nLab syntactic site

Idea

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

Definition

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

Definition

• For $\mathbb{T}$ a cartesian theory, $J$ is the trivial coverage: the covering families consist of single isomorphisms.

• For $\mathbb{T}$ a regular theory, $J$ is the regular coverage: the covering families consist of single regular epimorphisms.

• For $\mathbb{T}$ a coherent theory, $J$ is the coherent coverage: the covering families consist of morphisms $\{U_i \to U\}$ such that the union $\cup_i U_i \simeq U$ equals $U$.

• For $\mathbb{T}$ a geometric theory, $J$ is the geometric coverage

Properties

Proposition

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

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

  $$\mathbb{T}-Model(\mathcal{E}) \simeq Func_\times(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.$$

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

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

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

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

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

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

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

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

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 $\mathcal{E} \to Sh(\mathcal{C})$ are equivalent to morphisms of sites $\mathcal{C} \to \mathcal{E}$ (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 $\mathcal{C}$ to epimorphisms in $\mathcal{E}$.

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

References

Section D3.1 of

• Peter Johnstone, Sketches of an Elephant