Thread: Push-forward ?
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Jun7-09, 02:18 PM
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As I said, functions are contravariant, so a function on B
Functions are contravariant on their domain, but covariant on their codomain.

Well, that's somewhat of an abuse of language. More accurately, "the set of functions from X to Y" is a functor contravariant in the variable X and covariant in the variable Y.

If I denote it by Hom(X,Y), then:

For any function f:Y->Z, I have f*:Hom(X,Y)->Hom(X,Z) given by f*(g) = f o g

For any function f:W->X, I have f*:Hom(X,Y)->Hom(W,Y) given by f*(g) = g o f

For an example in the setting of manifolds, recall that for a manifold M, we define a "curve on M" to be a continuous function [0,1]->M. This is a case where we fix the domain and vary the codomain, so curves get pushed around covariantly: given any continuous function f:M->N and curve c on M, we have a pushforward curve f*(c) on N, given by f*(c) = f o c.