nLab spectrum object

Idea

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

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

Definition

Via $\Omega$-spectrum objects

Definition

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

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

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

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

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

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

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

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

this induces maps

$$\alpha_n : \Sigma X[n] \to X[n+1]$$

$$\beta_{n+1} : X[n] \to \Omega X[n+1] \,.$$

A prespectrum object is

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

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

(StabCat)

One writes

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

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

Via excisive functors

Write $\infty Grpd_{fin}$ for the (∞,1)-category of finite homotopy types, hence those freely generated by finite (∞,1)-colimits from the point. Write $\infty Grpd_{fin}^{\ast/}$ for the pointed finite homotopy types.

Definition

Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-limits. Then a spectrum object in $\mathcal{C}$ is a reduced (i.e. terminal object-preserving) excisive (∞,1)-functor of the form

$$\infty Grpd_{fin}^{\ast/} \longrightarrow \mathcal{C} \,.$$

(HigherAlg, def. 1.4.2.8 and around p. 823).

Remark

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

In an ordinary category

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

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

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

A $\Phi$-symmetric $F$-spectrum in $C$ is a $\Phi$-symmetric sequence $(X_n)_{n \in \mathbf{N}}$ together with assembly morphisms

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

such that the composite morphism

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

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

When $\Phi = \Sigma$, the graded monoid of symmetric groups, $\Sigma$-symmetric $F$-spectra are called simply symmetric $F$-spectra. When $\Phi = 1$, $1$-symmetric $F$-spectra are called simply nonsymmetric $F$-spectra. When the endofunctor $F$ is given by $T \otimes -$ for some object $T \in C$, $F$-spectra are called $T$-spectra.

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

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

Properties

As a model for stabilization

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

Reflection into pre-spectrum objects

For $\mathcal{C}$ 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).

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

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

• For $C = Top$, $Stab(C)$ is the $(\infty,1)$-category version of the classical stable homotopy category of spaces: the stable (infinity,1)-category of spectra.

• In the equivariant homotopy theory of G-spaces a spectrum object is a spectrum with G-action.

• see also at motivic spectrum

Related concepts

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

• stable homotopy type

• cospectrum

References

Discussion in terms of stable (infinity,1)-categories is in

• Jacob Lurie, section 8 of Stable Infinity-Categories

• Jacob Lurie, section 1.4.2 Higher Algebra

• André Joyal, section 35 Notes on Logoi, 2008 (pdf)

Discussion of model structures for spectrum objects includes

• Mark Hovey, Spectra and symmetric spectra in general model categories (arXiv:0004051)

• Bjørn Ian Dundas, Oliver Röndigs, Paul Arne Østvær, Enriched functors and stable homotopy theory, Doc. Math., 8:409–488, 2003 (EuDML)

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

• Andrew Blumberg, Continuous functors as a model for the equivariant stable homotopy category (arXiv:math.AT/0505512)