logical functor


Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




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 Ω\Omega \in \mathcal{E} for the subobject classifier. For each object AA \in \mathcal{E} write

PA:=Ω A P A := \Omega^A

for the exponential object. Write

APA×A \in_A \hookrightarrow P A \times A

for the subobject classified by the evaluation map ev:PA×AΩev : P A \times A \to \Omega.

This exhibits PAP A as a power object for AA.


A functor F:F : \mathcal{E} \to \mathcal{F} between elementary toposes is called a logical morphism if

  1. FF preserves finite limits;

  2. for every object AA \in \mathcal{E}

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. 1 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:F : \mathcal{E} \to \mathcal{F} a logical functor, we have by definition a diagram

op F op op P P F \array{ \mathcal{E}^{op} &\stackrel{F^{op}}{\to}& \mathcal{F}^{op} \\ {}^{\mathllap{P}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{P}} \\ \mathcal{E} &\stackrel{F}{\to}& \mathcal{F} }

in Cat. This satisfies the assumptions of the adjoint lifting theorem and hence FF has a right adjoint precisely if F opF^{op} does. But a right adjoint of F opF^{op} is a left adjoint of FF, and vice versa.

Preserved structures

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.

Relation to geometric morphisms

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.


If a logical functor is right adjoint to a cartesian functor, then it is an equivalence of categories.

This appears as (Johnstone, scholium 2.3.9).


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 and hence is in particular a cartesian morphism

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.


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 AobA \in ob \mathcal{E} are subobjects of AA. 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.



Section A2.1 in

Section IV.2, page 170 of

Revised on July 27, 2016 08:36:10 by Dexter Chua (