$Lang$ of a Hyperdoctrine

Question

$$Form \dashv Lang : Hyp \rightarrow FOL$$

We consider an object of $FOL$ to be a signature made of sorts, function symbols and relation symbols, together with a set of axioms of first order logic.

We consider an object of $Hyp$ to be a first order hyperdoctrine formed by a cartesian category $C$ and a functor $C\rightarrow HA'$ where $HA'$ is the category of Heyting algebras with heyting algebra morphisms, with left and right adjoints adjointed (unlucky naming) such that these adjoints satisfy Beck-Chevalley and Frobenius conditions.

The functor $Form$ (assuming we know how to make $FOL$ a category ---I appreciate if someone has a comment on this) on objects constructs the free hyperdoctrine out of a signature and quotients a la Lindebaum.

For that we make $C$ the category of contexts out of our signature, and our functor $F : C \rightarrow HA'$ constructs the free Heyting algebra where $F\ [x_1:\sigma_1,\dots,x_n:\sigma_n]$ is thought of as the set of formulas that depend on $x_1,\dots,x_n$. (And it's easy to see what it does on terms.)

Question:

How do we define $Lang$?

We can consider the arrows of $C$ to be the function symbols of our theory (plus some quotenting), but what about the relation symbols?

Leaving the set of relation symbols empty seems wrong.

A little bit of self-promotion, but I wrote a paper about how these categories are equivalent (well, you haven't fully specified the categories in your post, but the paper does make this precise). There is even a 2-categorical structure. See here: https://doi.org/10.1007/s00153-022-00825-7. The paper is in terms of positive logic and type space functors, but the former is just a generalisation and can be specialised back to full first-order logic and the latter is just a dual perspective on hyperdoctrines (see Corollary 1.2 and Section 5 respectively, in that paper). — Mark Kamsma, Mar 09 '24 at 13:48

score 1 · Accepted Answer · answered Mar 08 '24 at 04:39

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You include one relation symbol of sort $X$ for each element of the Heyting algebra $F(X)$.

The arrows in the category FOL are interpretations between theories.

answered Mar 08 '24 at 04:39

Alex Kruckman

76,357

Thanks. Do you have a reference for an explicit and complete definition of interpretation between theories? – Julián Mar 08 '24 at 12:31
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@Julián The definition is given in Hodges' book Model Theory. – Alex Kruckman Mar 08 '24 at 12:47
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See also my answer here for the categorical logic view of interpretations. – Alex Kruckman Mar 08 '24 at 12:50
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Noah Schweber gave a definition of interpretation between theories is this recent answer. – Alex Kruckman Mar 08 '24 at 12:55

$Lang$ of a Hyperdoctrine

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