So, in another thread I mentioned the pull-back in passing. Let's do it properly now, as it is just so much fun.

We will suppose that are vector spaces, and that the linear operators (aka transformations) . Then we know that the composition as shown here.

Notice the rudimentary (but critical) fact, that this only makes sense because the codomain of is the domain of

Now, it is a classical result from operator theory that the set of all operators is a vector space (you can take my word for it, or try to argue it for yourself).

Let's call the vector space of all such operators etc. Then I will have that are vectors in these spaces.

The question naturally arises: what are the linear operators that act onthesespaces? Specifically, what is the operator that maps onto ?

By noticing that here the "" is a fixed domain, and that , we may suggest the notation . But, for reasons which I hope to make clear, I will use a perfectly standard alternative notation .

Now, looking up at my diagram, I can think of this as "pushing" the tip of the "f-arrow" along the "g-arrow" to become the "composite arrow". Accordingly, I will call this thepush-forwardof on , or, by a horrid abuse of English as we normally understand it, thepush-forward of

So, no real shocks here, right? Ah, just wait, the fun is yet to begin, but this post is already over-long, so I'll leave you to digest this for a while.........