$\operatorname{Sp}(\mathcal{S}_*) = \operatorname{Sp}$, the $\infty$-category of spectra. The adjunction $\Sigma^\infty \dashv \Omega^\infty$ sends a pointed space $X$ to its suspension spectrum $\Sigma^\infty X$ and a spectrum $E$ to its infinite loop space $\Omega^\infty E = E_0$.

The counit $\Sigma^\infty \Omega^\infty E \to E$ is an equivalence when $E$ is a suspension spectrum. The unit $X \to \Omega^\infty \Sigma^\infty X$ corresponds to the stable Hurewicz map.

For a commutative ring $R$, consider the $\infty$-category $\operatorname{Mod}_R^{\geq 0}$ of connective $R$-module spectra (or equivalently, simplicial $R$-modules). Its stabilization is:

$\operatorname{Sp}(\operatorname{Mod}_R^{\geq 0}) \simeq D(R)$

the derived $\infty$-category of $R$. The adjunction sends a connective module to its inclusion in $D(R)$, and the right adjoint is the connective cover $\tau_{\geq 0}$.

For an $\infty$-topos $\mathcal{X}$, the $\infty$-category of abelian group objects $\operatorname{Ab}(\mathcal{X})$ (equivalently, grouplike $E_\infty$-monoids) has stabilization:

$\operatorname{Sp}(\operatorname{Ab}(\mathcal{X})) \simeq \operatorname{Sp}(\mathcal{X})$

the $\infty$-category of spectrum objects in $\mathcal{X}$. When $\mathcal{X} = \mathcal{S}$, this gives $\operatorname{Sp}(\operatorname{Ab}(\mathcal{S})) \simeq \operatorname{Sp}$.

The $\infty$-category of connective spectra $\operatorname{Sp}_{\geq 0}$ is equivalent to the $\infty$-category of grouplike $E_\infty$-spaces (infinite loop spaces). The equivalence is given by:

$\Omega^\infty: \operatorname{Sp}_{\geq 0} \xrightarrow{\simeq} \operatorname{Mon}_{E_\infty}^{\mathrm{gp}}(\mathcal{S})$

This is not a stabilization statement per se, but it shows that the connective part of $\operatorname{Sp}$ has a purely space-level description. The full $\operatorname{Sp}$ is obtained by inverting the suspension.

In motivic homotopy theory, the motivic stable $\infty$-category $\operatorname{SH}(S)$ over a scheme $S$ is the stabilization of the pointed motivic $\infty$-category $\mathcal{H}_\bullet(S)$ with respect to the $\mathbb{P}^1$-suspension:

$\operatorname{SH}(S) = \operatorname{Sp}^{\mathbb{P}^1}(\mathcal{H}_\bullet(S)) = \operatorname{colim}\left(\mathcal{H}_\bullet(S) \xrightarrow{\Sigma^{\mathbb{P}^1}} \mathcal{H}_\bullet(S) \xrightarrow{\Sigma^{\mathbb{P}^1}} \cdots\right)$

The universal property says: for any presentable stable $\infty$-category $\mathcal{D}$, colimit-preserving functors $\operatorname{SH}(S) \to \mathcal{D}$ correspond to $\mathbb{P}^1$-stable functors $\mathcal{H}_\bullet(S) \to \mathcal{D}$.

For a site $(\mathcal{C}, \tau)$, the $\infty$-topos $\operatorname{Shv}(\mathcal{C})$ of sheaves of spaces can be stabilized:

$\operatorname{Sp}(\operatorname{Shv}(\mathcal{C})_*) \simeq \operatorname{Shv}(\mathcal{C}; \operatorname{Sp})$

the $\infty$-category of sheaves of spectra on $\mathcal{C}$. This is a stable presentable $\infty$-category that serves as the natural home for sheaf cohomology.

For the etale site of a scheme $X$: $\operatorname{Sp}(\operatorname{Shv}_{\mathrm{et}}(X)_*) \simeq \operatorname{Shv}_{\mathrm{et}}(X; \operatorname{Sp})$, the $\infty$-category of etale sheaves of spectra.

