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Journal Article•DOI•

The two-square lemma.

Temple H. Fay, K.A. Hardie, Peter Hilton
01 Jan 1989-Publicacions Matematiques (Servei de Publicacions)-Vol. 33, Iss: 1, pp 133-137
About: This article is published in Publicacions Matematiques.The article was published on 1989-01-01 and is currently open access. It has received 19 citations till now. The article focuses on the topics: Lemma (mathematics).

Summary (1 min read)

Jump to:  and [Consider now the commutative diagram]

Consider now the commutative diagram

  • Here the exactness of the top row is obtained by dualizing the argument just given, noting that the definition of 77 is evidently self-dual .
  • The final ingredient for the Snake Lemma is the following Triangle Lemma .

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Publicacions
Matemátiques,
Vol
33
(1989),
133-137
.
Abstract
THE
TWO-SQUARE
LEMMA
TEMPLE
H
.
FAY,
KEITH
A
.
HARDIE,
AND
PETER
J
.
HILTON
A
new
proof
is
given
of
the
connecting
homomorphism
.
One
of
the
most
useful,
and
hence
important,
lemmas
of
homological
algebra
in
an
abelian category
(say,
the category
of
R-modules)
is
the
Snake
Lemma
(see
the
references
at
the
end
of
this
note)
.
This
result
is
used
to
produce
connecting
homomorphisms,
building
long
exact
sequences
such
as
the
familiar
ones involving
the
Hom
-
Ext and
®
-
Tor
functors
.
According
to
Mac
Lane
[M1],
early
proofs
of
this
lmma
were obscure
.
One
technique,
used
by
Popescu
[P],
is
to
prove
the
result
for
abelian
groups
and
use
the
Mitchell
Embedding
Theorem
to
obtain
the
result in
other
abelian
categories
.
The
essential
point
is
the construction
of
the
homomorphism
to
do
the
connecting, not
the exactness
which
comes
from
simple
diagram
chasing
.
In
this
note
we
give
an
interesting
new
result,
which
we
call
the
Two-Square
Lemma,
that
provides
a
simple
and
completely
categorical
construction of the
snake
connecting
morphism
.
This
has
the
pleasant
property
of
avoiding
all
references
to
zig-zags
or
Switchback
formulas
and
to such
powerful
tools
as
Mitchell's
Embedding
Theorem
.
We
shall
do
all
our
work
in
an
abelian
category,
although
some
relaxing
of
the
axioms
might
be
possible,
say
to
an
exact
category
.
We
begin
by
stating
two
lemmas
which
are
well-known
and
fundamental
.
Indeed
apart
from
these
two
lemmas
we
require
only
the
notions
of
exactness,
pullback
and
pushout
in
a
balanced
category
.
We
record
that
Beyl
[B]
has
also
given
a
purely
categorical
description
of
the
connecting
homomorphism
.
He
bases
his
work on
the
Product
Lemma
which
is
our
Lemma
4,
and
does
not
enunciate
a
Two-Square
Lemma
.
Lemma
1
.
Consider
the
commutative
diagram
05
B
al
pb
m'
A'
--~
B'
.
C
7
C'
134
