Lemma 19.11.4 (079E)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 19: Injectives
Section 19.11: Injectives in Grothendieck categories
Lemma 19.11.4 (cite)

Lemma 19.11.4. Let $\mathcal{A}$ be a Grothendieck abelian category with generator $U$.

If $|M| \leq \kappa $, then $M$ is the quotient of a direct sum of at most $\kappa $ copies of $U$.
For every cardinal $\kappa $ there exists a set of isomorphism classes of objects $M$ with $|M| \leq \kappa $.

Proof. For (1) choose for every proper subobject $M' \subset M$ a morphism $\varphi _{M'} : U \to M$ whose image is not contained in $M'$. Then $\bigoplus _{M' \subset M} \varphi _{M'} : \bigoplus _{M' \subset M} U \to M$ is surjective. It is clear that (1) implies (2). $\square$

Comments (1)

Comment #9496 by Elías Guisado on July 08, 2024 at 09:55

I was a little confused by the way (2) is phrased. At the beginning I understood that what was being asserted is that for each cardinal $\kappa$ there is an object $M$ in $\mathcal{A}$ with $|M|\leq\kappa$ . Maybe it's me alone, but I would phrase it as "the collection of isomorphisms classes of objects $M$ with $|M|\leq\kappa$ is a set" or something like that.

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