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Shawn Henry
Shawn Henry
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Let $A$ and $B$ be objects in a topos $\mathcal{E}$ with natural numbers object $\mathbb{N}$. If there existsis a monomorphism $m: \mathbb{N}\rightarrow A\times B$, is it necessarily the case that there is either a monomorphism from $\mathbb{N}$ to $A$ or a monomorphism from $\mathbb{N}$ to $B$?

This is clearly true when $\mathcal{E}$ is Boolean, although the proof I've seen is more involved than one might expect. It freely uses the Boolean assumption in several places.

Let $A$ and $B$ be objects in a topos $\mathcal{E}$ with natural numbers object $\mathbb{N}$. If there exists a monomorphism $m: \mathbb{N}\rightarrow A\times B$, is it necessarily the case that there is either a monomorphism from $\mathbb{N}$ to $A$ or a monomorphism from $\mathbb{N}$ to $B$?

This is clearly true when $\mathcal{E}$ is Boolean, although the proof I've seen is more involved than one might expect. It freely uses the Boolean assumption in several places.

Let $A$ and $B$ be objects in a topos $\mathcal{E}$ with natural numbers object $\mathbb{N}$. If there is a monomorphism $m: \mathbb{N}\rightarrow A\times B$, is it necessarily the case that there is either a monomorphism from $\mathbb{N}$ to $A$ or a monomorphism from $\mathbb{N}$ to $B$?

This is clearly true when $\mathcal{E}$ is Boolean, although the proof I've seen is more involved than one might expect. It freely uses the Boolean assumption in several places.

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Shawn Henry
Shawn Henry
  • 93
  • 5

Monomorphisms from natural numbers objects into products.

Let $A$ and $B$ be objects in a topos $\mathcal{E}$ with natural numbers object $\mathbb{N}$. If there exists a monomorphism $m: \mathbb{N}\rightarrow A\times B$, is it necessarily the case that there is either a monomorphism from $\mathbb{N}$ to $A$ or a monomorphism from $\mathbb{N}$ to $B$?

This is clearly true when $\mathcal{E}$ is Boolean, although the proof I've seen is more involved than one might expect. It freely uses the Boolean assumption in several places.