nLab orthogonal spectrum

Idea

Orthogonal spectra are one version of highly structured spectra that support a symmetric monoidal smash product of spectra. An orthogonal spectrum is a sequence of pointed topological spaces $\{X_n\}_{n \in \mathbb{N}}$ equipped with maps $X_n \wedge S^1 \longrightarrow X_{n+1}$ from the suspension of one into the next, but such that the $n$th topological space is equipped with an action of the orthogonal group $O(n)$ and such that the induced structure maps. $X_n \wedge S^k \longrightarrow X_{n+k}$ are all $O(n)\times O(k)$-equivariant, hence are action homomorphisms. There is a natural homotopy theory of such orthogonal spectra and it is equivalent to the standard stable homotopy theory (MMSS 98).

The category of orthogonal spectra is a presentation of the symmetric monoidal (∞,1)-category of spectra, with the special property that it implements the smash product of spectra such as to yield itself a symmetric monoidal model category of spectra: the model structure on orthogonal spectra.

This implies in particular that with respect to this symmetric smash product of spectra an E-∞ ring is presented simply as a plain commutative monoid in orthogonal spectra (“highly structured ring spectrum”). See at orthogonal ring spectrum.

Other presentations sharing this property are symmetric spectra and S-modules. In contrast to symmetric spectra, orthogonal spectra need to consist of topological spaces instead of simplicial sets.

One advantages of orthogonal spectra over symmetric spectra is that for them the naive definition of homotopy groups comes out homotopically correct, while for symmetric spectra an intransparent replacement is needed first (see symmetric spectrum – Homotopy groups).

Another advantage is that orthogonal spectra support a similarly good model for equivariant stable homotopy theory with equivariance by compact Lie groups, while symmetric spectra share this property only for equivariance under finite groups.

Definition

Orthogonal spectra

Definition

An orthogonal spectrum $X$ consists of for each $n \in \mathbb{N}$

1. a sequence of pointed topological spaces $X_n$ (the $n$th level);

2. a base-point preserving continuous action of the topological orthogonal group $O(n)$ on $X_n$;

3. based-point preserving continuous functions $\sigma_n \colon X_n \wedge S^1 \longrightarrow X_{n+1}$ from the smash product with the 1-sphere (the $n$th structure map)

such that for all $n,k \in \mathbb{N}$ with $k \geq 1$

• the continuous functions $\sigma^k \colon X_n \wedge S^k \longrightarrow X_{n+k}$ given as the compositions

  $$\sigma^k \colon X_n \wedge S^k \stackrel{\sigma_n \wedge S^{k-1}}{\longrightarrow} X_{n+1} \wedge S^{k-1} \stackrel{\sigma_{n-1} \wedge S^{k-2}}{\longrightarrow} X_{n+2} \wedge S^{k-2} \stackrel{\sigma_{n-2} \wedge S^{k-3}}{\longrightarrow} \cdots \stackrel{\sigma_{n+k-2} \wedge S^{1}}{\longrightarrow} X_{n+k-1} \wedge S^1 \stackrel{\sigma_{n+k-1} }{\longrightarrow} X_{n+k}$$

  is $O(n) \times O(k)$-equivariant

  (with respect to the $O(k)$-action on $S^k$ regarded as the representation sphere of the defining action on $\mathbb{R}^k$ and via the diagonal embedding $O(n)\times O(k) \hookrightarrow O(n+k)$).

A homomorphism $f \colon X \longrightarrow Y$ of orthogonal spectra is a sequence of $O(n)$-equivariant based continuous functions $f_n \colon X_n \longrightarrow Y_n$ commuting with the structure maps

$$\array{ X_n \wedge S^1 & \stackrel{\sigma_n^X}{\longrightarrow} & X_{n+1} \\ \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{f_{n+1}}} \\ Y_n \wedge S^1 & \stackrel{\sigma_n^Y}{\longrightarrow} & Y_{n+1} } \,.$$

We write $OrthSpectra$ for the category of orthogonal spectra with homomorphisms between them.

