nLab
spectrum object

Contents

Idea

Every (∞,1)-category 𝒞\mathcal{C} with finite (∞,1)-limits has a stabilization to a stable (∞,1)-category Stab(𝒞)Stab(\mathcal{C}). This stabilization may be defined by abstract properties, but it may also be constructed explicitly as the category of spectrum objects in 𝒞\mathcal{C}.

In the special case that 𝒞=\mathcal{C} = ∞Grpd L whe\simeq L_{whe}Top, a spectrum object in 𝒞\mathcal{C} is a spectrum in the traditional sense. There is an evident generalization of the traditional notion of Omega-spectrum from Top to any (,1)(\infty,1)-category 𝒞\mathcal{C} with finite (∞,1)-limits: a spectrum object X X_\bullet is essentially a list of pointed objects X iX_i together with equivalences X iΩX i+1X_i \to \Omega X_{i+1}, from every object in the list to the loop space object of its successor.

Definition

Via Ω\Omega-spectrum objects

Definition

For 𝒞\mathcal{C} an (∞,1)-category, a prespectrum object of 𝒞\mathcal{C} is

  • a (,1)(\infty,1)-functor X:×𝒞X : \mathbb{Z} \times \mathbb{Z} \to \mathcal{C}

  • such that for all integers iji \neq j we have X(i,j)=0X(i,j) = 0 a zero object of 𝒞\mathcal{C}

Notice that this definition is highly redundant. The point is that writing X[n]X(n,n)X[n] \coloneqq X(n,n) a spectrum object is for all nn \in \mathbb{Z} a (homotopy) commuting diagram

X[n+1] 0 0 X[n]. \array{ X[n+1] &\to& 0 \\ \downarrow && \downarrow \\ 0 &\to& X[n] } \,.

Recalling that in an (infinity,1)-category with zero object

  • ΩX[n+1]\Omega X[n+1] denotes the pullback of such a diagram;

  • ΣX[n]\Sigma X[n] denotes the pushout of such a diagram

this induces maps

α n:ΣX[n]X[n+1] \alpha_n : \Sigma X[n] \to X[n+1]
β n+1:X[n]ΩX[n+1]. \beta_{n+1} : X[n] \to \Omega X[n+1] \,.

A prespectrum object is

  • a spectrum object if β m\beta_m is an equivalence for all for all mm \in \mathbb{Z} (a spectrum below nn, if β m\beta_m is an equivalence for all mnm \leq n);

  • a suspension spectrum if α m\alpha_m is an equivalence for all mm \in \mathbb{Z} (a suspension spectrum above nn, if α m\alpha_m is an equivalence for all mnm \geq n).

(StabCat)

One writes

  • Sp(𝒞)Sp(\mathcal{C}) for the full sub-(∞,1)-category of Fun(×,C)Fun(\mathbb{Z} \times \mathbb{Z},C) on spectrum objects in CC;

  • Stab(𝒞)Sp(𝒞 *)Stab(\mathcal{C}) \coloneqq Sp(\mathcal{C}_*) – the stabilization of CC for the (,1)(\infty,1)-category of spectrum objects in the (,1)(\infty,1)-category C *C_* of pointed objects of 𝒞\mathcal{C}.

Via excisive functors

Write Grpd fin\infty Grpd^{fin} for the (∞,1)-category freely generated by finite (∞,1)-colimits from a single object.

Definition

Let 𝒞\mathcal{C} be an (∞,1)-category with finite (∞,1)-limits. Then a spectrum object in 𝒞\mathcal{C} is an excisive (∞,1)-functor

Grpd fin𝒞. \infty Grpd^{fin} \longrightarrow \mathcal{C} \,.

(HigherAlg, def. 1.4.2.8).

Remark

This generalizes for instance to G-spectra (Blumberg 05).

In an ordinary category

One can define Φ\Phi-symmetric FF-spectra in a category CC, where Φ\Phi is a graded monoid in the category of groups and F:CCF : C \to C is a Φ\Phi-symmetric endofunctor of CC. Here we follow Ayoub.

(One recovers the classical case described at spectrum by taking CC to be the category of pointed spaces, Φ\Phi to be the trivial graded monoid, and FF to be the suspension functor.)

Let Φ\Phi be a graded monoid in the category of groups. Write Seq(Φ,C)Seq(\Phi, C) for the category of Φ\Phi-symmetric sequences. Let F:CCF : C \to C be a Φ\Phi-symmetric endofunctor of CC. (Usually FF will be the functor T CT \otimes_C - induced by tensor product with some object TT.)

A Φ\Phi-symmetric FF-spectrum in CC is a Φ\Phi-symmetric sequence (X n) nN(X_n)_{n \in \mathbf{N}} together with assembly morphisms

γ n:F(X n)X n+1 \gamma_n : F(X_n) \to X_{n+1}

such that the composite morphism

F m(X n)F m1(X n+1)X m+n F^m(X_n) \to F^{m-1}(X_{n+1}) \to \cdots \to X_{m+n}

is (Φ m×Φ n)(\Phi_m \times \Phi_n)-equivariant. (Note that Φ m\Phi_m acts on F mF^m by the definition of symmetric endofunctor, and Φ n\Phi_n acts on X nX_n by the definition of symmetric sequence.) A morphism of Φ\Phi-symmetric TT-spectra X=(X n) nY=(Y n) nX = (X_n)_n \to Y = (Y_n)_n is a morphism of Φ\Phi-symmetric sequences making the obvious diagrams commute. We write Spect F Φ(C)Spect^{\Phi}_F(C) for the category of Φ\Phi-symmetric FF-spectra in CC.

When Φ=Σ\Phi = \Sigma, the graded monoid of symmetric groups, Σ\Sigma-symmetric FF-spectra are called simply symmetric FF-spectra. When Φ=1\Phi = 1, 11-symmetric FF-spectra are called simply nonsymmetric FF-spectra. When the endofunctor FF is given by TT \otimes - for some object TCT \in C, FF-spectra are called TT-spectra.

When CC is a symmetric monoidal category, there is an induced symmetric monoidal structure on spectrum objects.

When CC is a sufficiently nice model category, there are induced model structures on spectrum objects?.

Properties

As a model for stabilization

If CC is a pointed (,1)(\infty,1)-category with finite limits, then Sp(C)Sp(C) is a stable (infinity,1)-category.

Reflection into pre-spectrum objects

For 𝒞\mathcal{C} an (∞,1)-category with (∞,1)-pullbacks and (∞,1)-colimits, then the inclusion of spectrum objects into prespectum objects should be a left exact reflective sub-(∞,1)-category inclusion (Joyal 08, section 35).

Spec(𝒞)lexPreSpec(𝒞). Spec(\mathcal{C}) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PreSpec(\mathcal{C}) \,.

This implies in particular that the tangent (∞,1)-category of an (∞,1)-topos is itself again an (∞,1)-topos (Joyal 08, section 35.5), see at tangent (∞,1)-category – Tangent (∞,1)-topos .

Examples

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq Γ-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

References

section 8 of

Section 1.4.2 of

Section 35 of

A detailed treatment of the 1-categorical case is in the last chapter of

  • Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I. Astérisque, Vol. 314 (2008). Société Mathématique de France. (pdf)

Generalization to G-spectra is in

Revised on April 14, 2014 06:25:30 by Urs Schreiber (88.128.80.70)