Introduction to sheaves using categorical approach

Question

When I first started to learn about sheaves, it was a very geometric approach. This is nice, but it seems like knowing more abstract categorical approach is very useful.

For example, sheafification $(\mathcal{F})^+$ of a presheaf $\mathcal{F}$ became much more clear to me when I realized that it can be defined as the left adjoint functor $(-)^+\colon \mathsf{PSh}(X)\to \mathsf{Sh}(X)$ to the forgetful functor $i\colon \mathsf{Sh}(X)\to \mathsf{PSh}(X)$, where $i$ views any sheaf as a presheaf. Here $X$ is a topological space, and $\mathsf{Sh}(X)$ and $\mathsf{PSh}(X)$ are categories of sheaves and presheaves on $X$ respectively.

So my question is: do you know any nice introduction to sheaves that would not be too crazy abstract, but which will use categorical definitions, constructions and proofs (where possible)?

Thank you very much!

Answer 1

I have three suggestions:

Mac Lane, S., and Moerdijk, I., "Sheaves in Geometry and Logic: A First Introduction to Topos Theory"

Kashiwara, M., and Schapira, P., "Categories and Sheaves"

The first is my favorite. The latter is more advanced, and doesn't really start talking about sheaves until late in the book. It's a quality text nonetheless.

Finally, Angelo Vistoli's notes on descent theory have a nice discussion of sheaves (with algebraic geometry in mind) in the second chapter.

Answer 2

I suppose what you are looking for is an introduction to classical topos theory. A topos is a category which can be thought of as a category of sheaves. One thing to keep in mind when reading about topos is that there are three equivalent definitions. The first is a left exact localization of a presheaf category. The second constructs sheaves from a Grothendiek topology on a site. The last way is more intrinsic and characterizes a topos category using Giraud's axioms. Depending on the author, these objects could be introduced using any one of these definitions. The above suggestions are good, but I might add

Robert Goldblatt, "Topoi, the Categorial Analysis of Logic."

Michael Barr and Charles Wells, "Toposes, Triples and Theories."

Also, John Baez has a non technical overview in a short exposition called "Topos Theory in a Nutshell" which you can find online.

If you get through these and want to see where the theory has gone, you could try to tackle Jacob Lurie's book "Higher Topos Theory" (chapters 6-8). This is still an active area of research and has applications to string theory, algebraic and differential K theory and a lot of other areas.