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edited Apr 13, 2017 at 12:58

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After reading your other question other question, I think I understand what you're looking for.

In ZF, one of the key purposes of the Union Axiom is to prove that the universe of sets (viewed as a topos) is closed under internal coproducts, i.e., that one can always form the disjoint union of a set-indexed family of sets.

Any elementary topos $\mathcal{E}$ is closed under internal coproducts and internal products: for any morphism $f:A \to B$, the functor $f^*:\mathcal{E}/B \to \mathcal{E}/A$ has both a left adjoint $\sum_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal coproduct) and a right adjoint $\prod_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal product). Note that internal coproducts are conceptually stronger than internal unions, though the existence proof for either in elementary toposes is essentially the same.

One can also formulate this in terms of indexed categories, where the topos $\mathcal{E}$ is used to index itself. This is closer to the usual notion of (set-indexed) coproduct, but working with slice categories is equivalent and technically simpler.

After reading your other question, I think I understand what you're looking for.

In ZF, one of the key purposes of the Union Axiom is to prove that the universe of sets (viewed as a topos) is closed under internal coproducts, i.e., that one can always form the disjoint union of a set-indexed family of sets.

Any elementary topos $\mathcal{E}$ is closed under internal coproducts and internal products: for any morphism $f:A \to B$, the functor $f^*:\mathcal{E}/B \to \mathcal{E}/A$ has both a left adjoint $\sum_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal coproduct) and a right adjoint $\prod_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal product). Note that internal coproducts are conceptually stronger than internal unions, though the existence proof for either in elementary toposes is essentially the same.

One can also formulate this in terms of indexed categories, where the topos $\mathcal{E}$ is used to index itself. This is closer to the usual notion of (set-indexed) coproduct, but working with slice categories is equivalent and technically simpler.

After reading your other question, I think I understand what you're looking for.

In ZF, one of the key purposes of the Union Axiom is to prove that the universe of sets (viewed as a topos) is closed under internal coproducts, i.e., that one can always form the disjoint union of a set-indexed family of sets.

Any elementary topos $\mathcal{E}$ is closed under internal coproducts and internal products: for any morphism $f:A \to B$, the functor $f^*:\mathcal{E}/B \to \mathcal{E}/A$ has both a left adjoint $\sum_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal coproduct) and a right adjoint $\prod_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal product). Note that internal coproducts are conceptually stronger than internal unions, though the existence proof for either in elementary toposes is essentially the same.

One can also formulate this in terms of indexed categories, where the topos $\mathcal{E}$ is used to index itself. This is closer to the usual notion of (set-indexed) coproduct, but working with slice categories is equivalent and technically simpler.

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created Nov 21, 2011 at 19:34

François G. Dorais

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After reading your other question, I think I understand what you're looking for.

In ZF, one of the key purposes of the Union Axiom is to prove that the universe of sets (viewed as a topos) is closed under internal coproducts, i.e., that one can always form the disjoint union of a set-indexed family of sets.

Any elementary topos $\mathcal{E}$ is closed under internal coproducts and internal products: for any morphism $f:A \to B$, the functor $f^*:\mathcal{E}/B \to \mathcal{E}/A$ has both a left adjoint $\sum_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal coproduct) and a right adjoint $\prod_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal product). Note that internal coproducts are conceptually stronger than internal unions, though the existence proof for either in elementary toposes is essentially the same.

One can also formulate this in terms of indexed categories, where the topos $\mathcal{E}$ is used to index itself. This is closer to the usual notion of (set-indexed) coproduct, but working with slice categories is equivalent and technically simpler.