The $\infty$-category $\mathcal{S}$ of spaces is presentable. The compact objects are the finite CW complexes (spaces with finitely many cells). Every space is a filtered colimit of finite CW complexes.

$\mathcal{S}$ is the free presentable $\infty$-category on one generator (the point $*$): $\mathcal{S} \simeq \operatorname{Fun}(\{*\}^{\mathrm{op}}, \mathcal{S})$.

For a ring $R$, the derived $\infty$-category $D(R)$ (the $\infty$-categorical enhancement of the derived category) is presentable. The compact objects are the perfect complexes: bounded chain complexes of finitely generated projective $R$-modules.

$D(R)$ is generated under colimits by the single object $R[0]$ (the free module of rank $1$ in degree $0$). In fact, $D(R)$ is the free stable presentable $\infty$-category generated by $R$: $D(R) \simeq \operatorname{Mod}_R(\operatorname{Sp})$.

For any small $\infty$-category $\mathcal{A}$, the presheaf $\infty$-category $\operatorname{Fun}(\mathcal{A}^{\mathrm{op}}, \mathcal{S})$ is presentable. The compact objects include the representable presheaves $\operatorname{Map}_{\mathcal{A}}(-, a)$ for $a \in \mathcal{A}$.

This is the $\infty$-categorical analogue of the category of presheaves of sets, which is always a Grothendieck topos.

For a Grothendieck site $(\mathcal{C}, \tau)$, the $\infty$-category $\operatorname{Shv}_\tau(\mathcal{C})$ of sheaves valued in $\mathcal{S}$ is presentable (it is an accessible left exact localization of the presheaf category). This is an $\infty$-topos.

For a scheme (or algebraic stack) $X$, the $\infty$-category $\operatorname{QCoh}(X)$ of quasi-coherent sheaves is presentable. When $X = \operatorname{Spec}(R)$ is affine, $\operatorname{QCoh}(X) \simeq D(R)$.

The compact objects in $\operatorname{QCoh}(X)$ are the perfect complexes (when $X$ is quasi-compact and quasi-separated).

The $\infty$-category $\operatorname{Sp}$ of spectra is presentable. The compact objects are the finite spectra (e.g., the sphere spectrum $\mathbb{S}$, suspension spectra of finite CW complexes).

$\operatorname{Sp}$ is the stabilization of $\mathcal{S}_*$: it is obtained by inverting the suspension functor.

Every presentable $\infty$-category has all small limits and colimits. This is because accessible categories with colimits automatically have limits (by the special adjoint functor theorem).

Explicitly: colimits exist by definition; limits are constructed from products and equalizers, which exist in any accessible $\infty$-category with colimits.

A functor between presentable $\infty$-categories $F: \mathcal{C} \to \mathcal{D}$ has:

• A right adjoint if and only if $F$ preserves small colimits.
• A left adjoint if and only if $F$ preserves small limits and is accessible.

This is the $\infty$-categorical adjoint functor theorem (Lurie, HTT 5.5.2.9). It is one of the most powerful tools in higher category theory, reducing the existence of adjoint functors to easily checkable conditions.

A presentable $\infty$-category $\mathcal{C}$ always has a set of generators: a small set $S$ of objects such that if $\operatorname{Map}(s, x) \simeq *$ for all $s \in S$, then $x$ is a zero object. Every object of $\mathcal{C}$ can be written as a colimit of objects from $S$.

For $\mathcal{S}$: the single generator $* = \Delta[0]$. For $D(R)$: the single generator $R[0]$. For $\operatorname{Sp}$: the single generator $\mathbb{S}$ (the sphere spectrum).

Any accessible localization of a presentable $\infty$-category is presentable. Concretely, if $\mathcal{C}$ is presentable and $S$ is a set of morphisms, the localization $S^{-1}\mathcal{C}$ (the full subcategory of $S$-local objects) is presentable.

Example: $n$-truncated spaces $\mathcal{S}_{\leq n}$ is a localization of $\mathcal{S}$ (localize at the maps $\partial\Delta[k] \hookrightarrow \Delta[k]$ for $k > n$).

• $\mathcal{S} \otimes \mathcal{C} \simeq \mathcal{C}$ for any presentable $\mathcal{C}$ ($\mathcal{S}$ is the unit).
• $D(R) \otimes D(S) \simeq D(R \otimes S)$ for commutative rings $R, S$.
• $\operatorname{Sp} \otimes \mathcal{C} \simeq \operatorname{Sp}(\mathcal{C})$ (stabilization) for a pointed presentable $\mathcal{C}$.