If $\mathcal{C}$ is already a stable $\infty$-category, then $\operatorname{Sp}(\mathcal{C}) \simeq \mathcal{C}$. The stabilization functor is idempotent:

$\operatorname{Sp}(\operatorname{Sp}(\mathcal{C})) \simeq \operatorname{Sp}(\mathcal{C})$

This is because $\Omega$ is already an equivalence on $\operatorname{Sp}(\mathcal{C})$, so the sequential limit $\lim(\cdots \xrightarrow{\Omega} \operatorname{Sp}(\mathcal{C}) \xrightarrow{\Omega} \operatorname{Sp}(\mathcal{C}))$ stabilizes immediately.

The tangent $\infty$-category $T_X \mathcal{C}$ at an object $X$ of a presentable $\infty$-category $\mathcal{C}$ is defined as:

$T_X \mathcal{C} = \operatorname{Sp}(\mathcal{C}_{/X})$

the stabilization of the slice category. When $\mathcal{C} = \operatorname{CAlg}_k$ (commutative $k$-algebras) and $X = A$:

$T_A \operatorname{CAlg}_k \simeq \operatorname{Mod}_A$

the $\infty$-category of $A$-modules. The cotangent complex $L_{A/k}$ is the image of $A \to A$ under $\Sigma^\infty: (\operatorname{CAlg}_k)_{/A} \to \operatorname{Mod}_A$. This is the foundation of derived deformation theory.

In Goodwillie calculus, the first derivative of a homotopy functor $F: \mathcal{S}_* \to \mathcal{S}_*$ is a functor $D_1 F: \mathcal{S}_* \to \operatorname{Sp}$, obtained by stabilizing $F$. The stabilization of the identity functor gives:

$D_1(\operatorname{id}_{\mathcal{S}_*}) \simeq \Sigma^\infty: \mathcal{S}_* \to \operatorname{Sp}$

More generally, for a functor $F: \mathcal{C} \to \mathcal{D}$ between presentable $\infty$-categories, the derivative at $X \in \mathcal{C}$ is a functor $D_X F: T_X \mathcal{C} \to T_{F(X)} \mathcal{D}$ between tangent (stable) categories. This is the $\infty$-categorical analogue of the differential of a smooth map.

A functor $F: \mathcal{S}_* \to \operatorname{Sp}$ is excisive (or $1$-excisive) if it sends pushout squares to pullback squares. The $\infty$-category of excisive functors $\operatorname{Exc}(\mathcal{S}_*, \operatorname{Sp})$ is equivalent to $\operatorname{Sp}$:

$\operatorname{Exc}(\mathcal{S}_*, \operatorname{Sp}) \simeq \operatorname{Sp}$

This is another manifestation of the universal property: $\operatorname{Sp}$ is the universal target for "linear" (excisive) functors from pointed spaces.

For a ring spectrum $R$ (an $E_1$-ring), the $\infty$-category $\operatorname{Mod}_R$ is a presentable stable $\infty$-category. By the universal property of $\operatorname{Sp}$, giving a colimit-preserving functor $\operatorname{Sp} \to \operatorname{Mod}_R$ is the same as giving an object of $\operatorname{Mod}_R$. The functor corresponding to $R \in \operatorname{Mod}_R$ is $E \mapsto R \wedge E$ (the smash product with $R$), which is the "free $R$-module" functor:

$\operatorname{Free}_R = R \wedge (-): \operatorname{Sp} \to \operatorname{Mod}_R$

This left adjoint corresponds to the forgetful functor $\operatorname{Mod}_R \to \operatorname{Sp}$ (which preserves limits).

For a bifunctor $\otimes: \mathcal{C} \times \mathcal{D} \to \mathcal{E}$ between pointed $\infty$-categories, the bilinear stabilization produces a bifunctor $\operatorname{Sp}(\mathcal{C}) \times \operatorname{Sp}(\mathcal{D}) \to \operatorname{Sp}(\mathcal{E})$. Applied to the smash product of pointed spaces:

$\wedge: \mathcal{S}_* \times \mathcal{S}_* \to \mathcal{S}_*$

stabilization in both variables gives the smash product of spectra $\wedge: \operatorname{Sp} \times \operatorname{Sp} \to \operatorname{Sp}$, recovering the symmetric monoidal structure on $\operatorname{Sp}$.