### Logical and Sublogical Functors

#### Posted by John Baez

I’m trying to understand sublogical functors, so I’m looking for examples of sublogical functors between presheaf categories, preferably direct images (if that’s possible).

Just to make this post a bit more interesting, I’ll explain that sentence! This will give beginners a chance to learn something, and experts a chance to catch mistakes in what I’m saying, so that beginners can learn something *true*.

Let’s write $\widehat{C}$ for the category of presheaves on $C$:

$\widehat{C} = Set^{C^op}$

This is an example of an elementary topos. Precomposition with a functor $f \colon C \to D$ gives a functor

$f^\ast \colon \widehat{D} \to \widehat{C}$

This is an example of the ‘inverse image’ part of a ‘geometric morphism’ because it has a left adjoint that preserves finite limits. But this particular example of a geometric morphism is better than average since it also has a right adjoint.

We say $f^\ast$ is a ‘logical functor’ if it preserves finite limits (which it does, since it’s a right adjoint) and also the natural map

$\phi \colon f^\ast P A \to P f^\ast A$

is an isomorphism for all $A \in \widehat{D}$, where $P A$ is the power object of $A$, namely $\Omega^A$ where $\Omega$ is the subobject classifier.

Johnstone says $f^\ast$ is a **sublogical functor** if it preserves finite limits and $\phi$ is a *monomorphism*.

So, I’m really looking for examples where $f^\ast$ is sublogical… and not logical.

## Re: Logical and Sublogical Functors

Hello, I’m not sure whether there’s a typo and you meant “inverse image” rather than “direct image”. Anyway, the Elephant has quite a lot: Lemma C3.1.2 characterizes sublogical inverse images between presheaf categories (induced by precomposition), and sublogical direct images are characterized in Proposition C3.1.8, and logical inverse images are in C3.5.