continuous or simplicial functor T yields a multiplication on the symmetric spectrumT(S). Indeed, using the assembly map twice and the triple structure map produces multiplication maps
T(K)∧T(L) −→ T(K∧T(L)) −→ T(T(K∧L)) −→ T(K∧L) ; here K andLare pointed spaces. If we apply this to spheres, we get Σp×Σq-equivariant maps
T(Sp)∧T(Sq) −→ T(Sp+q)
which provide the multiplication. The unit maps come from the natural transformation Id −→ T by evaluating on spheres. Here are some examples.
• The identity triple gives the sphere spectrum as a symmetric ring spectrum.
• LetGr be the reduced free group triple, i.e., it sends a pointed setKto the free group generated byKmodulo the normal subgroup generated by the basepoint. SinceGr(Sn) is weakly equivalent to ΩSn+1, which in the stable range is equivalent toSn, the unit maps form a π∗-isomorphism
S−→Gr(S). The same conclusion would hold with the free reduced monoid functor, also known
as the ‘James construction’J, sinceJ(Sn) is also weakly equivalence to ΩSn+1 as soon asn≥1.
• LetM be a topological monoid and consider the pointed continuous functorK7→M+∧K. The multiplication and unit of M make this into a triple whose algebras are pointed sets with left
M-action. The associated symmetric ring spectrum is the spherical monoid ringS[M].
• Let Abe a ring and consider the free reducedA-module triple ˜A[K] =A[K]/A[∗]. Then ˜A[S] =
HA, the Eilenberg-Mac Lane ring spectrum. We shall see later [ref] that for every symmetric spectrum of simplicial setsX the symmetric spectrum ˜A[X] isπ∗-isomorphic to the smash product
HA∧X.
• Let B be a commutative ring and consider the tripleX 7→ I( ˜B(X)), the augmentation ideal of the reduced polynomial algebra over B, generated by the pointed setX. The algebras over this triple are non-unital commutativeB-algebras, or augmented commutative B-algebras (which are equivalent categories). The ring spectrum associated to this triple is denotedDB, and it is closely related to topological Andr´e-Quillen homology for commutative B-algebras. The ring spectrum
DB is rationally equivalent to the Eilenberg-Mac Lane ring spectrumHB, butDB has torsion in higher homotopy groups.
More generally, if we evaluate a tripleT on a symmetric ring spectrumR, then the resulting spectrumT(R) is naturally a ring spectrum with multiplication maps
T(Rn)∧T(Rm) −→ T(Rn∧Rm)
T(µn,m)
−−−−−→ T(Rn+m).
Example 2.39 (Γ-spaces). Many continuous or simplicial functors arise from so called Γ-spaces, and
then the associated symmetric spectra have special properties. The category Γ is a skeletal category of the category of finite pointed sets: there is one objectn+={0,1, . . . , n} for every non-negative integern, and morphisms are the maps of sets which send 0 to 0. (Γ is really equivalent to the opposite of Segal’s category Γ, cf. [55]). A Γ-spaceis a covariant functor from Γ to the category of spaces or simplicial sets taking 0+ to a one point space (simplicial set). A morphism of Γ-spaces is a natural transformation of functors. We follow the established terminology to speak of Γ-spaceseven if the values are simplicial sets.
A Γ-spaceXcan be extended to a continuous (respectively simplicial, depending on the context) functor by a coend construction. IfXis a Γ-space andKa pointed space or simplicial set, the value of the extended functor onK is given by
Z n+∈Γ
Kn∧ X(n+),
where we use thatKn= map(n+, K) is contravariantly functorial inn+. We will not distinguish notationally between the original Γ-space and its extension. The extended functor is continuous respectively simplicial. In the simplicial context, the extension of a Γ-space admits the following different (but naturally isomorphic) description. First, X can be prolonged, by direct limit, to a functor from the category of
pointed sets, not necessarily finite, to pointed simplicial sets. Then if K is a pointed simplicial set we get a bisimplicial set [k]7→X(Kk) by evaluating the (prolonged) Γ-space degreewise. The simplicial setX(K) defined by the coend above is naturally isomorphic to the diagonal of this bisimplicial set.
