# nLab compactly generated (∞,1)-category

Contents

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

#### Compact objects

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

# Contents

## Definition

For $\kappa$ a regular cardinal, an (∞,1)-category is called $\kappa$-compactly generated if it is $\kappa$-accessible and locally presentable.

Terminology The terms “$\kappa$-compactly generated (∞,1)-category” and “locally $\kappa$-presentable (∞,1)-category” have the same meaning. There are differences in usage, though.

• If we “drop the $\kappa$”, then a locally presentable (∞,1)-category is a an (∞,1)-category which is locally $\kappa$-presentable for some $\kappa$, but a compactly generated (∞,1)-category is an (∞,1)-category which is locally finitely presentable, i.e. locally $\kappa$-presentable for $\kappa = \aleph_0$.

• If we “leave the $\kappa$ in”, the terms “$\kappa$-compactly generated (∞,1)-category” and “locally $\kappa$-presentable (∞,1)-category” have the same meaning. Some authors choose one term over the other. For example, in Higher Topos Theory, “$\kappa$-compactly generated (∞,1)-category” is preferred. Albeit, Lurie uses $\mathrm{Pr}_\kappa$ to denote the (∞,1)-category of $\kappa$-compactly generated (∞,1)-categories.

## Properties

### Recognition for stable $(\infty,1)$-categories

Compact generation for stable (∞,1)-categories is detected already on their triangulated homotopy category as discussed at compactly generated triangulated category

## References

Last revised on March 18, 2024 at 22:13:02. See the history of this page for a list of all contributions to it.