Category Theory - what is its current state?

mardi 30 juillet 2013

When I was very young, I read a book from the 60's about classical mathematics. I was far too young to understand anything in it of course, but I remember it mentioned that category theory was the new thing that was threatening to become a new foundation for mathematics.

So I was wondering, has this happened? Is this happening? Has it become ubiquitous? And has it reached a point like ZFC has in that it is canonical and well understood? Or has it remained a sort of ad hoc merging of different areas ostensibly to allow one to carry results across from one area to another?

Quote:

A natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so-called functor categories.

This makes it sound like I can understand natural transformations better by seeing what they inherit from the category of functors. Or another example, by looking at what functors inherit in the category (2-category?) of small categories, this should help me to better understand functors. Does category theory have this trickle-down structure? Or does it perhaps have a trickle-up structure? I ask because of sentences like this next quote:

Quote:

For example, the 2-category Cat of categories, functors, and natural transformations is a doctrine. One sees immediately that all presheaf categories are categories of models.

Thanks.