The Stacks project

Lemma 19.11.1. Let $\mathcal{A}$ be an abelian category with a generator $U$ and $X$ and object of $\mathcal{A}$. If $\kappa $ is the cardinality of $\mathop{\mathrm{Mor}}\nolimits (U, X)$ then

  1. There does not exist a strictly increasing (or strictly decreasing) chain of subobjects of $X$ indexed by a cardinal bigger than $\kappa $.

  2. If $\alpha $ is an ordinal of cofinality $> \kappa $ then any increasing (or decreasing) sequence of subobjects of $X$ indexed by $\alpha $ is eventually constant.

  3. The cardinality of the set of subobjects of $X$ is $\leq 2^\kappa $.

Proof. For (1) assume $\kappa ' > \kappa $ is a cardinal and assume $X_ i$, $i \in \kappa '$ is strictly increasing. Then take for each $i$ a $\phi _ i \in \mathop{\mathrm{Mor}}\nolimits (U, X)$ such that $\phi _ i$ factors through $X_{i + 1}$ but not through $X_ i$. Then the morphisms $\phi _ i$ are distinct, which contradicts the definition of $\kappa $.

Part (2) follows from the definition of cofinality and (1).

Proof of (3). For any subobject $Y \subset X$ define $S_ Y \in \mathcal{P}(\mathop{\mathrm{Mor}}\nolimits (U, X))$ (power set) as $S_ Y = \{ \phi \in \mathop{\mathrm{Mor}}\nolimits (U,X) : \phi )\text{ factors through }Y\} $. Then $Y = Y'$ if and only if $S_ Y = S_{Y'}$. Hence the cardinality of the set of subobjects is at most the cardinality of this power set. $\square$

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