Recently I've been reading "Sheaves in Geometry and Logic (1)" which defines pre-sheaves as the contravariant functor from C^op to Set. I didn't understand why the author made this choice, as it seemed to make a lot of statements very awkward.After reading "Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity (2)", this choice is much clearer.
The key seems to be about sieves. In (2), pre-sheaves are defined covariantly. Consequently, the sieve is those functions from a which are the first in a chain that get somewhere. This is quite annoying, as it seems to put a lot of emphasis on the somewhere, which shouldn't be important. By defining pre-sheaves contravariantly, the arrows reverse. We get a sieve as all the functions that get to a, putting the emphasis on the object we are studying. Defining pre-sheaves contravariantly makes them conceptually easier to think about, at least with respect to sieves.
Are their other concepts in the theory of pre-sheaves which benefit from a contravariant definition? What other ways does the language I use to think about pre-sheaves (and categories more generally) affect how I understand the underlying concepts?
On a separate note, I am beginning to appreciate (1) a little more. My first impressions of the book were not favorable, and I only stuck with it because of the lack of alternatives. However, it gets a lot better after the first two chapters and I am beginning to appreciate the subtle knowledge of the author. Note to self: when writing a book, put all the info dumping in an appendix. Dry definitions are no way to hook a reader.
