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There obviously seems to be a connection between the push-forward and the pull-back of a smooth function $f:M \to N$ between smooth manifolds, and the Hom-maps from category theory $f^*=Mor_C(f,\mathbb{R})$ and $f_*=Mor_C(f,\mathbb{R})$ or some other Hom-maps, where $C$ is the category of differentiable manifolds with differentiable maps.

Since I'm not very good at category theory and the whole thing seems a bit intricate to me, I can't really figure it out. Any thoughts/ more precise statement on that topic would be very helpful

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    $\begingroup$ A pullback square can be reinterpreted as a pullback functor. In your case, it would be a functor $f^*: \mathbf{Man}/N \to \mathbf{Man}/M$ taking a morphism of smooth manifolds to it's pullback along $f$. $\endgroup$ – gian May 20 '18 at 15:34
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    $\begingroup$ Do you mean to talk about the pushforward of tangent vectors and the pullback of differential forms? $\endgroup$ – Kevin Carlson May 21 '18 at 18:19
  • $\begingroup$ @KevinCarlson Yes, I meant those. $\endgroup$ – Intergalakti May 22 '18 at 17:44

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