A logical morphism or logical functor is a homomorphism between elementary toposes that preserves the structure of a topos as a context for logic: a functor which preserves all the elementary topos structure, including in particular power objects, but not necessarily any infinitary structure (such as present additionally in a sheaf topos).
If instead a topos is regarded as a context for geometry or specifically geometric logic, then the notion of homomorphism preserving this is that of a geometric morphism.
Since all the elementary topos structure follows from having finite limits and power objects, it suffices to define a logical functor to preserve these, up to isomorphism. It then follows that it is also a locally cartesian closed functor, a Heyting functor, etc.
Let $\mathcal{E}$ be an elementary topos. Write $\Omega \in \mathcal{E}$ for the subobject classifier. For each object $A \in \mathcal{E}$ write
for the exponential object. Write
for the subobject classified by the evaluation map $ev : P A \times A \to \Omega$.
This exhibits $P A$ as a power object for $A$.
A functor $F : \mathcal{E} \to \mathcal{F}$ between elementary toposes is called a logical morphism if
$F$ preserves finite limits;
for every object $A \in \mathcal{E}$
the canonical morphism
is an isomorphism; this is the name of the relation
(using that cartesian functors preserve both products as well as monomorphism)
equivalently: $F(P A)$ equipped with the relation $F(\in_A)$ is a power object for $F(A)$ in $\mathcal{F}$.
The notion of logical functors between toposes is in contrast to geometric morphisms between toposes: the former preserve the structure of an elementary topos, the latter those of a sheaf topos.
But both can be combined:
A geometric morphism whose inverse image is a logical functor is called an atomic geometric morphism.
The other case, that the direct image of a geometric morphism is a logical functor is not of interest. See cor. 4 below.
A logical functor has a left adjoint precisely if it has a right adjoint.
This appears as (Johnstone, cor. A2.2.10).
For $F : \mathcal{E} \to \mathcal{F}$ a logical functor, we have by definition a diagram
in Cat. This satisfies the assumptions of the adjoint lifting theorem and hence $F$ has a right adjoint precisely if $F^{op}$ does. But a right adjoint of $F^{op}$ is a left adjoint of $F$, and vice versa.
If a logical functor $F : \mathcal{E} \to \mathcal{F}$ has a left adjoint $L$, then $L$ preserves monomorphisms, and indeed induces a bijection between subobjects of $A\in \mathcal{F}$ and subobjects of $L A\in \mathcal{E}$.
Since $F(\Omega_{\mathcal{E}}) = \Omega_{\mathcal{F}}$, we have a bijection
It remains to check that this bijection is implemented by the action of $L$, which can be done with partial map classifiers; see (Johnstone, Lemma A2.4.8).
For a logical functor $F : \mathcal{E} \to \mathcal{F}$ having a left adjoint $L$, the following are equivalent:
This appears as (Johnstone, Prop. A2.3.8).
The left adjoint of pullback has all the other properties. Any pullback-preserving functor preserves equalizers. An equalizer-preserving functor is faithful as soon as it reflects invertibility of monomorphisms, which follows from Proposition 2 above. A faithful functor reflects monos and epis, hence is conservative once its domain is balanced.
Finally, if $L$ is conservative then so is $L'$, and its right adjoint $F'$ (given by $F$ followed by pullback along $\eta : 1 \to F L 1$) is also logical, hence cartesian closed. Thus, $L'\dashv F'$ is a Hopf adjunction for the cartesian monoidal structures, and $L'$ preserves the terminal object. Now the counit of $L'\dashv F'$ can be factored as
so it is an isomorphism. By the triangle identity, $L'(\eta)$ is an isomorphism, but $L'$ is conservative, hence the unit $\eta$ is also an isomorphism, and $L'\dashv F'$ is an equivalence.
This appears as (Johnstone, cor. A2.2.10).
If a logical functor is right adjoint to a left exact functor, then it is an equivalence of categories.
This appears as (Johnstone, scholium 2.3.9).
In particular, a logical functor preserves the truth of all sentences in the internal logic. If it is moreover conservative, then it also reflects the truth of such sentences. For example, the transfer principle of nonstandard analysis can be stated as the fact that a certain functor is logical and conservative.
The difference between geometric and logical functors between toposes is, in a certain sense, a categorification of the difference between a homomorphism of frames and a homomorphism of Heyting algebras. When the latter are complete, these are the same objects with the same isomorphisms but different morphisms.
However, while frame homomorphisms naturally categorified by geometric functors, a more precise categorification of Heyting algebra homomorphisms would be Heyting functors, which preserve the internal first-order logic, but not the higher-order logic as logical functors do.
A logical functor is the direct image of a geometric morphism precisely if it is an equivalence.
Since by definition the direct image of a geometric morphism has a left adjoint that preserves finite limits.
But logical inverse images are of interest. Recall from def. 3 above that a geometric morphism with logical inverse image is called an atomic geometric morphism.
Every atomic geometric morphism is an essential geometric morphism.
By prop. 1.
The following is the main source of examples of atomic geometric morphisms.
The inverse image of any base change geometric morphism, hence in particular of any etale geometric morphism, is a logical morphism.
The inverse image is given by pullback along the given morphism.
When considering the internal logic of a given topos $\mathcal{E}$ relations, predicates/propositions about variables of type $A \in ob \mathcal{E}$ are subobjects of $A$. Application of function symbols to such expressions corresponds to pullback along the morphism representing the function symbol. The above says that this is, indeed, a logical operation.
More generally, for any small category $C$ the inclusion
into the presheaf topos is logical.
For $G$ a group and $\mathbf{B}G$ its delooping groupoid, the forgetful functor
from permutation representations to Set is logical.
Section A2.1 in
Section IV.2, page 170 of