Western University (Ontario)  Also known as University of Western Ontario
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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
Homotopy Theory
MATH 9151B

Winter 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