Could not include topos theory - contents
|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination? for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
For a theory, the syntactic site of a syntactic category is the structure of a site on such that geometric morphisms into the sheaf topos over the syntactic site are equivalent to models for the theory in , hence such that is the classifying topos for .
This appears as (Johnstone), theorem 3.1.1, 3.1.4, 3.1.9, 3.1.12.
For cartesian theories this is the statement of Diaconescu's theorem.
The other cases follow from this by using this discussion at classifying topos, which says that geometric morphism are equivalent to morphisms of sites (for the canonical coverage on ). This means that in addition to preserving finite limits, as in Diaconescu's theorem, these functors also send covers in to epimorphisms in .
Section D3.1 of