Mal'cev, Protomodular, Homological and Semi-Abelian Categories (Mathematics and Its Applications) - DRAFT

484 Pages·2004·0.91 MB·English

by  Francis Borceux

Mal’cev, protomodular, homological
and semi-abelian categories
Mathematics and its Applications
(List of titles)
MATHEMATICS AND ITS APPLICATIONS
Mal’cev, protomodular,
homological
and semi-abelian categories
Francis Borceux and Dominique Bourn
Kluwer Academic Publishers
2004
(copyright page)
Contents
Preface vii
Metatheorems 1
0.1 The Yoneda embedding . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Pointed categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Intrinsic centrality 10
1.1 Spans and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Unital categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Cooperating and central morphisms . . . . . . . . . . . . . . . . . 28
1.4 Commutativeobjects . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5 Symmetrizable morphisms . . . . . . . . . . . . . . . . . . . . . . . 49
1.6 Regular unital categories . . . . . . . . . . . . . . . . . . . . . . . . 57
1.7 Associated abelian object . . . . . . . . . . . . . . . . . . . . . . . 67
1.8 Strongly unital categories . . . . . . . . . . . . . . . . . . . . . . . 74
1.9 Gregarious objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
1.10 Linear and additive categories . . . . . . . . . . . . . . . . . . . . . 99
1.11 Antilinear and antiadditivecategories . . . . . . . . . . . . . . . . 107
1.12 Complemented subobjects . . . . . . . . . . . . . . . . . . . . . . . 118
2 Mal’cev categories 123
2.1 Slices, coslices and points . . . . . . . . . . . . . . . . . . . . . . . 123
2.2 Mal’cev categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2.3 Abelian objects in Mal’cev categories . . . . . . . . . . . . . . . . . 149
2.4 Naturally Mal’cev categories . . . . . . . . . . . . . . . . . . . . . . 156
2.5 Regular Mal’cev categories. . . . . . . . . . . . . . . . . . . . . . . 166
2.6 Connectors in Mal’cev categories . . . . . . . . . . . . . . . . . . . 173
2.7 Connector and cooperator . . . . . . . . . . . . . . . . . . . . . . . 188
2.8 Associated abelian object and commutator . . . . . . . . . . . . . 191
2.9 Protoarithmetical categories . . . . . . . . . . . . . . . . . . . . . . 206
2.10 Antilinear Mal’cev categories . . . . . . . . . . . . . . . . . . . . . 217
2.11 Abelian groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
v
vi CONTENTS
3 Protomodular categories 227
3.1 De(cid:12)nition and examples . . . . . . . . . . . . . . . . . . . . . . . . 227
3.2 Normal subobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
3.3 Couniversal property of the product . . . . . . . . . . . . . . . . . 260
4 Homological categories 266
4.1 The short (cid:12)ve lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 266
4.2 The nine lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
4.3 The Noether isomorphismtheorems . . . . . . . . . . . . . . . . . 282
4.4 The snake lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4.5 The longexact homologysequence . . . . . . . . . . . . . . . . . . 302
4.6 Examples of homologicalcategories . . . . . . . . . . . . . . . . . . 305
5 Semi-abelian categories 311
5.1 De(cid:12)nition and examples . . . . . . . . . . . . . . . . . . . . . . . . 311
5.2 Semi-direct products . . . . . . . . . . . . . . . . . . . . . . . . . . 316
5.3 Semi-associative Mal’cev varieties . . . . . . . . . . . . . . . . . . . 323
6 Strongly protomodular categories 336
6.1 Centrality and normality. . . . . . . . . . . . . . . . . . . . . . . . 336
6.2 Normal subobjects in the (cid:12)bres . . . . . . . . . . . . . . . . . . . . 343
6.3 Normal functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
6.4 Strongly protomodularcategories . . . . . . . . . . . . . . . . . . . 352
6.5 A counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
6.6 Connector and cooperator . . . . . . . . . . . . . . . . . . . . . . . 358
7 Essentially a(cid:14)ne categories 362
