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.
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
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
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 $m \leq n$);
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 $\mathcal{C}$.
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.
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
(HigherAlg, def. 1.4.2.8 and around p. 823).
This generalizes for instance to G-spectra (Blumberg 05).
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
such that the composite morphism
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?.
If $C$ is a pointed $(\infty,1)$-category with finite limits, then $Sp(C)$ is a stable (infinity,1)-category.
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).
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 .
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
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
Generalization to G-spectra is in