If you’re comfortable with the language of schemes and representable functors, you can find a nice introduction to Grothendieck topologies here: Notes on Grothendieck topologies, fibered categories and descent theory, by Angelo Vistoli.

A few universal algebra links, collected by Ralph Freese. (Universal algebra is the branch of mathematics that studies generalizations of groups, rings, and algebras. The hallmark of each family of objects that can be studied by universal algebra is that it satisfies a version of Noether’s isomorphism theorems.)

Another triumph for Wikipedia. Every time I do a web search, I’m impressed with the progress Wikipedia is making. Check out their dry but complete summary of Grothendieck topologies, and their more evocative Background and genesis of topos theory.

Is category theory the savior of mathematics, or its destroyer? Discuss.

I have unaccountably forgotten to mention that the June/July Notices of the AMS are available. In addition to the feature articles (about Ramanujan and the Clay Institute), there is a review of Roger Penrose’s The Road to Reality. Penrose attempts the obviously impossible task of providing a complete survey of all of modern physics (including string theory and loop quantum gravity) accessible to a nontechnical audience, but with mathematical details. I thought the review was oddly negative: Penrose is attempting something of immense ambition, ambition that cannot help but fall short of its goal, so it’s not surprising that he doesn’t fully succeed.

This month’s What is… is about coarse spaces, which are a new concept to me.