---
date: 2022-01-15 21:49
modification date: Sunday 13th February 2022 16:35:01
title: compact object of a category
aliases: [compact object of a category, compact, compactly generated, generators of a category, "generated", "generator", "generating object", "compact generators", "compactly generated", "compact generator"]

---


---

- Tags 
	- #higher-algebra/category-theory #lie-theory 
- Refs:
	- #todo/add-references
- Links:
	- [Balmer spectrum](Unsorted/Balmer%20spectrum.md)
	- [thick subcategory](thick%20subcategory.md)
	- [modern category theory](Unsorted/modern%20category%20theory.md)
	- [conservative functor](Unsorted/conservative%20functor.md)
	- [Grothendieck category](Grothendieck%20category.md)

---

# compact object of a category

$\cat{C}(X, \wait)$ preserves filtered colimits, so $\colim C(X, Y_i) \iso C(X, \colim Y_i)$.

Idea:
![](attachments/Pasted%20image%2020220317230918.png)

Motivations:

- Compact objects of the [derived category](derived%20category.md) are typically easier to construct as we may extend them over [open immersions](Unsorted/open%20immersion%20of%20schemes.md) using **Thomason’s localization theorem**
- $\derivedcat{\Coh(X)}_\qc$ (the unbounded derived category of $\OO_X\dash$modules with [quasicoherent cohomology](quasicoherent%20cohomology)) is compactly generated when $X\in\Sch$ is [quasicompact](quasicompact) and [separated](Unsorted/separated.md).
- Coherence:
	![](attachments/Pasted%20image%2020220323190736.png)
![](attachments/Pasted%20image%2020220323190747.png)


# Definitions

- For [infinity categories](Unsorted/infinity%20categories.md)
	- An object $X \in \cat{C}$ is **compact** iff $\operatorname{Map}_{\cat{C}}(X,\wait)\in \Cat(\cat C, \Set)$ preserves filtered homotopy colimits.
	- $\cat C$ is **compactly generated** iff there is an equivalence $\cat C \homotopic \cat \Ind(\cat D)$ for some $\cat D \in \inftycat$ small.

- For [locally small](locally%20small) categories:
	- An object $X \in \cat{C}$ is **compact** iff $\operatorname{Map}_{\cat{C}}(X,\wait)\in \Cat(\cat C, \Set)$ preserves filtered colimits.
	- $\cat C$ is **compactly generated** iff the following non-degeneracy condition holds: setting $F_X(\wait) \da \Map_{\cat C}(X, \wait)$, if $F_X(M) =0$ for all compact $X$ then $M=0$.

- For [triangulated categories](Unsorted/triangulated%20categories.md):
	- An object $X\in \cat C$ is **compact** iff homs commute with arbitrary direct sums, i.e.
	  $$\bigoplus_{i\in I} \cat{C}\qty{X, Y_i} \iso \cat{C}\qty{X, \coprod_{i\in I} Y_i}$$
	- An object $X$ is a **generator** if every object $Y\in \cat{C}$ is a cokernel of the form 
$$
\coker \left( X\sumpower{n} \to X\sumpower{m} \right) \iso Y, \qquad \iff \exists\quad X\sumpower{n} \to X\sumpower{m} \to Y \
$$
	- $\cat C$ is **compactly generated** iff there is a set of compact objects $S$ such that for every $S \in \cat{S}$, $\cat{C}\complex{(S, X)} = 0 \implies X=0$.


# Examples

- **Warning**: a compactly generated triangulated category need not have a compact generator!
- Note that finite colimits of compact objects are again compact
- In $\Set$, compact objects are finite sets
- In $\Top$, compact objects are finite sets with the discrete topology.
- in $\rmod$, compact objects are finitely presented modules
- $\Spaces\in \inftycat$ is compactly generated
- Categories of module spectra over an $\EE_1\dash$rings have a compact generator, namely the free module of rank 1.
- Proving an infinity category is compactly generated: exhibit it as a siftted-colimit completion.
- A well-known fact: for $X\in \Sch\slice S$ quasicompact and separated, $\derivedcat{\mods{\OO_X}}_\qc$ the derived category of quasicoherent $\OO_X\dash$modules has a compact generator.
- Full subcategories of compact objects form a [thick subcategory](Unsorted/thick%20subcategory.md).

# Compact Generation

## For [infty-categories](Unsorted/infinity%20categories.md)


![](attachments/Pasted%20image%2020220318104706.png)

## For [triangulated categories](Unsorted/triangulated%20categories.md)

![](attachments/Pasted%20image%2020220323191712.png)

# Misc
![](attachments/Pasted%20image%2020220325164212.png)
![](attachments/Pasted%20image%2020220325164459.png)