T
.H
.
FAY,
K
.A
.
HARDIE,
P
.J
.
HILTON
with
¡he
botiom
roca
exact
.
Then
there
exists
a
unique
morphism
:
A'
--
.>
B
such
that
(ZJ
N
Y'
=
Y',~
and
(ii)
A'
P)
B
-~-+
C
is
exact
The
dual
of
this
result
is,
of
course,
also
valid
.
Lemma
2
.
(The
Sharp
3 x 3
Lemma)
.
Consider
¡he
following
commutative
diagram
having
exact
columns
and
rocas
.
is
exact
.
0
>
A2
There
are uniquely
determined
morphisms
K
l
+
K2
and
K
2
-->
K3
such
that
the
completed
diagram
is
commutative
and
the
sequence
>B
2
i
.
C
2
K
3
Again, the dual
of
this
result
is
valid
.
Lemma
2
and
its
dual
provide
the
two
halves of
the
long exact
sequence
which
the
Snake
Lemma
connects
up
.
The
next
lemma
is
the
one
cae
wish
to
emphasize
.
Lemma
3
.
(The
Two-Square
Lemma)
Let
the
following
commutative
dia-
gram
have
exact
rocas
.
Form
the
pulí-back
of
y
and
0',
and
the
push-out
of
0
and
a,
thus
O
T
al
p,
Q
fil
oi
17
,,~~
..
%
~
p,
m,
pb
A'
3
B'
)
C'
TWO-SQUARE
LEMMA
13
5
Then
(i)
there
exists
a
unique
0
:
Q
-+
B'
such
that
9r
=
/3,
Br'
(ii)
there
exists
a
unique
p
:
B
-~
P
such
that
op
=
0,
o
'
p
=
/3
;
(iii)
there
exists
a
unique
rl
:
Q
-~
P
such
that
r/r
=
p,
o'i
=
0,
or7T'
=
0
.
Moreover,
rl
is
monic
if
zo'
is
monic,
and
17
is
epic
if
0
is
epic
.
Proof
Assertions
(i)
and
(ii)
are
obvious
.
To
prove
(iii),
we
first
use
Lemma
1
to find
a
unique
tt
:
A'
-->
P
such
that
o'p
=
'
and
A'
'>
P
-°>
C
is
exact
.
We
then
claim
that
aa
=
po
.
ForotccY=0=00=opo,
ando'pa=Tp'a=Po=o'pO
.
Thus
there
exists
a
unique
77
:
Q
-->
P
such
that
y/T
=
p,
i7r'
=
ti
.
But
then
o77T'
=
op
=
0,
and
o'r/r
=
o'p
=
(~
=
Br,
o'77r'
=
o'u
_
o'
=
Br', so
that
o'ir/
=
0
.
Conversely,
if
rl
:
Q
+
P
satisfies rlr
=
p, o'ij
=
0,
orlr'
=
0,
then
ortr'
=
0
=
op,
o'i7r'
=
BT'
=
o'
=
o'M,
so that
177-'
=
Es,
and
the
uniqueness
of
rl
satisfying
(iii) is
established
.
Consider
now
the
commutative
diagram
Here
the
exactness
of
the
top
row
is
obtained
by
dualizing the
argument
just
given,
noting
that
the
definition of
77
is
evidently
self-dual
.
If
0'
is
monic,
so
is
p
=
71r'
.
Hence,
by
the
above
diagram,
so
is
rl
.
Dually,
if
0
is
epic,
so
is
77
.
The
final
ingredient
for
the
Snake
Lemma
is
the
following
Triangle
Lemma
.
Lemma
4
.
(Triangle
Lemma)
Given
the
Trangle
there
is
an
exact
sequence
ker
(~
->
ker
>
coker
l
;
->
coker
Here
all
morphisms,
except
possibly
w,
are
obvious
and
w
is
¡he
composite
ker
(
-)
L
->
coker
136
T
.H
.
FAY,
K
.A
.
HARDIE,
P
.J
.
HILTON
of
¡he
embedding
and
the
projection
.
Proof
.
Exactness
at ker
(
follows
from
Lemma
2
applied
to
the
diagram
is
exaci
.
a
By
duality
we
have
also
exactness
at
coker
1
.
A'
)
B'
>
C'
We
are
now
ready
to
show
how
the
Snake
Lemma
follows
easily
from
the
Two-Square
Lemma
.
Lemma
5
.
'(The
Snake
Lemma)
Given
the
