Pushforward

The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different, but closely related things.

• Pushforward (differential): the differential of a smooth map between manifolds, and the "pushforward" operations it defines.
• Direct image sheaf: the pushforward of a sheaf by a map.
• Pushforward (homology): the map induced in homology by a continuous map between topological spaces.
• Fiberwise integral: the direct image of a differential form or cohomology by a smooth map, defined by "integration on the fibres".
• Pushout (category theory): the categorical dual of pullback.
• Pushforward measure: measure induced on the target measure space by a measurable function.
• The transfer operator is the pushforward on the space of measurable functions; its adjoint, the pull-back, is the composition or Koopman operator.