14.3 Geometric morphisms between topoi

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

These morphisms play a crucial role in relating different topoi, like sheaves on spaces or sets. They allow us to transfer information between categories while maintaining important structural relationships.

Geometric Morphisms and Their Properties

Components of geometric morphisms

• A geometric morphism $f: \mathcal{F} \to \mathcal{E}$ between topoi consists of two functors:
  • The direct image functor $f_*: \mathcal{F} \to \mathcal{E}$ preserves finite limits, exponentials (if they exist in $\mathcal{F}$), and subobject classifiers
  • The inverse image functor $f^*: \mathcal{E} \to \mathcal{F}$ preserves finite limits, finite colimits, exponentials (if they exist in $\mathcal{E}$), and subobject classifiers, and is left exact

Adjunction in geometric morphisms

• The functors $f_*$ and $f^*$ form an adjunction, with $f^*$ being the left adjoint and $f_*$ being the right adjoint
  • For any objects $A$ in $\mathcal{E}$ and $B$ in $\mathcal{F}$, there is a natural isomorphism: $\text{Hom}_{\mathcal{F}}(f^*(A), B) \cong \text{Hom}_{\mathcal{E}}(A, f_*(B))$
• The unit of the adjunction $\eta: 1_{\mathcal{E}} \to f_* \circ f^*$ is called the unit of the geometric morphism
  • Measures how close $f^*$ is to being a left inverse of $f_*$ ($\text{Sh}(X)$ and $\text{Set}$)
• The counit of the adjunction $\varepsilon: f^* \circ f_* \to 1_{\mathcal{F}}$ is called the counit of the geometric morphism
  • Measures how close $f_*$ is to being a right inverse of $f^*$ ($\text{FinSet}$ and $\text{Set}$)

Examples of geometric morphisms

• The global sections functor $\Gamma: \text{Sh}(X) \to \text{Set}$ for a topological space $X$ is a geometric morphism
  • The direct image functor is the global sections functor $\Gamma$ which sends a sheaf to its global sections
  • The inverse image functor is the constant sheaf functor $\Delta$ which sends a set to the constant sheaf with that set as stalks
• For any continuous map $f: X \to Y$ between topological spaces, there is an induced geometric morphism $\text{Sh}(f): \text{Sh}(X) \to \text{Sh}(Y)$
  • The direct image functor is the pushforward sheaf functor $f_*$ which sends a sheaf on $X$ to its pushforward sheaf on $Y$
  • The inverse image functor is the pullback sheaf functor $f^*$ which sends a sheaf on $Y$ to its pullback sheaf on $X$
• The inclusion functor $i: \text{FinSet} \to \text{Set}$ induces a geometric morphism between the topos of finite sets and the topos of sets
  • The direct image functor is the inclusion functor $i$ which sends a finite set to itself as a set
  • The inverse image functor sends a set to its poset of finite subsets ordered by inclusion

Properties of geometric morphisms

• A geometric morphism is an equivalence of topoi if and only if:
  • The unit $\eta: 1_{\mathcal{E}} \to f_* \circ f^*$ is a natural isomorphism ($f^*$ is fully faithful)
  • The counit $\varepsilon: f^* \circ f_* \to 1_{\mathcal{F}}$ is a natural isomorphism ($f_*$ is fully faithful)
• The composition of geometric morphisms is a geometric morphism ($\text{Sh}(X) \to \text{Sh}(Y) \to \text{Set}$)
• The identity functor on a topos induces the identity geometric morphism ($1_{\mathcal{E}}: \mathcal{E} \to \mathcal{E}$)