Classifying Toposes, Geometric Theories, and Syntax-Semantics¶
Job: develop the classifying topos \(\mathbf{B}[T]\), the geometric theories / geometric logic, the syntax-semantics duality (Caramello), and the Morita equivalence — with the \(\mathbf{B}[\mathrm{Obj}]=\mathbf{Set}\) as the worked example.
Core¶
The classifying topos¶
| Classifying / \(\mathbf{B}[T]\) / \(\mathrm{Mod}\) / Caramello | Description |
|---|---|
| Classifying topos \(\mathbf{B}[T]\) | \(\mathrm{Mod}(T;\mathcal{E})\cong\mathbf{GeomMorph}(\mathcal{E},\mathbf{B}[T])\) (the Caramello) |
| \(\mathrm{Mod}(T;\mathcal{E})\) | the \(T\)-models in \(\mathcal{E}\) (the semantics) |
The geometric theories and the geometric logic¶
| Geometric theory / \(\exists,\bigvee\) / categorical logic | Description |
|---|---|
| Geometric theory | \(\exists,\bigvee,\bigwedge\) the geometric, the finite limits (the category theory / categorical logic) |
| Geometric logic | \(\exists,\bigvee\) the geometric (the category theory) |
The syntax-semantics and the duality¶
| Syntax / semantics / Caramello / bridges | Description |
|---|---|
| Syntax-semantics | the theory \(T\) the syntax, \(\mathbf{B}[T]\) the semantics (the Caramello) |
| Bridges | the Caramello the bridges (the category theory) |
The Morita equivalence and the classifying¶
| Morita / \(T_1,T_2\) / \(\mathbf{B}\) / category theory | Description |
|---|---|
| Morita equivalence | \(T_1,T_2\) Morita equivalent \(\Leftrightarrow\) \(\mathbf{B}[T_1]\simeq\mathbf{B}[T_2]\) (the Caramello) |
See it¶
Classifying topos of the theory of objects and the geometric morphisms. The \(\mathbf{B}[\mathrm{Obj}]\) the classifying topos of the theory of objects (the \(\mathrm{Mod}(\mathrm{Obj};\mathcal{E})\cong\mathbf{GeomMorph}(\mathcal{E},\mathbf{Set})\), the \(\mathbf{Set}\) the classifying). The geometric: the \(\exists,\bigvee\) the geometric, the category theory / categorical logic. The key point: the \(\mathbf{B}[\mathrm{Obj}]=\mathbf{Set}\), the \(\mathrm{Mod}\cong\mathbf{GeomMorph}\) — the classifying topos the syntax / semantics.
Check¶
Explain the classifying topos \(\mathbf{B}[T]\) and the universal property.
The \(\mathbf{B}[T]\): the classifying topos of the geometric theory \(T\) (the universal property \(\mathrm{Mod}(T;\mathcal{E})\cong\mathbf{GeomMorph}(\mathcal{E},\mathbf{B}[T])\), the Caramello). The \(\mathrm{Mod}(T;\mathcal{E})\): the \(T\)-models in \(\mathcal{E}\) (the semantics, the \(\mathcal{E}\) the topos). The \(\mathbf{GeomMorph}\): the geometric morphisms (the \(f^*\dashv f_*\), the foundations). The \(\mathbf{B}[\mathrm{Obj}]=\mathbf{Set}\) (the theory of objects the \(\mathbf{Set}\), the \(\mathrm{Mod}(\mathrm{Obj};\mathcal{E})\cong\mathbf{GeomMorph}(\mathcal{E},\mathbf{Set})\)). So the \(\mathbf{B}[T]\) the universal, the \(\mathrm{Mod}\cong\mathbf{GeomMorph}\).
Explain the Morita equivalence and the Caramello bridges.
The Morita: the \(T_1, T_2\) the Morita equivalent \(\Leftrightarrow\) the \(\mathbf{B}[T_1]\simeq\mathbf{B}[T_2]\) (the classifying toposes the equivalent, the Caramello). The bridges: the Caramello the "bridges" (the topos the bridge the different theories the Morita, the category theory / categorical logic). The syntax-semantics: the \(T\) the syntax, the \(\mathbf{B}[T]\) the semantics (the Caramello the duality). So the Morita the \(\mathbf{B}\), the bridges the Caramello.
Pitfalls¶
- The classifying topos \(\mathbf{B}[T]\) (the universal, the \(\mathrm{Mod}\cong\mathbf{GeomMorph}\)) — the \(\mathbf{B}[T]\) the classifying, the Caramello.
- The geometric theory (\(\exists,\bigvee\) the geometric, the category theory) — the geometric the \(\exists,\bigvee\), the finite limits.
- The Morita the \(\mathbf{B}\) (the \(T_1,T_2\) the Morita, the \(\mathbf{B}[T_1]\simeq\mathbf{B}[T_2]\)) — the Morita the equivalence, the Caramello.
- The syntax-semantics (the \(T\) the syntax, the \(\mathbf{B}[T]\) the semantics) — the Caramello the duality, the bridges.
Refs¶
- Caramello, Theories, Sites, Toposes (2018); the classifying / the bridges
- Mac Lane–Moerdijk, Sheaves in Geometry and Logic (1992); the classifying topos
- Johnstone, Sketches of an Elephant (2002); the geometric theory
- Makkai–Reyes, First Order Categorical Logic (1977); the geometric logic
Next¶
TT index · Foundations · Boolean & Forcing · Cohesive & Higher · Synthetic & Infinitesimal · Category theory / categorical logic