Mon Jan 29

The category R-Mod of left R-modules and R-linear maps.
Z-Mod = abelian groups.
C-Mod = complex vector spaces.
C[x]-Mod = complex vector spaces equipped with specific endomorphisms.
Isomorphism classes in this last category correspond to Jordan canonical forms.

Submodules, quotient modules. Infinite products vs. infinite direct sums.
Left ideals.
Given m in an R-module M, we get an R-linear map R -> M, whose kernel is a left ideal.
Conversely, given a left ideal I, we get a map R -> R/I of R-modules (not rings).

I claimed that R is a (R x R^op)-module. Not so. It is a left R-module, and a left R^op-module, but we can't glue them together so easily; this will await tensor products.

Mon Jan 29

The category R-Mod of left R-modules and R-linear maps.
Z-Mod = abelian groups.
C-Mod = complex vector spaces.
C[x]-Mod = complex vector spaces equipped with specific endomorphisms.
Isomorphism classes in this last category correspond to Jordan canonical forms.

Submodules, quotient modules. Infinite products vs. infinite direct sums.
Left ideals.
Given m in an R-module M, we get an R-linear map R -> M, whose kernel is a left ideal.
Conversely, given a left ideal I, we get a map R -> R/I of R-modules (not rings).

I claimed that R is a (R x R^op)-module. Not so. It is a left R-module, and a left R^op-module, but we can't glue them together so easily; this will await tensor products.

HW #3 due Fri Feb 2 -- corrected

1a. Let f:R->S be a ring homomorphism, and M a left S-module. Show that M has also the structure of an R-module.

b. Even better... show that there is a functor S-Mod -> R-Mod (the categories of left modules), where the above says what to do with the objects.

c. Even better... show that there is a functor Ring^op -> Cat, taking R |-> R-Mod.

2. Let M be a left R-module, and A,B two submodules. Show that the abelian subgroups A+B, A intersect B are also submodules.

3a. Let b be an element of the ring R. Show that the smallest 2-sided ideal containing b is RbR := {sum_i r_ibs_i}.

b. Let R = M_n(C), nxn complex matrices, and b a nonzero matrix. Show that RbR = R, i.e. this R is a "simple" ring.

4. Let A,B be two abelian groups. Show that Hom(A,B) is not only an abelian group, but a left End(A)^op x End(B) module End(B)-module, and also a left End(A)^op module.
(We will later show it to be a left End(A)^op tensor End(B) module.)

5. Recall the definitions of initial (resp. terminal) object X in a category; there exists a unique morphism out of X to (resp. into X from) any other object Y.

Find an initial and a terminal object in the category Ring = {rings with units, s.t. homomorphisms preserve the units}. (As the example Set shows, they need not be the same object.)

Office hours

Wednesday after class, 2-2:45
Thursday morning 11-12
Anytime you find me (10-2 most likely) if you're lucky
7450 APM

Wed Jan 24

(2-sided) ideals are exactly the kernels of ring homomorphisms.

The "multiplicative group" functor Ring -> Grp. Note that this only makes sense if ring homomorphisms are required to preserve the unit.

3 defs of G-set:
1. functor from C_G to Set
2. action map G x X -> X satisfying some properties
3. group homomorphism G -> Sym(X).

3 defs of left R-module:
1. additive functor from C_R to Ab
2. action map R x M -> M satisfying some properties
3. ring homomorphism R -> End(M).

Examples of modules. Sections of the Mobius bundle over the circle.

Mon Jan 22

The category Ab of abelian groups is an "additive category".
The inclusion Ab -> Grp has a left adjoint, "abelianization", G |-> G/G'.

Categorical def: a ring is a 1-object additive category.
Unpacked, we get the usual definition of ring-with-unit.

Def: additive functor. Consequently, we get a def: ring homomorphism. (Preserving unit!)

Some examples of rings: M_n(C), Z[i], Fun(X -> Reals), Z[x-hat, d/dx].

HW due Friday Jan 26 -- corrected

1. Define a category Graph whose objects are digraphs, and define a forgetful functor Cat -> Graph.

2. Show that the "free category on a digraph" (from last HW) extends to a functor Graph -> Cat.

3. Show that these two functors (in #1 and #2) are adjoint.

4. Call a category a pre-poset if any two objects A,B have at most one morphism A->B. This defines a subcategory Pre-poset of Cat. Show that the inclusion functor Pre-poset -> Cat has a right left adjoint Cat -> Pre-poset. (In particular, figure out how to assign a pre-poset to a category.)

