orthogonal spectrum




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\{X_n\}_{n \in \mathbb{N}} equipped with maps X nS 1X n+1X_n \wedge S^1 \longrightarrow X_{n+1} from the suspension of one into the next, but such that the nnth topological space is equipped with an action of the orthogonal group O(n)O(n) and such that the induced structure maps. X nS kX n+kX_n \wedge S^k \longrightarrow X_{n+k} are all O(n)×O(k)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.


Orthogonal spectra


An orthogonal spectrum XX consists of for each nn \in \mathbb{N}

  1. a sequence of pointed topological spaces X nX_n (the nnth level);

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

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

such that for all n,kn,k \in \mathbb{N} with k1k \geq 1

  • the continuous functions σ k:X nS kX n+k\sigma^k \colon X_n \wedge S^k \longrightarrow X_{n+k} given as the compositions

    σ k:X nS kσ nS k1X n+1S k1σ n1S k2X n+2S k2σ n2S k3σ n+k2S 1X n+k1S 1σ n+k1X n+k \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)×O(k)O(n) \times O(k)-equivariant

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

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

X nS 1 σ n X X n+1 f n f n+1 Y nS 1 σ n Y Y n+1. \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 OrthSpectraOrthSpectra for the category of orthogonal spectra with homomorphisms between them.

Homotopy groups and Weak homotopy equivalences


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

ι n,k:π n+kX n()S 1π n+k+1(X nS 1)(σ n) *π n+k+1X n+1. \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} \,.

Given an orthogonal spectrum XX, def. , then for kk \in \mathbb{Z} its kkth stable homotopy group is the colimit

π kXlim nπ n+kX n \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. .


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

π (f):π (X)π (Y) \pi_\bullet(f) \;\colon\; \pi_\bullet(X) \longrightarrow \pi_\bullet(Y)

on all stable homotopy groups, def. .


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

See at model structure on orthogonal spectra.

Smash product


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

XYOrthSpectrum X \wedge Y \in OrthSpectrum

whose nnth level space is the coequalizer

(p+1+q=nO(n) +O(p)×1×O(q)X pS 1X q)(p+q=nO(n) +O(p)×O(q)X pX q)(XY) n \left( \underset{p+1+q = n}{\bigvee} O(n)_+ \underset{O(p)\times 1 \times O(q)}{\wedge} X_p \wedge S^1 \wedge X_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 X_q \right) \longrightarrow \left(X\wedge Y\right)_{n}

of the two maps whose components are σ p XY q\sigma_p^X \wedge Y_q and X pσ q YX pbraid S 1,Y qX_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 pσ q YX_p\wedge \sigma_q^Y.


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

()():OrthSpectra×OrthSpectraOrthSpectra (-)\wedge (-) \;\colon\; OrthSpectra \times OrthSpectra \longrightarrow OrthSpectra

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


For X,Y,ZOrthSpectrumX,Y,Z \in OrthSpectrum, def. , a bilinear-homomorphism

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

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

b p,q:X pX qZ p+q b_{p,q} \;\colon\; X_p \wedge X_q \longrightarrow Z_{p+q}

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

X pX qS 1 b p,qS 1 Z p+qS 1 X pσ q σ p+q X pY q+1 b p,q+1 Z p+q+1,X pX qS 1 b p,qS 1 Z p+qS 1 X pbraid X q,S 1 id X pS 1X q Z p+qS 1 σ pY q σ p+q X pY q+1 b p,q+1 Z p+q+1. \array{ X_p \wedge X_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 X_q \wedge S^1 &\stackrel{b_{p,q} \wedge S^1}{\longrightarrow}& Z_{p+q} \wedge S^1 \\ \downarrow^{\mathrlap{X_p \wedge braid_{X_q, S^1}}} && \downarrow^{\mathrlap{id}} \\ X_p \wedge S^1 \wedge X_q && Z_{p+q} \wedge S^1 \\ \downarrow^{\mathrlap{\sigma_p \wedge Y_q}} && \downarrow^{\mathrlap{\sigma_{p+q}}} \\ X_p \wedge Y_{q+1} &\stackrel{b_{p,q+1}}{\longrightarrow}& Z_{p+q+1} } \,.

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



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

𝕊 n=S n \mathbb{S}_n = S^n

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

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


Relation to the J-homomorphism

relation to the J-homomorphism:

see (Schwede 15, example 4.22)

Relation to the cobordism hypothesis


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)O(n)-∞-action for all nn, coming from the action O(n)O(n) on the n-framings of the point in the (∞,n)-category of cobordisms. This O(n)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.

model structure on spectra, symmetric monoidal smash product of spectra


Reviews include

  • Knut Berg, Orthogonal spectra (pdf)

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

Lecture notes include

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

Orthogonal spectra were introduced around

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

and for equivariant spectra in

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

and in global equivariant stable homotopy theory:

See also

Last revised on September 21, 2018 at 10:36:56. See the history of this page for a list of all contributions to it.