Convention. geometric morphisms and inverse image [fgap-000S]
Convention. geometric morphisms and inverse image [fgap-000S]
For a geometric morphism \[ f:\mathcal {E}\longrightarrow \mathcal {F}, \] we write \[ f^*:\mathcal {F}\longrightarrow \mathcal {E}, \qquad f_*:\mathcal {E}\longrightarrow \mathcal {F}, \qquad f^*\dashv f_*. \] Thus the morphism and its inverse-image functor point in opposite directions. The functor \(f^*\) preserves finite limits.
Two examples fix the convention: \[ \begin {array}{c|c|c} \text {input map}&\text {geometric morphism}&\text {inverse image}\\ \hline i:\{x\}\hookrightarrow X& \mathsf {Set}\to \mathsf {Sh}(X)& i^*F=F_x=\displaystyle \varinjlim _{x\in U}F(U)\\[3pt] \varphi :H\to G& \mathsf {B}H\to \mathsf {B}G& \operatorname {Res}^G_H:\mathsf {B}G\to \mathsf {B}H. \end {array} \] In the first row, the stalk is the filtered colimit of all neighborhood sections, not the value on one chosen neighborhood.
Mac Lane and Moerdijk construct inverse-image sheaves through pulled-back étale spaces in [maclane1992sheaves, sec. II.9], especially printed pp. 99--101.
In the second row, right actions are presheaves. Precomposition with \(B\varphi ^{\mathrm {op}}\) is restriction of actions. It has both Kan-extension adjoints, so it is the inverse-image part of the displayed geometric morphism. The homomorphism and geometric morphism point from \(H\) to \(G\); restriction points from \(G\)-objects to \(H\)-objects.