200B begins, and first day's topics

200B will focus on [Hungerford] chapters 10 (categories), 3,4,8 (commutative rings and modules). It is no longer in the Halkin room but in 7421.

First day (1/8):

Before starting categories...
Groups of permutations are certain special collections of functions.
Abstract groups aren't -- you don't "evaluate them on elements". You can only think about the group multiplication (and identity and inverses).

Definition of a category. Examples, consisting of collections of objects and certain special collections of functions, e.g. Set, Vec, Top, Grp, Top_*.
More abstract examples: any poset (where each hom-set has at most one element), any group (a single-object category), X/G where X is a G-set.

Definition: a product of two objects in a category.
We pretty nearly proved that products are unique up to unique isomorphism, but it got really rushed, so we'll attack it anew and slightly differently next time.