nLab
spectrum object

Idea

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

In the special case that $C=$ Top, a spectrum object in $C$ is just an ordinary spectrum. There is an evident generalization of the notion of spectrum from Top to any $(\mathrm{\infty }\mathrm{,1})$-category $C$ with finite limits: a spectrum object $X_{•}$ 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

For $C$ an (infinity,1)-category, a prespectrum object of $C$ is

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

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

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

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

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

$${\alpha }_n:\Sigma X[n]\to X[n+1]$$\alpha_n : \Sigma X[n] \to X[n+1]

$${\beta }_{n+1}:X[n]\to \Omega X[n+1].$$\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 for all $m\in \mathbb{Z}$ (a spectrum below $n$, if ${\beta }_m$ is an equivalence for all $m\le n$);

• 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 $m\ge n$).

One writes

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

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

Properties

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

Examples

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

References

section 8 of

• Jacob Lurie, Stable Infinity-Categories