commutative
diagram
with
exact
rows
there
is
a
connecting
morphism
w
:
ker
y
->
coker
a
such
that
the
sequence
ker
a
-->
ker
f3
--
.>
ker
y
w
>
coker
a
->
coker
/i
-+
coker
y
Proof
:
The
exactness
of
ker
a
->
ker
a
-+
ker
y
and
coker
a
-->
coker
/l
->
coker
-1
are
just
Lemma
2
and
its
dual
.
Adopting
the
notation of
the
Two-
Square
Lemma,
we
first
apply
Lemma
1 to
obtain
a
(monic)
te
:
ker
y
->
P
such
that
urc
is
the
embedding
ker
y
->
C
and
ker
y
r
>
P->
B'
is
exact
.
Dually
we
find
an
epic
A
:
Q
+
coker
a
such
that
A-r'
is
the projection
A' ->
coker
a
and
B
T
>
Q
IX)
coker
a
is
exact
.
Exploiting
the
isomorphism
il
:
Q
->
P
and
the
commutative
triangle
B
~
277-
p
=
6rl-1
0 0 0
ker
ker
coker
11
K
L
coker
M M
-
0
.
and
the
proof
is
complete
.
T
WO-SQUARELEMMA
13
7
we
have
coker
p
=
coker
T
=
coker
a,
ker
u'
=
ker
y,
so
that
Lemma
4
yields
the
exactness
of
ker
,(3
-+
ker
,y
coker
a
-3
coker
P,
Acknowledgements
.
The
second
author
acknowledges
a
grant
to
the
To-
pology
Research
Group
from
the
South
African
Council
for
Scientific
and
In-
dustrial
Research
.
References
[B]
F
.
R
.
BEYL,
The
connecting
morphism
in
the
Kernel-Cokernel
sequence,
Archiv
de
Math
.
32
(1979),
305-308
.
[HS]
P
.
J
.
HILTON
AND
U
.
STAMMBACH,
"A
Course
in
Homological
Álge-
bra,"
Springer,
New
York,
Heidelberg,
Berlin,
1970
.
[M1]
S
.
MAC
LANE,
"Categoriesfor
¡he
Working
Mathematicáan,"
Springer,
New
York,
Heidelberg,
Berlin,
1971
.
[M2]
S
.
MAC
LANE,
"Homology,"
Springer,
New
York,
Heidelberg,
Berlin,
1963
.
[M]
B
.
MITCHELL,
Theory
of
Categories,
Academic
Press
(1965),
London,
New
York
.
[P]
N
.
POPESCU,
Abelian
Categories
with
Applications
to
Rings
and
Mo-
dules,
Academic
Press
(1973),
London,
New
York
.
[H]
H
.
SCHUBERT,
"Categories,"
Springer,
New
York,
Heidelberg,
Berlin,
1972
.
Temple
H
.
Fay
:
Department
of
Mathematics
Univ
.
of
Southern
Mississippi
Hattiesburg,
MS
39406
USA
Peter
J
.
Hilton
:
Department
of
Math
.
Sciences
State
Univ
.
of
New
York
Binghamton,
NY
13901
USA
Keith
A
.
Hardie
:
Department
of
Mathematics
Univ
.
of
Cape
Town
Rondebosch
7700
SOUTH
ÁFRICA
Rebut
el
16 de
juny
de
1988
Citations
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Maciej Borodzik1, Stefan Friedl2, Mark Powell3•
University of Warsaw1, University of Regensburg2, Université du Québec à Montréal3
19 Jul 2016-Journal of The Mathematical Society of Japan
TL;DR: In this article, the authors used invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers.
Abstract: Given a link in S3S3 we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi. We give an application restricting the knot types which can arise from a sequence of splitting operations on a link. This allows us to answer a question asked by Colin Adams in 1996.

