Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 010 (April 12, 2011)
24
Rigidity
We will prove Suslins rst rigidity theorem [7]:
Theorem 24.1. Suppose that i : k L is an
inclusion of algebraically closed elds, and that
n is a number such th
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 008 (March 9, 2011)
19
Ktheory with coecients
Suppose that n is some positive number. There is
a natural cobre sequence
p
n
E E E/n
in the category of spectra (or symmetric spectra),
where
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 005 (April 12, 2011)
10
Group completion
I say that a commutative diagram
/
X
Y
f
/
Z
W
of simplicial set maps is homology cartesian (for
some theory h) if there is a factorization
X BB
i /
V
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 004 (March 9, 2011)
8
The Ktheory spectrum
Recall that there is a poset isomorphism
0 n n + 1,
=
and write
=0 :m+10m0nn+1
=
=
for each ordinal number map : m n.
Let X be a simplicial set, an
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 002 (January 25, 2008)
2
Exact categories
Ill start with the canonical example.
Suppose that R is a unitary ring and let Mod(R)
of be the category of (left, right) Rmodules. Let
P(R) be the f
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 002 (January 27, 2014)
4
Spaces and homotopy groups
Write CGWH for the category of compactly generated weak Hausdor spaces.
A space X is compactly generated if a subset Z is
closed if and only
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 003 (January 26, 2011)
5
Waldhausens s construction
The basic denitions and results of this section
rst appeared in Waldhausens seminal paper [3].
Many of the tricks in the proofs which are g
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 001 (January 17, 2014)
1
Chain complexes
Suppose that R is a commutative ring with 1. In
most cases, R is either the integers Z or some eld
k.
First, we recall some basic denitions and facts
a
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 003 (January 29, 2014)
6
Example: Chain homotopy
Suppose that C is an ordinary chain complex. Let
C I be the complex with
I
Cn = Cn Cn Cn+1
for n > 0, and with
I
C0 = cfw_(x, y, z) C0C0C1  (x
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 001 (January 21, 2008)
1
Categorical homotopy theory
Much of the material in this section was invented
by Quillen to describe the Ktheory spaces, and
originally appeared in [3]. The theory ha
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 008 (April 7, 2014)
21
Bisimplicial sets
A bisimplicial set X is a simplicial object X :
op sSet in simplicial sets, or equivalently a
functor
X : op op Set.
Note that op op = ( )op. I shall w
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 009 (April 7, 2014)
24
24.1
Bisimplicial abelian groups, derived functors
Homology
Suppose that A : I Ab is a diagram of abelian
groups, dened on a small index category I. There
is a simplicia
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 006 (February 24, 2014)
14
Simplicial groups
Recall that a simplicial group G : op Grp
is a contravariant functor dened on the ordinal
number category, and taking values in the category
of gro
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 009 (April 12, 2011)
21
Some algebra: ltered and graded rings
A ring A is said to be ltered if the underlying
abelian group has a ltration
0 = F1(A) F0(A) i1 Fi(A) = A,
such that
Fp(A) Fq (A)
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 004 (February 10, 2014)
9
Simplicial sets
A simplicial set is a functor X : op Set,
ie. contravariant setvalued functor on the ordinal number category . Such things are usually
written n Xn,
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 006 (March 14, 2008)
13
Homotopy bres
Here is another consequence of the Additivity Theorem:
Theorem 13.1. Suppose that f : M N is
an exact functor. Then the square
s(S(M) S(N) ES(N)
sS(M)
f
/
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 005 (February 12, 2014)
12
Kan brations
Say that a map p : X Y is a Kan bration if
it has the right lifting property with respect to all
inclusions n n of horns in simplices.
k
A bration, as d
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 011 (April 7, 2014)
32
Postnikov towers
Suppose that X is a simplicial set, and that x, y :
n X are nsimplices of X. I say that x is
kequivalent to y and write x k y if there is a
commutativ
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 007 (January 29, 2014)
17
Proper model structures
Heres the governing principle in what follows:
Lemma 17.1. Suppose that f : X Y is a
morphism of a closed model category M, such
that both X a
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 007 (April 13, 2011)
16
Abelian category localization
Suppose that A is a small abelian category, and let
B be a full subcategory such that in every exact
sequence
0a aa 0
in A, a is an object
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 011 (April 12, 2011)
26
Ktheory of nite elds
We will sketch a proof of the following wellknown
result of Quillen [4]:
Theorem 26.1. Suppose that Fq is the eld
with q = pn elements. Then ther
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2014
Lecture 010 (April 7, 2014)
28
The fundamental groupoid
The path category PX for a simplicial set X is
the category generated by the graph X1
X0
of 1simplices x : d1(x) d0(x), subject to the
relation
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2009
Lecture 10 (January 6, 2011)
22
Localization for simplicial presheaves
Suppose that C is a small Grothendieck site, and
that S is a set of cobrations A B in the category s Pre(C) of simplicial preshea
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2009
Lecture 12 (March 27, 2009)
25
T spectra
Suppose that T is a pointed simplicial presheaf on
a small site C.
A T spectrum X is a collection of pointed simplicial presheaves X n, n 0, with pointed map
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2009
Lecture 006 (November 9, 2009)
11
Cohomology
Suppose that C is a chain complex of Rmodules.
1) There is an isomorphism
=
hom(R[n], C) Zn(C)
which is dened by the assigment f f (1), where
1 R = R[n]n
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2009
Lecture 04 (November 1, 2010)
10
Local weak equivalences
Suppose that C is a small Grothendieck site. Recall
that s Pre(C) and s Shv(C) denote the categories
of simplicial presheaves and simplicial sh
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2009
Lecture 005 (October 30, 2009)
10
Tensor products of chain complexes
Suppose that C is a chain complex of right Rmodules and that D is a complex of left Rmodules.
The tensor product C R D of these co
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2009
Lecture 02 (January 18, 2009)
5
Grothendieck topologies
A Grothendieck site is a small category C equipped
with a topology T .
A Grothendieck topology T consists of a collection
of subfunctors
R hom(
Western University (Ontario)  Also known as University of Western Ontario
Local Homotopy Theory & Homological Algebra
PHYSICS 2010

Spring 2009
Lecture 03 (September 23, 2010)
9
Rigidity
Suppose that k is an algebraically closed eld and
let be a prime which is distinct from the characteristic of k.
We will be working with the big tale site (S