Every (∞,1)-category with finite (∞,1)-limits has a stabilization to a stable (∞,1)-category . This stabilization may be defined by abstract properties, but it may also be constructed explicitly as the category of spectrum objects in .
In the special case that ∞Grpd Top, a spectrum object in is a spectrum in the traditional sense. There is an evident generalization of the traditional notion of Omega-spectrum from Top to any -category with finite (∞,1)-limits: a spectrum object is essentially a list of pointed objects together with equivalences , from every object in the list to the loop space object of its successor.
For an (∞,1)-category, a prespectrum object of is
such that for all integers we have a zero object of
Notice that this definition is highly redundant. The point is that writing a spectrum object is for all a (homotopy) commuting diagram
denotes the pullback of such a diagram;
denotes the pushout of such a diagram
this induces maps
A prespectrum object is
a spectrum object if is an equivalence for all for all (a spectrum below , if is an equivalence for all );
a suspension spectrum if is an equivalence for all (a suspension spectrum above , if is an equivalence for all ).
for the full sub-(∞,1)-category of on spectrum objects in ;
– the stabilization of for the -category of spectrum objects in the -category of pointed objects of .
Let be a graded monoid in the category of groups. Write for the category of -symmetric sequences. Let be a -symmetric endofunctor of . (Usually will be the functor induced by tensor product with some object .)
A -symmetric -spectrum in is a -symmetric sequence together with assembly morphisms
such that the composite morphism
is -equivariant. (Note that acts on by the definition of symmetric endofunctor, and acts on by the definition of symmetric sequence.) A morphism of -symmetric -spectra is a morphism of -symmetric sequences making the obvious diagrams commute. We write for the category of -symmetric -spectra in .
When , the graded monoid of symmetric groups, -symmetric -spectra are called simply symmetric -spectra. When , -symmetric -spectra are called simply nonsymmetric -spectra. When the endofunctor is given by for some object , -spectra are called -spectra.
For 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).
For , is the -category version of the classical stable homotopy category of spaces: the stable (infinity,1)-category of spectra.
|(∞,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 Γ-space||infinite loop space object|
|connective spectrum||connective spectrum object|
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
Generalization to G-spectra is in