Definition. geometric morphism, Topoi [kostecki2011introduction, 7.2] [tt-004Q]
Definition. geometric morphism, Topoi [kostecki2011introduction, 7.2] [tt-004Q]
If \({\cal E}_1\) and \({\cal E}_2\) are toposes, then a geometric morphism \(\mathscr {G}: {\cal E}_1 \to {\cal E}_2\) is a pair of adjoint functors \[ \mathscr {G}^*:{\cal E}_2\to {\cal E}_1,\qquad \mathscr {G}_*:{\cal E}_1\to {\cal E}_2,\qquad \mathscr {G}^*\dashv \mathscr {G}_*, \] such that the inverse-image functor \(\mathscr {G}^*\) preserves finite limits; that is, \(\mathscr {G}^*\) is left exact. See [maclane1992sheaves, sec. VII.1, pp. 348--352].
As a left adjoint, \(\mathscr {G}^*\) preserves every colimit that exists. As a right adjoint, \(\mathscr {G}_*\) preserves every limit that exists. The additional left-exactness condition on \(\mathscr {G}^*\) is not automatic for a left adjoint, and right adjointness does not in general make \(\mathscr {G}_*\) preserve colimits.
The category of toposes and their geometric morphisms is denoted \(\mathbf {Topoi}\).