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Abelian categories in dimension 2

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Mathieu Dupont
10 Sep 2008-arXiv: Category Theory
TL;DR: In this paper, the authors define a 2-dimensional version of abelian categories, where symmetric 2-groups play the role that abelians played in 1-dimensional algebra.
Abstract: The goal of this thesis is to define a 2-dimensional version of abelian categories, where symmetric 2-groups play the role that abelian groups played in 1-dimensional algebra. Abelian and 2-abelian groupoid enriched categories are defined and it is proved that homology can be developed in them, including the existence of a long exact sequence of homology corresponding to an extension of chain complexes. This generalises known results for symmetric 2-groups. The examples include, in addition to symmetric 2-groups, the 2-modules on a 2-ring, which form a 2-abelian groupoid enriched category. Moreover, internal groupoids, functors and natural transformations in an abelian category C (in particular, Baez-Crans 2-vector spaces) form a 2-abelian groupoid enriched category if and only if the axiom of choice holds in C.

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Abelian categories in dimension 2

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Mathieu Dupont
01 Sep 2008
TL;DR: In this paper, a 2-dimensional version of the notion of abelian categories was introduced, and the theory of exact sequences and homology in 2-dimensions was developed.
Abstract: In 2-dimensional algebra, symmetric 2-groups (symmetric monoidal groupoids in which every object has an inverse up to isomorphism) play a similar role to that of abelian groups in 1-dimensional algebra. Since abelian categories are defined in the image of the category of abelian groups, a 2-dimensional version of the notion of abelian category should be a solution to the equation abelian categories abelian groups = ?? symmetric 2-groups . I give two solutions to this equation. The first — abelian groupoid enriched categories — is a generalisation of ordinary abelian categories. In such a context, we can develop the theory of exact sequences and homology in a way close to homology in an abelian category: we prove several classical diagram lemmas as well as the existence of the long exact sequence of homology corresponding to an extension of chain complexes. This generalises known results for symmetric 2-groups [18, 26]. The other solution — 2-abelian groupoid enriched categories, which are also abelian in the sense of the previous paragraph — mimics the specifically 2dimensional properties of symmetric 2-groups, in particular the existence of two factorisation systems: surjective/full and faithful, and full and surjective/faithful [52]. Moreover, in a 2-abelian groupoid enriched category, the category of discrete objects is equivalent to that of connected objects, and these categories are abelian. This is close to the project of Marco Grandis of “developing homotopical algebra as an enriched version of homological algebra” [40]. The examples include, in addition to symmetric 2-groups, the “2-modules” on a “2-ring”, which form a 2-abelian groupoid enriched category. Moreover, internal groupoids, internal functors and internal natural transformations in an abelian category C (which includes as a special case Baez-Crans 2-vector spaces on a field K [4]) form a 2-abelian groupoid-enriched category if and only if the axiom of choice holds in C.

18 citations

Cites methods from "The two-square lemma."

  • ...To prove the snake lemma, we will follow the proofs of [12] and [33]....

    [...]

  • ...This has been done by Grandis [40]; the equivalence between the two constructions of homology is a form of the two-square lemma, as it was stated in [33]....

    [...]

  • ...Here is now an ad hoc 2-dimensional version of the two-square lemma of Fay, Hardie and Hilton [33], which is stated without name by Beyl [12], and was already in Mitchell’s book [63, section VII....

    [...]

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On diagram-chasing in double complexes

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George M. Bergman1•
University of California, Berkeley1
03 Aug 2011-arXiv: Category Theory
TL;DR: In this paper, for any double complex in an abelian category, certain short-distance maps and an exact sequence involving these, instances of which can be pieced together to give the "long-distance" maps and exact sequences of results such as the Snake Lemma.
Abstract: We construct, for any double complex in an abelian category, certain "short-distance" maps, and an exact sequence involving these, instances of which can be pieced together to give the "long-distance" maps and exact sequences of results such as the Snake Lemma. Further applications are given. We also note what the building blocks of an analogous study of triple complexes would be.

14 citations

Cites background from "The two-square lemma."

  • ...n the above proof fulfill the promise that “long” connections would be reduced to composites of “short” ones. The proofs of the next three lemmas continue this theme. Lemma 11 (Snake Lemma, [1, p.23], [4], [9, p.158], [11, p.50]). If in the commuting diagram at left below, both rows are exact, and we append a row of kernels and a row of cokernels to the vertical maps, as in the diagram at right, (16) ...

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The snail lemma

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Enrico Vitale
01 Jan 2016-Theory and Applications of Categories
TL;DR: The classical snake lemma produces a six terms exact sequence starting from a commutative square with one of the edge being a regular epimorphism as discussed by the authors, which is a condition that cannot be removed.
Abstract: The classical snake lemma produces a six terms exact sequence starting from a commutative square with one of the edge being a regular epimorphism. We establish a new diagram lemma, that we call snail lemma, removing such a condition. We also show that the snail lemma subsumes the snake lemma and we give an interpretation of the snail lemma in terms of strong homotopy kernels. Our results hold in any pointed regular protomodular category.

9 citations

Cites background from "The two-square lemma."

  • ...What we prove is that we can get a six-term exact sequence (the snail sequence) starting from any commutative diagram like A f // α B β A0 f0 // B0 (2) Received by the editors 2014-07-05 and, in final form, 2016-06-07....

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References
More filters
Book•
Categories for the Working Mathematician

[...]

Saunders Mac Lane
01 Jan 1971
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.

