 4.39}).

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

\begin{itemize}%
\item [[Aldridge Bousfield]], [[Eric Friedlander]], \emph{Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets}, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf}{pdf})


\item [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, \emph{Equivariant stable homotopy theory}, Springer Lecture Notes in Mathematics, 1986 (\href{http://www.math.uchicago.edu/~may/BOOKS/equi.pdf}{pdf})


\item [[L. Gaunce Lewis]], \emph{Analysis of the passage from prespectra to spectra}, appendix in [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, \emph{Equivariant stable homotopy theory}, Springer Lecture Notes in Mathematics, 1986 (\href{http://www.math.uchicago.edu/~may/BOOKS/equi.pdf}{pdf})


\item [[Charles Weibel]], \emph{[[An introduction to homological algebra]]}, Cambridge Studies in Adv. Math. 38, CUP 1994


\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], \emph{[[Modern foundations for stable homotopy theory]]}, in [[Ioan Mackenzie James]] (ed.), \emph{[[Handbook of Algebraic Topology]]}, Amsterdam: North-Holland, 1995 pp. 213--253, (\href{http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf}{pdf})


\item [[Stefan Schwede]], \emph{Spectra in model categories and applications to the algebraic cotangent complex}, Journal of Pure and Applied Algebra 120 (1997) 77-104 (\href{http://www.math.uni-bonn.de/people/schwede/modelspec.pdf}{pdf})


\item [[Stefan Schwede]], around prop. 4.39 of \emph{[[Symmetric spectra]]} (2012)


\item [[André Joyal]], \emph{Notes on Logoi}, 2008 (\href{http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf}{pdf})


\item [[Cary Malkiewich]], \emph{Some facts about $Q X$} (\href{http://www.math.uiuc.edu/~cmalkiew/Q.pdf}{pdf})



\end{itemize}
[[!redirects spectrifications]]

[[!redirects Omega-spectrification]] [[!redirects Omega-spectrifications]]



\end{document}