Abstract

We reconcile the two different category-theoretic semantics of regular theories in predicate logic. A 2-category of regular fibrations is constructed, as well as a 2-category of regular proarrow equipments, and it is shown that the two are equivalent. A regular equipment is a cartesian equipment satisfying certain axioms, and a cartesian equipment is a slight generalization of a cartesian bicategory.

This is done by defining a tricategory $Biprof$ whose objects are bicategories and whose morphisms are category-valued profunctors, and then defining an equipment to be a pseudo-monad in this tricategory. The resulting notion of equipment is compared to several existing ones. Most importantly, this involves showing that every pseudo-monad in $Biprof$ has a Kleisli object. A strict 2-category of equipments, over locally discrete base bicategories, is identified, and cartesian equipments are defined to be the cartesian objects in this 2-category. Thus cartesian equipments themselves form a 2-category, and this is shown to admit a 2-fully-faithful functor from the 2-category of regular fibrations. The cartesian equipments in the image of this functor are characterized as those satisfying certain axioms, and hence a 2-category of regular equipments is identified that is equivalent to that of regular fibrations.

It is then shown that a regular fibration admits comprehension for predicates if and only if its corresponding regular equipment admits tabulation for morphisms, and further that the presence of tabulations for morphisms is equivalent to the existence of Eilenberg–Moore objects for co-monads. We conclude with a brief examination of the two different constructions of the effective topos, via triposes and via assemblies, in the light of the foregoing.

Outline

Regular theories and fibrations

This is standard stuff, mostly:

• regular logic?
• regular fibration

The tricategory $Biprof$

We want to define the tricategory $BiProf$ of bicategories and profunctors; we need to know what the following things are:

1. 2-extranatural transformation
2. 2-end
3. free 2-cocompletion

and then we define $BiProf$ as here, and prove that it has Kleisli objects for pseudo-monads.

The 2-category of equipments

We can define an equipment to be a pseudo-monad in $Biprof$, and an equipment functor to be a (representable) monad (op-)morphism. The existence of Kleisli objects in $Biprof$ ensures that these definitions recover those of Wood in Wood82 and Wood85. They also have a clear relationship to some of the later ideas in CKVW.

This is fine as far as it goes, but it gives us the wrong kind of 2-cell (what we get is not lax enough). So instead we show that our equipments and their morphisms form an ordinary category that is equivalent to the category underlying Shulman’s strict 2-category of framed bicategories and their morphisms. Then we can say that an equipment 2-cell is a framed transformation between corresponding framed functors.

That’s a bit of a swindle, but the alternative would be to develop a whole theory of ‘doubly-lax’ monad morphisms and lax monad 2-cells and so on. It ought to be done, certainly, but not here by us, simply because it would take too long.

Cartesian equipments and regular fibrations

We have a 2-category of equipments now, and it clearly has finite products, so we define a cartesian equipment to be a cartesian object in it. That’s a monoid object whose multiplication and unit maps are right adjoint to the diagonal and terminal maps. So we get a 2-category $CartEqt$. When we spell out what it means for an equipment to be cartesian, it becomes obvious that a bicategory $\mathcal{B}$ is a cartesian bicategory if and only if $Map(\mathcal{B}) \to \mathcal{B}$ is a cartesian equipment.

As in Shulman08, from a regular fibration $E$ over some category $B$ we can construct a cartesian equipment $Matr(E)$ (of ‘matrices’ in $E$): its objects are those of $B$, its vertical/tight maps are the morphisms of $B$ and the category of horizontal maps and (globular) cells from $b$ to $b'$ is $E(b \times b')$. The important point now is that $E \mapsto Matr(E)$ is a fully faithful functor $RegFib \to CartEqt$.

If $B \to \mathcal{B}$ is a cartesian equipment, then $Pred(\mathcal{B}) = \mathcal{B}(-_\bullet, \mathbf{1})$ is a bifibration, where $f_\bullet$ is the morphism in $\mathcal{B}$ arising from a morphism $f$ in $B$, and $\mathbf{1}$ is the terminal object of $B$. In fact, it is almost a regular fibration: there are two Beck–Chevalley conditions required, one corresponding exactly to the notion of a separable object in a cartesian bicategory. The other has the Frobenius condition familiar from cartesian bicategories as a special case, but seems to be strictly more general. Cartesian equipments satisfying these two conditions are called regular equipments, and the 2-category of them is equivalent to that of regular fibrations.

Tabulation and comprehension

coming soon

The effective topos

coming soon