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Let $I$ be a set and $\mathcal{U}$ an ultrafilter on $I$. Let $(X_i)_{i \in I}$ be an $I$-indexed family of sets. The ultraproduct of the family $(X_i)$ with respect to $\mathcal{U}$ is, everyone agrees, another set. But which set is it? There are two different definitions, and they sometimes give different results.

For the sake of discussion, I'll call them "Type 1" and "Type 2" ultraproducts.

Type 1 $\$ The type 1 ultraproduct of $(X_i)_{i \in I}$ with respect to $\mathcal{U}$ is $$\Bigl( \prod_{i \in I} X_i \Bigr) \Bigl/ \sim$$ where $$(x_i)_{i \in I} \sim (x'_i)_{i \in I} \iff \{ i \in I: x_i = x'_i \} \in \mathcal{U}.$$

Type 2 $\$ View the poset $(\mathcal{U}, \subseteq)$ as a category. The type 2 ultraproduct of $(X_i)_{i \in I}$ with respect to $\mathcal{U}$ is the colimit of the functor $(\mathcal{U}, \subseteq)^{\text{op}} \to \mathbf{Set}$ defined on objects by $$J \mapsto \prod_{j \in J} X_j$$ and on maps by projection. Explicitly, then, it's the Type 2 ultraproduct is $$\Bigl( \coprod_{J \in \mathcal{U}} \prod_{j \in J} X_j \Bigr) \Bigl/ \approx$$ where $$(x_j)_{j \in J} \approx (x'_k)_{k \in K} \iff \{ i \in J \cap K: x_i = x'_i \} \in \mathcal{U}.$$

The difference $\$ The two types of ultraproduct are the same if either none of the sets $X_i$ are empty or almost all of them are empty. But in the remaining case, where at least one $X_i$ is empty but the set of such $i$ is not large enough to belong to $\mathcal{U}$, they're different: the Type 1 ultraproduct is empty but the Type 2 ultraproduct is not.

The question $\$ I've read in a couple of texts (both coming from the point of view of categorical logic) that the Type 2 ultraproduct is really the right one. But why? On what criteria is Type 2 judged to be better than Type 1?

A vague guess at an answer $\$ I think I can guess very roughly what's going on. There's been a tradition in logic — maybe dying out now? — of taking all structures to be nonempty by definition. But when you move to the more general setting of categorical logic, that's no longer a satisfactory approach. Although in the category of sets, there's just a single object with no elements, in many other categories, there are lots of interesting objects with no (global) elements: e.g. there are lots of interesting sheaves with no global sections.

So categorical logic sometimes involves a recasting of classical, set-based logic, in order to handle empty sets/types satisfactorily. I imagine that something of the sort is going on here. (I only defined ultraproducts of sets, but you could of course define ultraproducts of objects of any other sufficiently complete category.) But still, I don't see clearly why Type 2 is the right choice.

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# There are two slightly different notions of ultraproduct. Why is one said to be better than the other?

Let $I$ be a set and $\mathcal{U}$ an ultrafilter on $I$. Let $(X_i)_{i \in I}$ be an $I$-indexed family of sets. The ultraproduct of the family $(X_i)$ with respect to $\mathcal{U}$ is, everyone agrees, another set. But which set is it? There are two different definitions, and they sometimes give different results.

For the sake of discussion, I'll call them "Type 1" and "Type 2" ultraproducts.

Type 1 $\$ The type 1 ultraproduct of $(X_i)_{i \in I}$ with respect to $\mathcal{U}$ is $$\Bigl( \prod_{i \in I} X_i \Bigr) \Bigl/ \sim$$ where $$(x_i)_{i \in I} \sim (x'_i)_{i \in I} \iff \{ i \in I: x_i = x'_i \} \in \mathcal{U}.$$

Type 2 $\$ View the poset $(\mathcal{U}, \subseteq)$ as a category. The type 2 ultraproduct of $(X_i)_{i \in I}$ with respect to $\mathcal{U}$ is the colimit of the functor $(\mathcal{U}, \subseteq)^{\text{op}} \to \mathbf{Set}$ defined on objects by $$J \mapsto \prod_{j \in J} X_j$$ and on maps by projection. Explicitly, then, it's $$\Bigl( \coprod_{J \in \mathcal{U}} \prod_{j \in J} X_j \Bigr) \Bigl/ \approx$$ where $$(x_j)_{j \in J} \approx (x'_k)_{k \in K} \iff \{ i \in J \cap K: x_i = x'_i \} \in \mathcal{U}.$$

The difference $\$ The two types of ultraproduct are the same if either none of the sets $X_i$ are empty or almost all of them are empty. But in the remaining case, where at least one $X_i$ is empty but the set of such $i$ is not large enough to belong to $\mathcal{U}$, they're different: the Type 1 ultraproduct is empty but the Type 2 ultraproduct is not.

The question $\$ I've read in a couple of texts (both coming from the point of view of categorical logic) that the Type 2 ultraproduct is really the right one. But why? On what criteria is Type 2 judged to be better than Type 1?

A vague guess at an answer $\$ I think I can guess very roughly what's going on. There's been a tradition in logic — maybe dying out now? — of taking all structures to be nonempty by definition. But when you move to the more general setting of categorical logic, that's no longer a satisfactory approach. Although in the category of sets, there's just a single object with no elements, in many other categories, there are lots of interesting objects with no (global) elements: e.g. there are lots of interesting sheaves with no global sections.

So categorical logic sometimes involves a recasting of classical, set-based logic, in order to handle empty sets/types satisfactorily. I imagine that something of the sort is going on here. (I only defined ultraproducts of sets, but you could of course define ultraproducts of objects of any other sufficiently complete category.) But still, I don't see clearly why Type 2 is the right choice.