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

Contents

### Context

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

(∞,1)-category theory

## Models

#### Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

# Contents

## 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.

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 $\mathcal{H}(k)$ and $DM^{eff}_-(k)$ (2001) (pdf)