nLab
syntactic site

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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

logical 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

</table>

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. , 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

Last revised on May 28, 2017 at 05:24:18. See the history of this page for a list of all contributions to it.