Geometric Morphism

A geometric morphism is a pair of adjoint functors f* ⊣ f* between two topoi that preserves finite limits — the correct notion of a 'map between spaces' when spaces are understood as generalized universes of sets rather than as collections of points. The inverse image functor f* preserves finite limits and arbitrary colimits, while the direct image functor f* preserves finite limits and satisfies an additional exactness condition. This definition, due to William Lawvere and Myles Tierney, subsumes ordinary continuous maps between topological spaces, maps between locales, and even logical interpretations between theories.

Geometric morphisms reveal that the notion of 'space' is more general than point-set topology admits. A continuous map between topological spaces induces a geometric morphism between their sheaf topoi; conversely, not every geometric morphism between sheaf topoi comes from a continuous map. This means topos theory captures spatial structure that point-set topology misses. The existence of geometric morphisms that are not spatially induced is evidence that the category of topoi is the more natural setting for geometry — one in which the logic of the space and the geometry of the space are not separate subjects but two faces of the same structure.