Chapter 10 Notes
Sex and Gender Differences
Sex refers to biological characteristicsthe genetic, _hormonal_ and _anatomical_
differences between males and females. Gender is a social status referring to social differences
between the sexes, specifically t
Chapter 8 Notes
Understanding Social Stratification
Inequality is the unequal access to scarce goods or resources. Inequality is found in most, if not
all, societies. Social stratification refers to the _ranking of people_ according to their
wealth, power
Chapter 7 Notes
Deviance is any variation from a set of norms or shared _social expectations_.
The opposite of deviance is conformity, when people follow the norms of their social group or
society.
Societies everywhere have social controls to discourage c
Chapter 4 Notes
DEFINING CULTURE
Culture is a system of ideas, _values_, beliefs, knowledge, norms, _customs_, and
technology shared by almost everyone in a particular society.
Ethnocentrism
Cultural Relativism
Xenocentrism
Temporocentrism
ETHNOCENTR
Chapter 2 Notes The Development of Sociology
European Sociology18th Century: Sociology was imported to the U.S. to solve problems
of urban decay, crime, and poverty.
Rise of Industrialization
Using a scientific approach to studying society
Auguste Comte (
Chapter 3 notes
Whoever trusts in his own mind is a fool, but he who walks in wisdom will be delivered.
Proverbs 28:26 ESV
TYPES OF KNOWLEDGE
AUTHORITY
Some people are assumed to be knowledgeable simply because of their
_experience_ or position.
EXPERI
Chapter 5 notes
The terms, status and role, are central to understanding social structure.
A _status_ is a socially defined position someone occupies.
Status Set, Ascribed Status, Achieved Status, Master Status
A role consists of the _social expectations
Why Do Sociologist Conduct Research Outside The Laboratory?
Sciences such as biology and chemistry have been traditionally confined their research to
the laboratory for quite a few reasons. The primary reason biology and chemistry can be
confined in a lab
Chapter 5 notes
The terms, status and role, are central to understanding social structure.
A _status_ is a socially defined position someone occupies.
Status Set, Ascribed Status, Achieved Status, Master Status
A role consists of the _social expectations
COLLEGE PLANS ESSAY
The thought of achieving any sort of higher education has often
been an overlooked, or just plain disregarded idea in my family for
generations. Ive come from a long line of ancestors that labored
throughout life to make ends meet, oft
A successful life in America essay
Have you ever felt that there is not a solution for any problem or that you just dont
know why things happen? Well, everything happens for a reason. At first you might
feel upset or that there is no solution but in the e
10.4 Intervals in Acyclic Categories
163
Furthermore, we say that the set of morphisms between (m1 , m2 ) and
! 2 ) is indexed by the morphisms , rather than just taking the mor(m
! 1, m
phisms themselves, because the same morphism may give rise to dieren
9.2 Prodsimplicial Complexes
143
Definition 9.25. For two directed graphs T and G, a directed graph
homomorphism from T to G is a map : V (T ) V (G) such that
( )(E(T ) E(G).
Let A and B be the sets of vertices of two directed graphs T and G, and
let M to
10.5 Homeomorphisms Associated with the Direct Product Construction
171
product of two simplices [m] [n] , without introducing new vertices. See
the right part of Figure 10.16.
011
01
11
21
110
010
00
01
101
001
11
111
10
10
100
000
I2
I
00
3
20
[2]
[1]
F
10
Acyclic Categories
10.1 Basics
Many results in Combinatorial Algebraic Topology have until now been formulated in the context of posets. In this chapter we would like to emphasize
the more general framework of acyclic categories. As we shall see in sub
11.2 Discrete Morse Theory for CW Complexes
195
of a cone with apex in 2, and hence pairing cfw_2 gives a well-defined
acyclic matching with one critical cell cfw_2 in dimension 0.
If n = 3k, then we again have a face poset of the join of k copies of S0 .
11
Discrete Morse Theory
11.1 Discrete Morse Theory for Posets
When the set of cells of a CW complex is given by means of a combinatorial
enumeration, and the cell attachment maps are not too complicated, for instance if the CW complex in question is regu
11.2 Discrete Morse Theory for CW Complexes
189
Case 2. h f1 = f2 .
In fact, if h f1 = f2 , then it is much simpler to describe the homotopy
equivalence map f : X1 f1 X2 f2 . We may simply set
!
h(x), for x X1 ;
f (x) :=
(11.3)
x,
for x Int .
11.2.2 The M
12.1 Shellability
213
(1) The generalized simplicial complex obtained by the removal of the interiors
! := \ "
of the spanning simplices, that is, the complex
Int , is
collapsible. Even stronger, this complex can be obtained from a simplex by
a sequence
10.4 Intervals in Acyclic Categories
161
x
z
y
(C)
(Bd C)
Fig. 10.8. Nerves of the categories in Figure 10.7.
|Bd K|
= |(F(K)|
= |K|.
(10.5)
In particular, as an immediate corollary of (10.5) we see that an arbitrary
regular CW complex is homeomorphic t
11.2 Discrete Morse Theory for CW Complexes
191
Case 2. The cell is not critical.
In this case we must have (d(), ) M . Note that d() is maximal in F() \
! = \ (Int Int d().
cfw_, and let
Clearly, removing the pair (d(), ) is a cellular collapse; in part
10.2 The Regular Trisp of Composable Morphism Chains
155
regular trisps (P ) are always abstract simplicial complexes. On the other
hand, when P is a poset, the abstract simplicial complex (P ) is always a flag
complex; see Section 9.1.1; hence, for examp
10.6 The M
obius Function
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
0
0
1
1
1
1
1
177
Fig. 10.20. M
obius functions on morphisms of acyclic categories from Figure 10.1.
for all f : O(C) C, and
(f ) = ( )f,
(10.26)
for all f : O(C) C and , : M(C) C. Let x O(C). Th
11.2 Discrete Morse Theory for CW Complexes
199
The critical simplices form a subcomplex of Xn , which we call XnC . By
Theorem 11.13, there exists a sequence of elementary collapses leading from
Xn to XnC . Observe that if A = (a1 , . . . , at ) and B =
11.1 Discrete Morse Theory for Posets
185
S1 <U <U Sp <U T1 <U <U Tq
yields a cycle, contradicting the assumption that our matching was acyclic; in
the second case such a cycle is given by
S1 <U <U Sp <U Y <U T1 <U <U Tq .
Part (2) is straightforward. If
9.4 Chain Complexes
147
indexed with the number partitions of n; these are precisely the Sn -orbits of
the set partitions of n. Several collapsing sequences are readily seen in this
special case. We examine the case of general n in greater detail in Secti