Homotopy groups and Weak homotopy equivalences

Definition

Given an orthogonal spectrum $X$, def. , then for $n,k \in \mathbb{N}$ the stabilization map $\iota_{n,k}$ on homotopy groups $\pi_\bullet(X_\bullet)$ of the level spaces $X_\bullet$ is

$$\iota_{n,k} \;\colon\; \pi_{n+k} X_n \stackrel{(-)\wedge S^1}{\longrightarrow} \pi_{n+k+1}(X_n \wedge S^1) \stackrel{(\sigma_n)_\ast}{\longrightarrow} \pi_{n+k+1} X_{n+1} \,.$$

Definition

Given an orthogonal spectrum $X$, def. , then for $k \in \mathbb{Z}$ its $k$th stable homotopy group is the colimit

$$\pi_k X \;\coloneqq\; \underset{\longrightarrow}{\lim}_n \pi_{n+k} X_n$$

of the homotopy groups of the level spaces, taken with respect to the stabilization maps, def. .

Definition

A homomorphism $f\colon X \longrightarrow Y$ of orthogonal spectra, def. , is a weak homotopy equivalence if it induces isomorphisms (of abelian groups)

$$\pi_\bullet(f) \;\colon\; \pi_\bullet(X) \longrightarrow \pi_\bullet(Y)$$

on all stable homotopy groups, def. .

Remark

The simplicial localization of the category of orthogonal spectra, def. , at the weak homotopy equivalences, def. , is equivalent to the (infinity,1)-category of spectra:

$$L_{whe} OrthSpectra \simeq Spectra \,.$$

See at model structure on orthogonal spectra.

Smash product

Definition

Given two orthogonal spectra $X,Y\in OrthSpectra$, def. , their smash product of spectra is the orthogonal spectrum

$$X \wedge Y \in OrthSpectra$$

whose $n$th level space is the coequalizer

$$\left( \underset{p+1+q = n}{\bigvee} O(n)_+ \underset{O(p)\times 1 \times O(q)}{\wedge} X_p \wedge S^1 \wedge Y_q \right) \stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}} \left( \underset{p+q = n}{\bigvee} O(n)_+ \underset{O(p)\times O(q)}{\wedge} X_p \wedge Y_q \right) \longrightarrow \left(X\wedge Y\right)_{n}$$

of the two maps whose components are $\sigma_p^X \wedge Y_q$ and $X_p \wedge \sigma_q^Y \circ X_p \wedge braid_{S^1, Y_q}$, respectively, and whose structure maps are induced, under the coequalizer, by the component maps $X_p\wedge \sigma_q^Y$.

Proposition

The smash product of spectra from def. naturally extends to a functor

$$(-)\wedge (-) \;\colon\; OrthSpectra \times OrthSpectra \longrightarrow OrthSpectra$$

which makes $OrthSpectra$ into a symmetric monoidal category with unit the orthogonal sphere spectrum $\mathbb{S}$, example .

Definition

For $X,Y,Z \in OrthSpectra$, def. , a bilinear-homomorphism

$$b \;\colon\; (X,Y) \longrightarrow Z \,,$$

is a collection of, for each $p,q\in \mathbb{N}$, base-point preserving $O(p) \times O(q)$-equivariant continuous functions

$$b_{p,q} \;\colon\; X_p \wedge Y_q \longrightarrow Z_{p+q}$$

(out of the smash product of pointed topological spaces) which are bilinear in that the following diagrams commutes:

$$\array{ X_p \wedge Y_q \wedge S^1 &\stackrel{b_{p,q} \wedge S^1}{\longrightarrow}& Z_{p+q} \wedge S^1 \\ \downarrow^{\mathrlap{X_p \wedge \sigma_q}} && \downarrow^{\mathrlap{\sigma_{p+q}}} \\ X_p \wedge Y_{q+1} &\stackrel{b_{p,q+1}}{\longrightarrow}& Z_{p+q+1} } \;\;\;\;,\;\;\;\;\; \array{ X_p \wedge Y_q \wedge S^1 &\stackrel{b_{p,q} \wedge S^1}{\longrightarrow}& Z_{p+q} \wedge S^1 \\ \downarrow^{\mathrlap{X_p \wedge braid_{Y_q, S^1}}} && \downarrow^{\mathrlap{\sigma_{p+q}}} \\ X_p \wedge S^1 \wedge Y_q && Z_{p+q+1} \\ \downarrow^{\mathrlap{\sigma_{p} \wedge Y_q}} && \downarrow^{\mathrlap{id \times \chi_{1,q}}} \\ X_{p+1} \wedge Y_{q} &\stackrel{b_{p+1,q}}{\longrightarrow}& Z_{p+1+q} } \,.$$

