topos theory

# Contents

## Idea

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

## Definition

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

###### Definition
• For $𝕋$ a cartesian theory, $J$ is the trivial coverage: the covering families consist of single isomorphisms.

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

• For $𝕋$ a coherent theory, $J$ is the coherent coverage: the covering families consist of morphisms $\left\{{U}_{i}\to U\right\}$ such that the union ${\cup }_{i}{U}_{i}\simeq U$ equals $U$.

• For $𝕋$ a geometric theory, $J$ is the geometric coverage

## Properties

###### Proposition

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

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

$𝕋-\mathrm{Model}\left(ℰ\right)\simeq {\mathrm{Func}}_{×}\left({𝒞}_{𝕋},ℰ\right)\simeq \mathrm{Topos}\left(ℰ,\mathrm{Sh}\left({𝒞}_{𝕋}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{T}-Model(\mathcal{E}) \simeq Func_\times(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
• For $𝕋$ a regular theory, regular functors ${𝒞}_{𝕋}\to ℰ$ are equivalent to geometric morphisms $ℰ\to \mathrm{Sh}\left({𝒞}_{𝕋}\right)$

$𝕋-\mathrm{Model}\left(ℰ\right)\simeq \mathrm{RegFunc}\left({𝒞}_{𝕋},ℰ\right)\simeq \mathrm{Topos}\left(ℰ,\mathrm{Sh}\left({𝒞}_{𝕋}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{T}-Model(\mathcal{E}) \simeq RegFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
• For $𝕋$ a coherent theory, coherent functors ${𝒞}_{𝕋}\to ℰ$ are equivalent to geometric morphisms $ℰ\to \mathrm{Sh}\left({𝒞}_{𝕋}\right)$

$𝕋-\mathrm{Model}\left(ℰ\right)\simeq \mathrm{CohFunc}\left({𝒞}_{𝕋},ℰ\right)\simeq \mathrm{Topos}\left(ℰ,\mathrm{Sh}\left({𝒞}_{𝕋}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{T}-Model(\mathcal{E}) \simeq CohFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E}) \simeq Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \,.
• For $𝕋$ a geometric theory, geometric functors ${𝒞}_{𝕋}\to ℰ$ are equivalent to geometric morphisms $ℰ\to \mathrm{Sh}\left({𝒞}_{𝕋}\right)$

$𝕋-\mathrm{Model}\left(ℰ\right)\simeq \mathrm{GeomFunc}\left({𝒞}_{𝕋},ℰ\right)\simeq \mathrm{Topos}\left(ℰ,\mathrm{Sh}\left({𝒞}_{𝕋}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $\mathrm{Topos}\left(ℰ,\mathrm{Sh}\left({𝒞}_{𝕋}\right)\right)\stackrel{\simeq }{\to }𝕋-\mathrm{Model}\left(ℰ\right)$ is given by sending a geometric morphism $f:ℰ\to \mathrm{Sh}\left({𝒞}_{𝕋}\right)$ to the precomposition of its inverse image ${f}^{*}$ with the Yoneda embedding $j$ and sheafification $L$:

$f\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left({𝒞}_{𝕋}\stackrel{j}{\to }\mathrm{PSh}\left({𝒞}_{𝕋}\right)\stackrel{L}{\to }\mathrm{Sh}\left({𝒞}_{𝕋}\right)\stackrel{{f}^{*}}{\to }ℰ\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $ℰ\to \mathrm{Sh}\left(𝒞\right)$ are equivalent to morphisms of sites $𝒞\to ℰ$ (for the canonical coverage on $ℰ$). This means that in addition to preserving finite limits, as in Diaconescu's theorem, these functors also send covers in $𝒞$ to epimorphisms in $ℰ$.

In the cases at hand this last condition means precisely that ${𝒞}_{𝕋}\to ℰ$ 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)