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 that (n, char(k) = 1. Then
the induced map
i : K(k, Z/n)
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 n is multiplication by n, meaning the map
in the stable
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
BB
BB
p
f BB!
W
with i a trivial cobration and p a bra
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, and write EX for the
simplicial set with
EXn = Xn+1 = X(0
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 full subcategory of Mod(R) whose
objects are the nitely
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 if Z K is closed for all maps
K X with K compact.
The
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 given
here appear in [1] and [2].
Suppose that C is some
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
about Rmodules.
Suppose that f : M N is an Rmodule
ho
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  (xy)+(z) = 0 .
I
I
The boundary map : Cn Cn1 is dened by
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 has been
rened and extended over the years (see [1], for
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 write
Xm,n = X(m, n)
for the set of bisimplices in bidgr
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 simplicial abelian group EI A, with
EI An =
A(i0)
:i0 in
and wit
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 groups. A morphism of simplicial groups is a
natural trans
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) Fp+q (A)
under the ring multiplication, and 1 F0(A). Th
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, and Xn is called the set of nsimplices of X.
A simplici
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
/
sES(N)
/
d
sS(N)
is homotopy cartesian.
Recall from Se
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 dened in Section 11 (Lecture 004),
is obviously a Kan br
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
commutative diagram
i
skk n
i
n
x
n
/
/
y
X
or in other words if
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 and Y are cobrant. Then f has a
factorization
X i GZ
2
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 of C if and only if b and a are
objects of B. Such a s
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 there are isomorphisms
Ki(Fq ) =
Z/(q j 1)
if i = 2j 1, j >
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
relations
d1() = d0() d2()
given by the 2simplices of X. There
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 presheaves on C.
Im going to assume throughout this section th
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 maps
: T X n X n+1. A map f : X Y of
T spectra consists
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 is the multiplicative identity of
the ring R.
2) A chai
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 sheaves on
C, respectively.
Recall that a simplicial set
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 complexes is
the chain complex with
(C R D)n =
p+q=n
(Cp
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( , U ),
U C,
called covering sieves, such that the follo
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 (Schk )et
e
over the eld k throughout this section. Note