10.1 Definition and properties of geometric morphisms

Geometric morphisms are special functors between topoi that preserve key structures. They consist of an inverse image functor and a direct image functor, forming an adjoint pair with specific preservation properties.

These morphisms play a crucial role in connecting different topoi, allowing us to transfer information and structures between them. Examples include relationships between sets, presheaves, and sheaves, highlighting their importance in various mathematical contexts.

Geometric Morphisms in Topos Theory

Definition of geometric morphisms

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• Geometric morphism comprises functor $f^*: \mathcal{F} \to \mathcal{E}$ between topoi with right adjoint $f_*: \mathcal{E} \to \mathcal{F}$, where $f^*$ preserves finite limits (pullbacks, terminal objects)
• Composition of geometric morphisms yields another geometric morphism, allowing chaining of morphisms between multiple topoi
• Identity functor serves as geometric morphism, providing reflexivity in topos relationships
• Inverse image functor $f^*$ and direct image functor $f_*$ form core components, each playing distinct roles in structure preservation
• Adjunction relationship $f^* \dashv f_*$ established through natural isomorphism $\text{[Hom](https://www.fiveableKeyTerm:hom)}_\mathcal{E}(f^*(A), B) \cong \text{Hom}_\mathcal{F}(A, f_*(B))$, connecting morphisms in source and target topoi

Preservation of categorical structures

• $f^*$ preserves finite limits by definition, crucial for maintaining topos structure (terminal objects, pullbacks)
• $f^*$ preserves finite colimits due to left adjoint property, extends to preservation of initial objects and pushouts
• Exponential preservation by $f^*$ shown through isomorphism $f^*(B^A) \cong (f^*B)^{f^*A}$, utilizing adjunction and limit preservation
• Proofs involve demonstrating preservation of specific constructions (limits, colimits, exponentials) using functor properties and adjunction relationships

Examples in common topoi

• Set and topos $\mathcal{E}$ connected by global sections functor $\Gamma: \mathcal{E} \to \text{Set}$ and constant objects functor $\Delta: \text{Set} \to \mathcal{E}$
• Presheaf topoi linked through functors between base categories, employing left and right Kan extensions
• Subtopos inclusion uses inclusion functor as direct image and sheafification functor as inverse image
• Sheaf topoi connected via continuous functions between topological spaces, utilizing direct and inverse image sheaf functors (restriction, extension)

Relation to adjoint functors

• Geometric morphisms form special adjoint pair $f^* \dashv f_*$ with additional finite limit preservation condition
• Left adjoint $f^*$ preserves colimits while right adjoint $f_*$ preserves limits, fundamental to structure preservation
• Not all topos adjunctions qualify as geometric morphisms, highlighting unique nature of these morphisms
• Frobenius reciprocity $f^*(A) \times_{\mathcal{E}} B \cong f^*(A \times_{\mathcal{F}} f_*(B))$ characterizes special interaction between inverse image and fiber products
• Topos equivalence occurs when both $f^*$ and $f_*$ are equivalences, preserving all topos-theoretic properties (subobject classifier, power objects)