---
tag: 1F833GKJ9V
kind: definition
name: Grothendieck category
msc: ["18E10"]
defines: A7W468BQSM
uses: [Y9CZ3MTNF1, BJS1CGTKQ5, 975PMY4FG8, S1G61TDW0K, CK08JW32WB, 5QRFTBF4PG, 0NZ8JA25WK, 082GHNPJ53]
sources:
  - url: https://en.wikipedia.org/wiki/Grothendieck_category
---

## Definition

A *Grothendieck category* is an [[abelian category|Y9CZ3MTNF1]] $\mathcal{A}$ such that:

1. $\mathcal{A}$ has all small [[colimits|BJS1CGTKQ5]];
2. filtered colimits are exact: for every small [[filtered category|975PMY4FG8]] $I$, the functor $\operatorname{colim} \colon \mathcal{A}^I \to \mathcal{A}$ is [[exact|S1G61TDW0K]];
3. $\mathcal{A}$ has a [[generator|CK08JW32WB]]: an object $G$ such that for any two distinct morphisms $f \ne g \colon X \to Y$ there is a morphism $h \colon G \to X$ with $f \circ h \ne g \circ h$.

## Remarks

Clauses 1 and 2 together form Grothendieck's axiom AB5.
Every Grothendieck category has enough [[injective objects|5QRFTBF4PG]], and it also has all small limits.

Example: for a ring $R$, the category of left [[$R$-modules|0NZ8JA25WK]] is a Grothendieck category.
It has all colimits, and a filtered colimit of short exact sequences of modules is exact.
The module $R$ is a generator: if $f \ne g \colon X \to Y$, pick $x$ with $f(x) \ne g(x)$ and let $h \colon R \to X$ be $r \mapsto r x$.

Non-example: the opposite of the category of [[abelian groups|082GHNPJ53]] is abelian, has all small colimits, and has the generator $\Q/\Z$, but it fails clause 2.
Filtered colimits in it are cofiltered limits of abelian groups, and these are not exact.
For a prime $p$, take the inverse system of short exact sequences $0 \to p^n \Z \to \Z \to \Z / p^n \Z \to 0$ for $n \ge 1$.
Its limit is $0 \to 0 \to \Z \to \Z_p \to 0$, where $\Z_p$ is the group of $p$-adic integers, and the map $\Z \to \Z_p$ is not surjective.