I’ve just had the finest accomplishment of my mathematical career. I was trying to remember the definition of Jategaonkar’s second layer condition, which arises in noncommutative ring theory when studying the analogue of localizing at a prime ideal. So I typed Jategaonkar into Google, and lo and behold, I spelled it right the first time. (I didn’t say this was much of an accomplishment; just my finest.)

(As for the actual definition, I had less luck. The best I could find was this Ph. D. thesis, by Paul Chong-Hyun Kim.)

Kenny Easwaran and David Corfield are discussing whether mathematical concepts have essences. Kenny’s example is that of a normal subgroup of a group, that while it can be defined several different ways, the essence of the notion is that it is the kernel of a group homomorphism. David relates the question to his larger program to redirect the philosophy of mathematics away from its traditional concerns towards elucidating the meaning of mathematical concepts.

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.