# nLab spectrification

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

The inclusion of actual spectra (e.g. sequential Omega-spectra or excisive functors on $(\infty Grpd_{fin}^{\ast/})^{op}$) into pre-spectra (e.g. sequential pre-spectra or all functors on on $(\infty Grpd_{fin}^{\ast/})^{op}$) has a (infinity-)left adjoint, “spectrification”.

## Existence

### For sequential spectra

An original account is (Lewis 86). The following is the approach due to (Joyal 08), in the generality of parameterized spectra (which happens to make the analysis easier instead of harder).

###### Definition

Let $seq$ be the diagram category as follows:

$seq \coloneqq \left\{ \array{ && \vdots && \vdots \\ && \downarrow && \\ \cdots &\to& x_{n-1} &\stackrel{p_{n-1}}{\longrightarrow}& \ast \\ &&{}^{\mathllap{p_{n-1}}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_n}} & \searrow^{\mathrm{id}} \\ &&\ast &\underset{i_n}{\longrightarrow}& x_n &\stackrel{p_n}{\longrightarrow}& \ast \\ && &{}_{\mathllap{id}}\searrow& {}^{\mathllap{p_n}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_{n+1}}} \\ && && \ast &\stackrel{i_{n+1}}{\longrightarrow}& x_{n+1} &\to& \cdots \\ && && && \downarrow \\ && && && \vdots } \right\}_{n \in \mathbb{N}} \,.$

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

###### Remark
$X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}$

is equivalently

1. a choice of object $B \in \mathbf{H}$ (the image of $\ast \in seq$);

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

3. 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.

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

$\left\{ \array{ X_n &\stackrel{f_n}{\longrightarrow}& Y_n \\ \downarrow^{\mathrlap{p_n^X}} && \downarrow^{\mathrlap{p_n^Y}} \\ B_1 &\stackrel{f_b}{\longrightarrow}& B_2 } \right\}$

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

$\mathbf{H}^{seq} \coloneqq Func(seq, \mathbf{H})$

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 equivalences. The full sub-(∞,1)-category

$T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}$

on the genuine spectrum objects is therefore the “fiberwise stabilization” of the self-indexing, hence the tangent $(\infty,1)$-category.

###### Lemma

(spectrification is left exact reflective)

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

$T \mathbf{H} \stackrel{\overset{L_{lex}}{\leftarrow}}{\hookrightarrow} \mathbf{H}^{seq} \,.$
###### 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

$\theta \;\colon\; id \longrightarrow \Omega \,.$

This in turn is compatible with $\Omega$ in that

$\theta \circ \Omega \simeq \Omega \circ \theta \;\colon\; \rho \longrightarrow \rho \circ \rho = \rho^2 \,.$

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 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 left exact reflective sub-(∞,1)-category.

## Construction

### For sequential spectra

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

$(L E)_n = \underset{\longrightarrow}{\lim}_k \Omega^k E_{n+k} \,.$

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 (Lewis 86).

For $E$ a sequential prespectrum in pointed simplicial sets the spectrification may be constructed by (Bousfield-Friedlander 78, section 2.3)

$(L E)_n = \underset{\longrightarrow}{\lim}_k Sing {\Omega^k {\vert E_{n+k} \vert}}$

(i.e. by the previous formula combined with geometric realization/Kan fibrant replacement).

Generally, for sequential spectra in any proper pointed simplicial model category which admits a small object argument, spectrification is discussed in (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 Bousfield-Friedlander model structure.

### For coordinate-free spectra

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

$E_V \hookrightarrow \Omega^{W-V}E_W$

then the spectrification is

$(L E)_V = \underset{V\subset W}{\lim} \Omega^{W-V}E_V \,-$

### For symmetric spectra

For symmetric spectra, see (Schwede 12, prop. 4.39).

## References

Last revised on June 30, 2016 at 04:31:30. See the history of this page for a list of all contributions to it.