9,254 citations

Book•
A Course in Homological Algebra

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Peter Hilton, Urs Stammbach
31 Dec 1971
TL;DR: In this paper, the authors propose an extension of the Kunneth Theorem for Abelian groups, which is based on the notion of double complexes, and they use it to define the (co-)homology of groups.
Abstract: I. Modules.- 1. Modules.- 2. The Group of Homomorphisms.- 3. Sums and Products.- 4. Free and Projective Modules.- 5. Projective Modules over a Principal Ideal Domain.- 6. Dualization, Injective Modules.- 7 Injective Modules over a Principal Ideal Domain.- 8. Cofree Modules.- 9. Essential Extensions.- II. Categories and Functors.- 1. Categories.- 2. Functors.- 3. Duality.- 4. Natural Transformations.- 5. Products and Coproducts Universal Constructions.- 6. Universal Constructions (Continued) Pull-backs and Push-outs.- 7. Adjoint Functors.- 8. Adjoint Functors and Universal Constructions.- 9. Abelian Categories.- 10. Projective, Injective, and Free Objects.- III. Extensions of Modules.- 1. Extensions.- 2. The Functor Ext.- 3. Ext Using Injectives.- 4. Computation of some Ext-Groups.- 5. Two Exact Sequences.- 6. A Theorem of Stein-Serre for Abelian Groups.- 7. The Tensor Product.- 8. The Functor Tor.- IV. Derived Functors.- 1. Complexes.- 2. The Long Exact (Co) Homology Sequence.- 3. Homotopy.- 4. Resolutions.- 5. Derived Functors.- 6. The Two Long Exact Sequences of Derived Functors.- 7. The Functors Extn? Using Projectives.- 8. The Functors % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqegm0B % 1jxALjharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY- % Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq % 0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaae % aaeaaakeaadaqdaaqaaGqaaiaa-veacaWF4bGaa8hDaaaadaqhaaWc % baacciGae43MdWeabaGaamOBaaaaaaa!40A3! $$ \overline {Ext} _\Lambda ^n $$ Using Injectives.- 9. Extn and n-Extensions.- 10. Another Characterization of Derived Functors.- 11. The Functor Torn?.- 12. Change of Rings.- V. The Kiinneth Formula.- 1. Double Complexes.- 2. The Kunneth Theorem.- 3. The Dual Kunneth Theorem.- 4. Applications of the Kunneth Formulas.- VI. Cohomology of Groups.- 1. The Group Ring.- 2. Definition of (Co) Homology.- 3. H0, H0.- 4. H1, H1 with Trivial Coefficient Modules.- 5. The Augmentation Ideal, Derivations, and the Semi-Direct Product.- 6. A Short Exact Sequence.- 7. The (Co) Homology of Finite Cyclic Groups.- 8. The 5-Term Exact Sequences.- 9. H2, Hopf's Formula, and the Lower Central Series.- 10. H2 and Extensions.- 11. Relative Projectives and Relative Injectives.- 12. Reduction Theorems.- 13. Resolutions.- 14. The (Co) Homology of a Coproduct.- 15. The Universal Coefficient Theorem and the (Co)Homology of a Product.- 16. Groups and Subgroups.- VII. Cohomology of Lie Algebras.- 1. Lie Algebras and their Universal Enveloping Algebra.- 2. Definition of Cohomology H0, H1.- 3. H2 and Extensions.- 4. A Resolution of the Ground Field K.- 5. Semi-simple Lie Algebras.- 6. The two Whitehead Lemmas.- 7. Appendix : Hubert's Chain-of-Syzygies Theorem.- VIII. Exact Couples and Spectral Sequences.- 1. Exact Couples and Spectral Sequences.- 2. Filtered Differential Objects.- 3. Finite Convergence Conditions for Filtered Chain Complexes.- 4. The Ladder of an Exact Couple.- 5. Limits.- 6. Rees Systems and Filtered Complexes.- 7. The Limit of a Rees System.- 8. Completions of Filtrations.- 9. The Grothendieck Spectral Sequence.- IX. Satellites and Homology.- 1. Projective Classes of Epimorphisms.- 2. ?-Derived Functors.- 3. ?-Satellites.- 4. The Adjoint Theorem and Examples.- 5. Kan Extensions and Homology.- 6. Applications: Homology of Small Categories, Spectral Sequences.- X. Some Applications and Recent Developments.- 1. Homological Algebra and Algebraic Topology.- 2. Nilpotent Groups.- 3. Finiteness Conditions on Groups.- 4. Modular Representation Theory.- 5. Stable and Derived Categories.

975 citations

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Theory of categories

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Barry Mitchell
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Abelian categories with applications to rings and modules

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The connecting morphism in the Kernel-Cokernel sequence

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F. Rudolf Beyl1•
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Frequently Asked Questions (1)
Q1. What are the contributions in this paper?

In this note the authors give an interesting new result, which they call the Two-Square Lemma, that provides a simple and completely categorical construction of the snake connecting morphism.