nLab compact object in an (infinity,1)-category

Idea

The notion of compact object in an $(\infty,1)$-category is the analogue in (∞,1)-category theory of the notion of compact object in category theory.

Definition

Definition

Let $\kappa$ be a regular cardinal and $C$ an (∞,1)-category with $\kappa$-filtered (∞,1)-colimits.

Then an object $c \in C$ is called $\kappa$-compact if the (∞,1)-categorical hom space functor

$$C(c,-) : C \to \infty Grpd$$

preserves $\kappa$-filtered (∞,1)-colimits.

For $\omega$-compact we just say compact.

This appears as (HTT, def. 5.3.4.5).

Properties

General

Let $\kappa$ be a regular cardinal.

Proposition

Let $C$ be an (∞,1)-category which admits small $\kappa$-filtered (∞,1)-colimits. Then the full sub-(∞,1)-category of $\kappa$-compact objects in closed under $\kappa$-small (∞,1)-colimits in $C$.

This is (HTT, cor. 5.3.4.15).

Presentation in model categories

If the (∞,1)-category $\mathcal{C}$ is a locally presentable (∞,1)-category, then it is the simplicial localization of a combinatorial model category $C$, and one may ask how the 1-categorical notion of compact object in $C$ relates to the $(\infty,1)$-categorical notion of compact in $\mathcal{C}$.

Since compactness is defined in terms of colimits, the question is closely related to the question which 1-categorical $\kappa$-filtered colimits in $C$ are already homotopy colimits (without having to derive them first).

General statements seem not to be in the literature yet, but see this MO discussion. For discussion of compactness in a model structure on simplicial sheaves, see for instance (Powell, section 4).

Examples

• In $C =$ (∞,1)Cat, for uncountable $\kappa$, the $\kappa$-compact objects are precisely the $\kappa$-essentially small (∞,1)-categories. (See there for more details.)

• In $C =$ ∞Grpd, for uncountable $\kappa$, the $\kappa$-compact objects are precisely the $\kappa$-essentially small ∞-groupoids. When $\kappa = \omega$, the compact objects in ∞Grpd are the retracts of the $\omega$-small ∞Grpds, i.e., the retracts of the finite homotopy types (finite CW-complexes). Not every such retract is equivalent to a $\omega$-small ∞-groupoid; the vanishing of Wall's finiteness obstruction is a necessary and sufficient condition for such an equivalence to exist.

Related concepts

• compact object

• compact topos

• compact object in an $(\infty,1)$-category

• relatively k-compact morphism in an (infinity,1)-category

• compactly generated (∞,1)-category

References

The general definition appears as definition 5.3.4.5 in

• Jacob Lurie, Higher Topos Theory

Compactness in presenting model categories of simplicial sheaves is discussed for instance in

section 4 of

• Geoffrey Powell, The adjunction between $\mathcal{H}(k)$ and $DM^{eff}_-(k)$ (2001) (pdf)