Symmetric spectra which arise from Γ-spaces have special properties. Here we restrict to Γ-spaces of simplicial sets, where things are easier to state. First, every simplicial functor which arises from a Γ-space
X preserves weak equivalences of simplicial sets, see [11, Prop. 4.9]. So iff :A−→B is a level equivalence of symmetric spectra of simplicial sets, thenX(f) :X(A)−→X(B) is again a level equivalence. We shall see later thatX(−) also preservesπ∗-isomorphisms and stable equivalences [ref]. Another special property
is that symmetric spectra of the formX(S) for Γ-spaces of simplicial setsX are connective and the colimit
systems for the stable homotopy groups stabilize in a uniform way. This is because for every Γ-space X, the simplicial set X(Sn) is always (n−1)-connected [11] and the structure mapX(Sn)∧S1−→X(Sn+1) is 2n-connected [34, prop. 5.21]. Moreover, up to π∗-isomorphisms, Γ-spaces model all connective spectra
(see Theorem 5.8 of [11] [also reference to [55]?])
A Γ-space X is called specialif the map X((k+l)+)−→X(k+)×X(l+) induced by the projections from (k+l)+∼=k+
∨l+ to k+ andl+ is a weak equivalence for allkandl. In this case, the weak map
X(1+)×X(1+) ←∼− X(2+) −−−→X(∇) X(1+)
induces an abelian monoid structure on π0(X(1+)). Here ∇: 2+ −→1+ is defined by ∇(1) = 1 =∇(2). The Γ-spaceXis calledvery specialif it is special and the monoidπ0(X(1+)) is a group. By Segal’s theorem ([55, Prop. 1.4] or [11, Thm. 4.2]), the spectrumX(S) associated to a special Γ-spaceX by evaluation on spheres is a positive Ω-spectrum.
If X is very special, then X(S) is even an Ω-spectrum (i.e., from the 0th level on). In particular, the homotopy groups of a very special Γ-space X are naturally isomorphic to the homotopy groups of the simplicial setX(1+).
Example 2.40 (Orthogonal spectra). Anorthogonal spectrumconsists of the following data:
• a sequence of pointed spaces Xn forn≥0
• a base-point preserving continuous left action of the orthogonal groupO(n) onXn for eachn≥0
• based mapsσn:Xn∧S1−→Xn+1 forn≥0.
This data is subject to the following condition: for all n, m≥0, the iterated structure map
σm : Xn∧Sm −→ Xn+m
isO(n)×O(m)-equivariant. The orthogonal group acts onSmsince this is the one-point compactification ofRn andO(n)×O(m) acts on the target by restriction, along orthogonal sum, of theO(n+m)-action.
A morphismf :X−→Y of orthogonal spectra consists ofO(n)-equivariant based mapsfn :Xn −→Yn forn≥0, which are compatible with the structure maps in the sense thatfn+1◦σn =σn◦(fn∧IdS1) for
alln≥0.
Anorthogonal ring spectrumRconsists of the following data:
• a sequence of pointed spaces Rn forn≥0
• a base-point preserving continuous left action of the orthogonal groupO(n) onRn for eachn≥0
• O(n)×O(m)-equivariantmultiplication mapsµn,m:Rn∧Rm−→Rn+mforn, m≥0, and
• O(n)-equivariantunit mapsιn:Sn −→Rn for alln≥0.
This data is subject to the same associativity and unit conditions as a symmetric ring spectrum (see Definition 1.3) and a centrality condition for every unit map ιn. In the unit condition, permutations such asχn,m∈Σn+mhave to be interpreted as permutation matrices inO(n+m). An orthogonal ring spectrum
Riscommutativeif for alln, m≥0 the relationχn,m◦µn,m=µm,n◦twist holds as mapsRn∧Rm−→Rm+n. A morphism f : R −→ S of orthogonal ring spectra consists of O(n)-equivariant based maps fn :
Rn−→Sn forn≥0, which are compatible with the multiplication and unit maps (in the same sense as for symmetric ring spectra).
Orthogonal spectra are ‘symmetric spectra with extra symmetry’ in the sense that every orthogonal spectrum X has anunderlying symmetric spectrumU X. Here (U X)n =Xn and the symmetric group acts
2. EXAMPLES 39