nLab logical functor

Idea

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.

Definition

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.

Definition

Let $\mathcal{E}$ be an elementary topos. Write $\Omega \in \mathcal{E}$ for the subobject classifier. For each object $A \in \mathcal{E}$ write

$$P A \coloneqq \Omega^A$$

for the exponential object. Write

$$\in_A \hookrightarrow P A \times A$$

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

This exhibits $P A$ as a power object for $A$.

Definition

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

1. $F$ preserves finite limits;

2. for every object $A \in \mathcal{E}$

   • the canonical morphism

     $$\phi_A : F(P A) \to P (F A)$$

     is an isomorphism; this is the name of the relation

     $$F(\in_A) \hookrightarrow F(P A \times A) \simeq F(P A) \times F A$$

     (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:

Definition

A geometric morphism whose inverse image is a logical functor is called an atomic geometric morphism.

Remark

The other case, that the direct image of a geometric morphism is a logical functor is not of interest. See cor. below.

Properties

General

Proposition

A logical functor has a left adjoint precisely if it has a right adjoint.

This appears as (Johnstone, cor. A2.2.10).

Proof

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

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

in Cat. This satisfies the assumptions of the adjoint lifting theorem and hence $F$ 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.

Proposition

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 $L A\in \mathcal{E}$.

Proof

Since $F(\Omega_{\mathcal{E}}) = \Omega_{\mathcal{F}}$, we have a bijection

$$Sub_{\mathcal{F}}(A) \cong \mathcal{F}(A,\Omega_{\mathcal{F}}) \cong \mathcal{E}(L A,\Omega_{\mathcal{E}}) \cong Sub_{\mathcal{E}}(L A).$$

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).

Proposition

For a logical functor $F : \mathcal{E} \to \mathcal{F}$ having a left adjoint $L$, the following are equivalent:

1. The induced functor $L':\mathcal{F} \to \mathcal{E}/L 1$ is an equivalence of categories, i.e. the adjunction $L\dashv F$ can be identified with the pullback adjunction $\mathcal{E}/B \rightleftarrows \mathcal{E}$ for some $B$ (namely $L1$).
2. $L$ is conservative.
3. $L$ is faithful.
4. $L$ preserves equalizers.
5. $L$ preserves pullbacks.

This appears as (Johnstone, Prop. A2.3.8).

Proof

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 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

$$L' F' A \cong L'(1\times F' A) \xrightarrow{\cong} L' 1 \times A \cong 1\times A\cong A$$

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).

Corollary

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).

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.

Proposition

A logical functor is the direct image of a geometric morphism precisely if it is an equivalence.

Proof

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. above that a geometric morphism with logical inverse image is called an atomic geometric morphism.

Corollary

Every atomic geometric morphism is an essential geometric morphism.

Proof

By prop. .

The following is the main source of examples of atomic geometric morphisms.

Proposition

The inverse image of any base change geometric morphism, hence in particular of any etale geometric morphism, is a logical morphism.

Proof

The inverse image is given by pullback along the given morphism.

Remark

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.

Examples

• The inclusion FinSet $\hookrightarrow$ Set is logical.

• More generally, for any finite category $C$ the inclusion

  $$[C^{op}, FinSet] \hookrightarrow [C^{op}, Set]$$

  into the presheaf topos is logical.

• For $G$ a group and $\mathbf{B}G$ its delooping groupoid, the forgetful functor

  $$[\mathbf{B}G, Set] \to Set$$

  from permutation representations to Set is logical.

References

Section A2.1 in

• Peter Johnstone, Sketches of an Elephant

Section IV.2, page 170 of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic