nLab geometric type theory

Idea

Geometric type theory is a conjectural extension of geometric logic to an extensional dependent type theory. That is, it is supposed to be a dependent type theory which under the relation between category theory and type theory corresponds to sheaf toposes, and is preserved by geometric morphisms.

At present it is not clear exactly how such a type theory should be defined.

A motivating example is a geometric theory as follows. Its signature has one sort, $N$, and two function symbols $0\colon 1 \to N$ and $s \colon N \to N$. Its axioms are -

• $0 = s(n) \vdash_{n} \bot$
• $s(n) = s(n') \vdash_{n n'} n=n'$
• $\top \vdash_{n} \bigvee_{i} n=s^{i}(0)$

(“$s^{i}(0)$” is not a term in the formal language, but $i$ indexes an inductively defined sequence of formulae “$n=s^{i}(0)$”.)

In any model (in a Grothendieck topos) the interpretation of $N$ is uniquely isomorphic to an nno, so the nno can be characterized uniquely up to isomorphism by geometry theory - something that would be impossible in finitary first-order logic. There is essentially only one model, and this theory is equivalent to the empty theory. What this illustrates is that a sort for the nno can be added “for free” to any theory - and this is consistent with the fact that nno’s are preserved by inverse image functors.

List objects ($List X$ is the set of finite sequences of elements from $X$) can similarly be added “for free”, and in fact the same goes for free constructions for cartesian theories. Thus such constructors can be added as a type theoretic adjunct to geometric logic without altering its expressive power. Since geometric type theory also includes quotient types, the constructions are similar to quotient inductive types.

As an example, the geometric theory of a real number can be rewritten in a form that more directly describes Dedekind sections of rationals, with sort $Q$ that is geometrically constrained to be the rationals. For further details, see Vickers 2007.

All this leads to a conjecture of a “geometric type theory”, geometric logic enhanced with the geometrically definable types.

An “arithmetic type theory” has now been formalized Vickers 2018 by adjoining such type constructors to coherent logic. Categorically, it corresponds to replacing Grothendieck toposes with arithmetic universes. Although it lacks the infinitary disjunctions of geometric logic, they can in many cases be provided by existential quantification over infinite sorts. Vickers 2017 shows how it can be used to prove base-free results for (relative) Grothendieck toposes.

Related concepts

• modal type theory

• geometric homotopy type theory

• synthetic geometry

• geometric mathematics

References

• Steve Vickers, Locales and toposes as spaces, Handbook of Spatial Logics, Springer 2007 (pdf)

• Steve Vickers, Arithmetic universes and classifying toposes, Cahiers de topologie et géométrie différentielle catégorique 58 (4) (2017), pp. 213-248 (pdf)

• Steve Vickers, Sketches for arithmetic universes, accepted 2018 for Journal of Logic and Analysis (pdf)