9/27
adjunction as initial object in a comma category
Suppose we have
categories C, D
functors
F: C -> D
U: D -> C
an adjunction
tau: F -> U
that is to say, for any c in |C|, d in |D|,
natural in c and d
Then for any c in |C|,
define
i_c = tau_cfw_c,Fc(1_
9/16
bundles
fibre bundles
A fibre bundle consists of
a total space E
a base space B
a projection pi: E -> B
a fibre F
a collection I of isos from F to F
a collection C of open subsets in B with, for each U in C, an iso
phi_U : pi^cfw_-1U -> U x F such th
8/28
We found that these categories have products for all pairs of objects:
Group
Top
Set
R-Mod
In each case what had to be checked was the following:
The following objects and maps exist in the category:
A <- C -> B
For any
A <- AxB -> B
the induced map
9/23
comma categories
For a category C and object c in C, we have the "comma" category
(C,c)
objects are maps b -> c in C which go to c
morphisms are maps b -> b' in C so that the triangle commutes
example:
VecBun = category of vector bundles pi: E -> B
m
8/26
A category C consists of:
a class of objects |C|
for each X and Y in |C|, a set of morphisms
C(X,Y)
hom(X,Y)
f: X -> Y
or
or
and a partially-defined binary operation, composition, represented by o,
satisfying
f o g is defined whenever g in C(X,Y) and
9/18
Gdel-Bernays axioms
The Gdel-Bernays model for set theory is also called the theory of classes.
In this model, the most general objects are classes, not sets.
There is an undefined propositional function Set( )
Set(x) is to be interpreted as "x is a
9/13
Lie brackets and maps
Proposition:
Let theta: M -> N be a map in C^k-Man.
Suppose X and Y are in X(M), X' and Y' in X(N) such that
for all p in M,
theta_*X_p = X'_cfw_theta p
theta_*Y_p = Y'_cfw_theta p
Then for all p in M,
theta_*[X,Y]_p = [X',Y']_c
10/2
V*(x)W and Hom(V,W)
Let V and W be vector spaces over a field k (or R-modules over a commutative
ring R).
First note that there is a naturality in the tensor construction:
For
f: V -> V'
g: W ->W'
we have
f(x)g: V(x)W -> V'(x)W'
defined by
(f(x)g)(v
9/20
sheaf of germs of functions
There is a contravariant functor Open from Top to Cat (so Open^op : Top^op ->
Cat):
Open(X) = category with objects cfw_open sets in X and morphisms
inclusion maps
for f: X -> Y in Top
Open(f) : Open(Y) -> Open(X) is the f
9/6
op categories
For any category C, we can define the category C^op:
The objects of C^op are the objects of C.
For every X and Y in |C|,
For every X in |C|,
C^op(X,Y) = C(Y,X)
(1_X)^op = 1_X
For every f in C^op(X,Y) and g in C^op(Y,Z)
so that means f in
9/25
graphs
a (directed) graph G consists of
a set of objects O_G
a set of arrows A_G
two maps
d_0: A_G -> O_G
d_1: A_G -> O_G
this gives the domain of each arrow
this gives the codomain of each arrow
note:
nothing in here about composition
nothing about
9/11
natural transformations
Suppose C and D are categories and F: C -> D and G : C -> D are functors.
A natural transformation t: F -> G is a collection of maps t_A: FA ->
GA in D for each A in |C| such that
for all f: A -> B in C,
t_B o Ff = Gf o t_A
ex