Proposition

The smash product of orthogonal spectra $X \wedge Y$, def. , is the universal recipient in $OrthSpectra$ of bilinear maps, def. , out of $(X,Y)$.

Examples

Example

The canonical incarnation of the sphere spectrum $\mathbb{S}$ as an orthogonal spectrum, def. , has $n$th level space

$$\mathbb{S}_n = S^n$$

the representation sphere of the defining linear representation of $O(n)$ on $\mathbb{R}^n$, and as structure maps the canonical smash product isomorphisms (homeomorphisms)

$$S^p \wedge S^1 \longrightarrow S^{p+1} \,.$$

Properties

Relation to the J-homomorphism

relation to the J-homomorphism:

see (Schwede 15, example 4.22)

Relation to the cobordism hypothesis

check

A connective spectrum is equivalently a grouplike E-∞ space, hence a Picard ∞-groupoid. As such it is an (∞,0)-category of fully dualizable objects. By the cobordism hypothesis this means that it is equipped with an $O(n)$-∞-action for all $n$, coming from the action $O(n)$ on the n-framings of the point in the (∞,n)-category of cobordisms. This $O(n)$-action is that which is encoded by the definition of orthogonal spectrum (Lurie 09, example 2.4.15).

Relation to global equivariant stable homotopy theory

Since orthogonal spectra are by definition equipped with orthogonal group actions, they serve as models for equivariant homotopy theory “for all groups at once”, called global stable homotopy theory.

Related concepts

• free orthogonal spectrum

• orthogonal ring spectrum

model structure on spectra, symmetric monoidal smash product of spectra

• symmetric spectrum, model structure on symmetric spectra

• orthogonal spectrum, model structure on orthogonal spectra

• S-module, model structure on S-modules

• global equivariant stable homotopy theory

References

Reviews include

• Knut Berg, Orthogonal spectra (pdf)

• Cary Malkiewich, section 2.3 of The stable homotopy category, 2014 (pdf, pdf)

Lecture notes include

• Stefan Schwede, chapter I, section 7.1 of Symmetric spectra (2012)

• Stefan Schwede, section 1 of Lectures on Equivariant Stable Homotopy Theory, 2014 (pdf)

and (Schwede 15) (take throughout $\mathcal{F} = \{1\}$ there to be the trivial family to restrict to the non-equivariant case).

Orthogonal spectra were introduced around

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Diagram spaces, diagram spectra, and FSP’s, 1998 (KTheory:0319)

and their homotopy theory and Quillen equivalences of model categories of spectra were discussed in

• Michael Mandell, Peter May, Orthogonal spectra and $S$-modules (K-theory:0318, pdf)

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, 1998 (KTheory:0320)

and for equivariant spectra in

• Michael Mandell, Peter May, Equivariant orthogonal spectra and $S$-modules, 2000 KTheory:0408

and for operads enriched over orthogonal spectra in

• Tore Kro, Model structure on operads in orthogonal spectra, Homology Homotopy Appl. Volume 9, Number 2 (2007), 397-412.(Euclid)

and in the context of equivariant stable homotopy theory in (Schwede 14) and in

• Michael Mandell, Peter May, Equivariant orthogonal spectra and S-modules. Preprint, April 29, 2000, (KTheory:0408)

and in global equivariant stable homotopy theory:

• Stefan Schwede, Global homotopy theory, 2015 (pdf)

See also

• Jacob Lurie, On the Classification of Topological Field Theories, Current Developments in Mathematics Volume 2008 (2009), 129-280 (arXiv:0905.0465)