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Homological algebra : the interplay of homology with distributive lattices and orthodox semigroups
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Homological algebra : the interplay of homology with distributive lattices and orthodox semigroups
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Description
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Title
Homological
algebra
: the
interplay
of
homology
with
distributive
lattices
and
orthodox
semigroups
Creator
Grandis, Marco.
Contributors
World Scientific (Firm)
DescriptionAbstract
In this
book
we
want
to
explore
aspects
of
coherence
in
homological
algebra
, that
already
appear
in the
classical
situation
of
abelian
groups
or
abelian
categories
.
Lattices
of
subobjects
are
shown
to
play
an
important
role
in the
study
of
homological
systems
, from
simple
chain
complexes
to
all
the
structures
that
give
rise
to
spectral
sequences
. A
parallel
role
is
played
by
semigroups
of
endorelations
. These
links
rest
on the
fact
that
many
such
systems
, but not
all
of them,
live
in
distributive
sublattices
of the
modular
lattices
of
subobjects
of the
system
. The
property
of
distributivity
allows
one
to
work
with
induced
morphisms
in an
automatically
consistent
way
, as
we
prove
in a '
Coherence
Theorem
for
homological
algebra
'.
(On
the
contrary
, a '
nondistributive
'
homological
structure
like
the
bifiltered
chain
complex
can
easily
lead
to
inconsistency
, if
one
explores
the
interaction
of its
two
spectral
sequences
farther
than
it
is
normally
done.)
The
same
property
of
distributivity
also
permits
representations
of
homological
structures
by
means
of
sets
and
lattices
of
subsets
,
yielding
a
precise
foundation
for the
heuristic
tool
of
Zeeman
diagrams
as
universal
models
of
spectral
sequences
.
We
thus
establish
an
effective
method
of
working
with
spectral
sequences
,
called
'
crossword
chasing
', that
can
often
replace
the
usual
complicated
algebraic
tools
and be of
much
help
to
readers
that
want
to
apply
spectral
sequences
in any
field
.
DescriptionTable Of Contents
Introduction
.
0.1
.
Homological
algebra
in a
nonabelian
setting
.
0.2
. The
coherence
problem
for
subquotients
.
0.3
. The
transfer
functor
.
0.4
.
Distributivity
and
coherence
.
0.5
.
Universal
models
and
crossword
chasing
.
0.6
.
Outline
.
0.7
.
Further
extensions
.
0.8
.
Literature
and
terminology
.
0.9
.
Acknowledgements

1
.
Coherence
and
models
in
homological
algebra
.
1.1
.
Some
basic
notions
.
1.2
.
Coherence
and
distributive
lattices
.
1.3
.
Coherence
and
crossword
diagrams
.
1.4
.
Coherence
and
representations
of
spectral
sequences
.
1.5
.
Introducing
pexact
categories
.
1.6
. A
synopsis
of the
projective
approach
.
1.7
.
Free
modular
lattices

2
.
Puppeexact
categories
.
2.1
.
Abelian
and
pexact
categories
.
2.2
.
Subobjects
,
quotients
and the
transfer
functor
.
2.3
.
Projective
pexact
categories
and
projective
spaces
.
2.4
.
Categories
with a
regular
involution
.
2.5
.
Relations
for
pexact
categories
.
2.6
.
Exact
squares
,
subquotients
,
induction
.
2.7
.
Coherence
,
distributivity
,
orthodoxy
.
2.8
.
Weak
induction
and the
distributive
expansion

3
.
Involutive
categories
.
3.1
.
ROcategories
.
3.2
. The
2category
of
ROcategories
.
3.3
.
Projectioncompletion
and
epimono
factorisations
.
3.4
.
REcategories
,
I
.
3.5
.
REcategories
,
II
.
3.6
.
REfunctors
and
REtransformations
.
3.7
.
Strict
completeness
of the
2category
of
REcategories

4
.
Categories
of
relations
as
REcategories
.
4.1
.
Puppeexact
categories
and
REcategories
.
4.2
.
Modular
relations
and
transfer
functors
of
REcategories
.
4.3
.
Complements
on
subquotients
and
regular
induction
.
4.4
.
Coherence
,
distributivity
and
orthodoxy
,
II
.
4.5
.
Idempotent
REcategories
.
4.6
.
Universal
distributive
and
idempotent
REcategories
.
4.7
.
Distributive
joins
in
inverse
categories

5
.
Theories
and
models
.
5.1
.
Graphs
and
REgraphs
.
5.2
.
REtheories
and
universal
models
.
5.3
.
Properties
of
REtheories
.
5.4
.
Universal
projective
models
.
5.5
.
Criteria
for
idempotent
theories
.
5.6
.
EXtheories
and
classifying
pexact
categories
.
5.7
.
Models
in
semitopological
spaces
.
5.8
.
Crossword
models
and the
Birkhoff
theorem

6
.
Homological
theories
and their
universal
models
.
6.1
. The
bifiltered
object
.
6.2
. The
sequence
of
relations
.
6.3
. The
bounded
filtered
chain
complex
.
6.4
.
Applications
.
6.5
. The
real
filtered
chain
complex
.
6.6
. The
double
complex
.
6.7
.
Eilenberg's
exact
system
.
6.8
. The
discrete
exact
system
and
Massey's
exact
couple
.
6.9
.
Some
nonidempotent
theories
.
Publisher
World Scientific Pub. Co.
Subject
Algebra, Homological.
Lattices, Distributive.
Identifier
9789814407076
(electronic
bk.)
;
9814407062
;
9789814407069
;
http://www.worldscientific.com/worldscibooks/10.1142/8483#t=toc
Language
eng
Type
Electronic books.
FormatExtent
xi, 369 p. : ill.
Date
c2012
.
OCLC number
874498009
CONTENTdm number
507
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