nLab
syntactic site

Context

Topos Theory

Could not include topos theory - contents

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Contents

Idea

For 𝕋\mathbb{T} a theory, the syntactic site of a syntactic category 𝒞 𝕋\mathcal{C}_{\mathbb{T}} is the structure of a site on 𝒞 𝕋\mathcal{C}_{\mathbb{T}} such that geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}}) into the sheaf topos over the syntactic site are equivalent to models for the theory 𝕋\mathbb{T} in \mathcal{E}, hence such that Sh(𝒞 𝕋)Sh(\mathcal{C}_{\mathbb{T}}) is the classifying topos for 𝕋\mathbb{T}.

Definition

For 𝕋\mathbb{T} a theory and 𝒞 𝕋\mathcal{C}_{\mathbb{T}} its syntactic category, we define coverages JJ on 𝒞 𝕋\mathcal{C}_{\mathbb{T}}. These depend on which type of theory mathbT\mathb{T} is (or is regarded to be).

Definition

Properties

Proposition

For 𝒯\mathcal{T} a cartesian theory, regular theory, etc. and 𝒞 𝕋\mathcal{C}_{\mathbb{T}} its syntactic site, according to def. 1, we have

  • For 𝕋\mathbb{T} a cartesian theory, left exact functors 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} are equivalent to geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})

    𝕋Model()Func ×(𝒞 𝕋,)Topos(,Sh(𝒞 𝕋)). \mathbb{T}-Model(\mathcal{E}) \simeq Func_\times(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
  • For 𝕋\mathbb{T} a regular theory, regular functors 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} are equivalent to geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})

    𝕋Model()RegFunc(𝒞 𝕋,)Topos(,Sh(𝒞 𝕋)). \mathbb{T}-Model(\mathcal{E}) \simeq RegFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
  • For 𝕋\mathbb{T} a coherent theory, coherent functors 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} are equivalent to geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})

    𝕋Model()CohFunc(𝒞 𝕋,)Topos(,Sh(𝒞 𝕋)). \mathbb{T}-Model(\mathcal{E}) \simeq CohFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
  • For 𝕋\mathbb{T} a geometric theory, geometric functors 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} are equivalent to geometric morphisms Sh(𝒞 𝕋)\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})

    𝕋Model()GeomFunc(𝒞 𝕋,)Topos(,Sh(𝒞 𝕋)). \mathbb{T}-Model(\mathcal{E}) \simeq GeomFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.

In each case the equivalence of categories Topos(,Sh(𝒞 𝕋))𝕋Model()Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \stackrel{\simeq}{\to} \mathbb{T}-Model(\mathcal{E}) is given by sending a geometric morphism f:Sh(𝒞 𝕋)f : \mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}}) to the precomposition of its inverse image f *f^* with the Yoneda embedding jj and sheafification LL:

f(𝒞 𝕋jPSh(𝒞 𝕋)LSh(𝒞 𝕋)f *). f \;\; \mapsto \;\; ( \mathcal{C}_\mathbb{T} \stackrel{j}{\to} PSh(\mathcal{C}_{\mathbb{T}}) \stackrel{L}{\to} Sh(\mathcal{C}_{\mathbb{T}}) \stackrel{f^*}{\to} \mathcal{E} ) \,.

This appears as (Johnstone), theorem 3.1.1, 3.1.4, 3.1.9, 3.1.12.

Definition

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 Sh(𝒞)\mathcal{E} \to Sh(\mathcal{C}) are equivalent to morphisms of sites 𝒞\mathcal{C} \to \mathcal{E} (for the canonical coverage on \mathcal{E}). This means that in addition to preserving finite limits, as in Diaconescu's theorem, these functors also send covers in 𝒞\mathcal{C} to epimorphisms in \mathcal{E}.

In the cases at hand this last condition means precisely that 𝒞 𝕋\mathcal{C}_{\mathbb{T}} \to \mathcal{E} is a regular functor or coherent functor etc., respectively.

References

Section D3.1 of

Revised on April 27, 2011 10:13:21 by Urs Schreiber (131.211.232.242)