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

Context

(,1)(\infty,1)-Category theory

Compact objects

Contents

Idea

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

Definition

Definition

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

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

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

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

For ω\omega-compact we just say compact.

This appears as (HTT, def. 5.3.4.5).

Properties

General

Let κ\kappa be a regular cardinal.

Proposition

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

This is (HTT, cor. 5.3.4.15).

Presentation in model categories

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

Since compactness is defined in terms of colimits, the question is closely related to the question which 1-categorical κ\kappa-filtered colimits in CC 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

References

The general definition appears as definition 5.3.4.5 in

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

section 4 of

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

Revised on February 15, 2014 04:58:49 by Urs Schreiber (89.204.154.124)