pushforwardrelated

Pushforwardrelated is a term used in mathematics to describe concepts and results connected to the pushforward construction. It is not a standard term in textbooks, but it can serve as an umbrella for the common pattern of transporting structures along a map between spaces.

In measure theory, the pushforward of a measure mu on X by a measurable map f: X -> Y is the measure f_* mu on Y defined by f_* mu(A) = mu(f^{-1}(A)). This enables a change of variables for integration: ∫ g d(f_* mu) = ∫ g∘f d mu. It also underpins how distributions of random variables transform under measurable mappings.

In differential geometry, the pushforward, or differential, is the map df_p: T_pX -> T_{f(p)}Y that carries tangent vectors forward along f. This concept extends to more general objects in geometric contexts and underlies constructions such as fiber integration and direct image of sections in bundles.

In category theory and algebraic geometry, the pushforward refers to the direct image functor f_* that sends objects on X to objects on Y, typically as a left or right adjoint to pullback functors. This viewpoint ties pushforward to sheaves, cohomology, and base change phenomena.

Related notions include the pullback, Jacobian transformations, and the disintegration of measures. Pushforwardrelated thus highlights a family of transport mechanisms that appear across analysis, geometry, and topology.