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Axiom of Choice

Table of Contents

1. For Category of Sets

The axiom of choice may be phrased as:

Every surjection in the category Set of sets splits.

2. In other categories

There are different ways to formulate the axiom of choice in other categories.

We say the category \(C\) satisfies the External Axiom of Choice if every epimorphism in \(C\) splits.

When the category we're working in has an internal logic (or it's a topos), we may "internalize" the axiom of choice asserting "every epimorphism has a section" is true in the internal logic. Another way to phrase this is exponentiation by an object preserves epimorphisms (every object is an "internally projective object").

Last Updated 2021-06-01 Tue 10:00.