nLab spectrification

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 $(\infty Grpd_{fin}^{\ast/})^{op}$) 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:

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

(Joyal 08, section 35.5)

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

Remark

An (∞,1)-functor

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

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.

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

(Joyal 08, section 35.1)

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.

See also at tangent (∞,1)-topos.

For excisive functors

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

Construction

For sequential spectra

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

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

(Lewis-May-Steinberger 86, p. 3, 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 (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

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

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

For symmetric spectra

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

See also

• spectrification of a sequential spectrum type

References

• Aldridge Bousfield, Eric Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)

• L. Gaunce Lewis, Peter May, M. Steinberger, Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics, 1986 (pdf)

• L. Gaunce Lewis, Analysis of the passage from prespectra to spectra, appendix in L. Gaunce Lewis, Peter May, M. Steinberger, Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics, 1986 (pdf)

• Charles Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, CUP 1994

• Anthony Elmendorf, Igor Kriz, Peter May, Modern foundations for stable homotopy theory, in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology, Amsterdam: North-Holland, 1995 pp. 213–253, (pdf)

• Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 77-104 (pdf)

• Stefan Schwede, around prop. 4.39 of Symmetric spectra (2012)

• André Joyal, Notes on Logoi, 2008 (pdf)

• Cary Malkiewich, Some facts about $Q X$ (pdf)