This is so much fun! Look
We will suppose thatare 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 ofis the domain of
Now, it is a classical result from operator theory that the set of all operatorsis 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 operatorsetc. Then I will have that are vectors in these spaces.
The question naturally arises: what are the linear operators that act on these spaces? Specifically, what is the operator that mapsonto ?
By noticing that here theis 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 the push-forward ofon , or, by a horrid abuse of English as we normally understand it, the push-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.........
Recall that, givenas a linear operator on vector spaces, we found as the linear operator that maps onto , and called it the push-forward of . In fact let's make that a definition: defines the push-forward.
This construction arose because we were treating the spaceas a fixed domain. We are, of course, free to treat as as fixed codomain, like this.
Using our earlier result, we might try to write the operator, but something looks wrong; is going "backwards"!
Nothing daunted, let's adopt the convention. (We will see this choice is no accident)
Looking up at my diagram, I can picture this a pulling the "tail" of the h-arrow back along the g-arrow onto the composite arrow, and accordingly (using the same linguistic laxity as before), callthe pull-back of , and make the definition: defines the pullback
(Compare with the pushforward)
This looks weird, right? But it all makes beautiful sense when we consider the following special case of the above.
where I have assumed that
Putting this all together I find that, forI will have as my pullback.
I say this is just about as nice as it possibly could be. What say you?