doctrines 2

some background on why i'm trying to (re?)invent a concept of "doctrine"...

in some of my semi-recent attempts to understand algebraic geometry, i found that an important aspect of what's going on is that the objects that algebraic geometers study tend to form themselves into 2-categories rather than 1-categories as you might naively expect.

(it could be that further study would show that n-categories for n>2 (or other sorts of higher categorical structures) are even more salient than 2-categories here, but 2 is at least a step on the way to n, and it could be in some sense the important step.)

thus naively you might expect that algebraic geometry is roughly the study of the objects in the category of projective varieties (and/or in a few related categories such as the category of "schemes"). to a projective variety x however is associated the category x# of so-called "coherent sheaves" over x, and it turns out that x# knows everything important there is to know about x, and that focusing on x# gives a purer and more illuminating picture of what's going on geometrically and conceptually than focusing on x does. then when you start axiomatizing what kind of object x# itself is, you realize that it's some kind of category-with-extra-structure and that this kind of categories-with-extra-structure will most usefully form a 2-category rather than a 1-category.

one reason