[This left adjoint is almost a right adjoint... there is a natural transformation from one of the relevant set-valued hom functors to the other; it's just not an isomorphism.]

5. Define the cardinality |C| of a category C as a sum over the isomorphism classes of objects, of 1 / |Aut(X)|. (This definition rarely makes sense -- Aut(X) may be infinite, or the sum may diverge -- but it's cool when it does.)

a. Check that |FinSet| = e, where FinSet is the category of finite sets.

b. Let X be a G-set (both finite), and X/G the category defined in class, whose objects are X and Hom(x,y) = {g : gx = y}. Show that |X/G| = |X|/|G|.

(If you think this stuff is cool, you might check out this chatty paper for lots more about it!)

6. Let C be a category that has products. For every X,Y objects in C, pick a product of X and Y; call this choice P : CxC -> C. Show that P can be extended to a functor.

7. Let Mon be the category of monoids, and Grp -> Mon the inclusion functor. Find left and right adjoints of this functor (two ways of assigning a group to a monoid).

Fri Jan 19

The "opposite category" is a functor Cat -> Cat.

The functors Hom(X,*) : C -> Set and Hom(*,X) : C^op -> Set.

Natural isomorphisms.

The product of two categories.

Given functors C->D or D->C, we get functors C^op x D -> Set.
If they're naturally isomorphic, call the functors adjoint.

Left adjoints of several forgetful functors. The right adjoint of forget : Top -> Set.

Wed Jan 17

Recall: group = a set + three operations + some axioms. Group homomorphism: a function preserving the operations.
Recall: category = two sets (objects and morphisms) + two operations + some axioms. Define a functor as two functions preserving the operations.

Some examples:
Forgetful functors (-> Set).
The vector space with given basis (Set -> Vec).
The one-object category made from a group (Grp -> Cat, G |-> C_G).
G-sets (C_G -> Set).

Natural transformations of functors.

TA office hours and section

"Office hours are 1-2pm Tuesday in AP&M 6446 this quarter. Also, graduate TAs are supposed to hold sections this quarter. I decided on 4-5 pm Wednesday as a time, and they said we could hold the section in the basement calc lab."

HW #1 due Friday Jan 19 -- major correction

1. Prove the assertion from class that "disjoint union with basepoints amalgamated" really is the coproduct in the category of sets-with-basepoints.

2. A directed graph or digraph (V,E,h,t) is a pair of sets V and E (called "vertices" and "edges") plus two functions h,t: E -> V called "head" and "tail".
(In particular, these digraphs may have repeated edges, and h(e)=t(e), whatever.)

a. Given a category, come up with a definition of "the underlying directed graph".

b. Given a directed graph, come up with a (more interesting) definition of "the free category on the directed graph".

You can check that you have the right notion using the following:

c. Given a poset P, define a directed graph G such that the free category on G is the category we associated to P. Oops, this isn't quite possible for infinite P. So assume P finite.
OOPS. Not any poset will do. Assume that there is at most one chain p > q > ... > s connecting any two elements p,s.

d. Let v,w be two vertices of a directed graph G. If the corresponding objects in the free category on G are isomorphic, show v=w.

3. Define a monomorphism f as one such that whenever f o g = f o h, it follows that g = h. (Here f,g,h are various morphisms in C for which o is defined.)

a. Let C = Set. Show f is monic (i.e. a monomorphism) iff f is 1:1.

b. (Harder.) Let C = Grp. Show f is monic iff f is 1:1.

Probably some more as I think of them; Monday at the latest.

Wed Jan 10 -- NO CLASS FRIDAY OR MONDAY

The opposite category to a category.
Terminal objects. Which are...
"Unique up to unique isomorphism." Some examples. Initial objects.
Given A,B in Obj(C), we defined a category C_A,B where a terminal object is a product of A and B.
Coproducts.
Coproduct in Set = disjoint union.
Coproduct in Set_* = disjoint union with basepoints amalgamated.
Coproduct in Grp = free product of groups.

No class on Friday (I am attending a funeral) or Monday (MLK birthday), so see you next Wednesday.

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.