nLab
syntactic site

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Topos Theory

topos theory

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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.