topos theory

# Contents

## 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 $\mathb{T}$ 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. 1, 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

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