nLab
spectrification

Contents

Contents

Idea

The inclusion of actual spectra (e.g. sequential Omega-spectra or excisive functors on (Grpd fin */) op(\infty Grpd_{fin}^{\ast/})^{op}) into pre-spectra (e.g. sequential pre-spectra or all functors on on (Grpd fin */) op(\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 seqseq be the diagram category as follows:

seq{ x n1 p n1 * p n1 i n id * i n x n p n * id p n i n+1 * i n+1 x n+1 } n. 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}} \,.

(Joyal 08, section 35.5)

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

Remark

An (∞,1)-functor

X :seqH X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}

is equivalently

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

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

  3. a sequence of morphisms X nΩ BX n+1X_n \longrightarrow \Omega_B X_{n+1} from X nX_n into the loop space object of X n+1X_{n+1} in the slice.

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

A natural transformation f:X Y f \;\colon \;X_\bullet \to Y_\bullet between two such functors with components

{X n f n Y n p n X p n Y B 1 f b B 2} \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:B 1B 2f_b \;\colon\; B_1 \longrightarrow B_2 in H\mathbf{H} together with morphisms X nf b *Y nX_n \longrightarrow f_b^\ast Y_n into the (∞,1)-pullback of the components of Y Y_\bullet along f bf_b.

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

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

is the codomain fibration of H\mathbf{H} with “fiberwise pre-stabilization”.

A genuine spectrum object is a prespectrum object for which all the structure maps X nΩ BX n+1X_n \stackrel{\simeq}{\longrightarrow} \Omega_B X_{n+1} are equivalences. The full sub-(∞,1)-category

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

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

Lemma

(spectrification is left exact reflective)

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

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

(Joyal 08, section 35.1)

Proof

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

θ:idΩ. \theta \;\colon\; id \longrightarrow \Omega \,.

This in turn is compatible with Ω\Omega in that

θΩΩθ:ρρρ=ρ 2. \theta \circ \Omega \simeq \Omega \circ \theta \;\colon\; \rho \longrightarrow \rho \circ \rho = \rho^2 \,.

Consider then a sufficiently deep transfinite composition ρ tf\rho^{tf}. By the small object argument available in the presentable (∞,1)-category H\mathbf{H} this stabilizes, and hence provides a reflection L:H seqTHL \;\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.

See also at tangent (∞,1)-topos.

For excisive functors

See at n-excisive functor – n-Excisive approximation and reflection

Construction

For sequential spectra

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

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

(Lewis-May-Steinberger 86, p. 3, Weibel 94, 10.9.6 and topology exercise 10.9.2)

In the special case that E=Σ XE = \Sigma^\infty X a suspension spectrum, i.e. with E n=Σ nXE_n = \Sigma^n X, then (LE) 0(L E)_0 is the free infinite loop space construction.

If the pre-spectrum EE is not a CW-spectrum then the construction of the spectrification is more involved (Lewis 86).

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

(LE) n=lim kSingΩ k|E n+k| (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 EE, if all the structure maps are inclusions

E VΩ WVE W E_V \hookrightarrow \Omega^{W-V}E_W

then the spectrification is

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

(Elmendorf-Kriz-May 95, p. 7)

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.