7.1 The (cid:12)bration of points . . . . . . . . . . . . . . . . . . . . . . . . . 362
7.2 Essentially a(cid:14)ne categories . . . . . . . . . . . . . . . . . . . . . . 370
7.3 Abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Appendix 391
A.1 Algebraic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
A.2 Internal relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
A.3 Internal groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
A.4 Variations on epimorphisms . . . . . . . . . . . . . . . . . . . . . . 419
A.5 Regular and exact categories . . . . . . . . . . . . . . . . . . . . . 427
A.6 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
A.7 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
Bibliography 457
Index of symbols 463
Index of de(cid:12)nitions 465
Preface
Themoststrikingsuccesses ofcategorytheory,asfarasclari(cid:12)cationofmathemat-
ical situations is concerned, are probably the theory of abelian categories and the
theory of toposes. This is not too amazing since both theories are closely related
to the development of sheaf theory, a context in which it is desirable to get rid of
the usual notion of element.
But up to recently, category theory did not provide any comparable insight
in General Algebra, a domain in which element-based mathematics remains the
slogan. In particular, category theory could not provide a structural tool able to
grasp, even in the most representative category of classical algebra { namely, the
categoryGpofgroups{thedeepessenceofthenotionofnormalsubobject: namely,
an equivalence class for a congruence and not just the kernel of a morphism.
And category theory could not grasp either the conceptual foundations of the
homological lemmas: the Nine Lemma, the Snake Lemma, which remain valid
and strongly meaningful in the category Gp of groups, even if this category does
not belong to the abelian setting in which these lemmasare generally proved in a
signi(cid:12)cant categorical way.
Of course, there have been since a long time attempts to provide an ax-
iomatic context in which to get the isomorphism theorems, the decomposition
theorems or the previous homologicallemmas for the varieties of Universal Alge-
bra: Baer (1947, [6]), Goldie (1952, [47]), Atiyah (1956, [5]), Higgins (1956,[52]),
Kurosh(1959,[70]),Hilton{Ledermann(1960,[53]),Eckmann{Hilton(1962,[40]),
Tsalenko (1967, [89]), but also Hofmannn (1960, [54]), Fro¨hlich (1961, [45]), Huq
(1968, [55]), Gerstenhaber (1970, [46]), Burgin (1970, [34]), Orzech (1972, [80]).
These(cid:12)rstattempts,despitetheirinterest,consistgenerallyinalonglistofaxioms
whose independence is certainly not clear. But more importantly, these axioms
look desperately heavy and complicated in comparison with the elegance of the
characterization of abelian models. We refer the reader to the introduction of the
paper by Janelidze{Mark(cid:18)(cid:16){Tholen(2002,[58])fora reliablehistoricalapproach to
this topic.
vii
viii PREFACE
Besides the masterworkthatLawvere’sthesis (1963,[71])onalgebraictheories
constitutes and the contribution of Linton (1969, [74] and [75]), the (cid:12)rst suc-
cesses of conceptual clari(cid:12)cation in this topic occurred only in the early nineties:
Carboni{Lambeck{Pedicchio (1991, [37]), Janelidze (1990, [57]), Bourn (1991,
[16]), Diers (1993,[39]). Here again,inparticular throughthe notionofaMal’cev
category inthe workofCarboni{Lambek{Pedicchio(1990,[37]),the inputofUni-
versal Algebra has been very signi(cid:12)cant. One should also certainly emphasize the
meeting onUniversal Algebra and Category Theoryheld injuly1993atthe MRSI
in Berkeley, which has been among other things at the origin of the pioneering
work of Pedicchio (1995, [82]) on the categorical notion of commutator.
Establishing an organic and synthetic connection between all these attempts
is the ambition of this book. To achieve this, an additional ingredient was nec-
essary, of purely categorical nature: the (cid:12)bration of points. This (cid:12)bration allows
representing every category as a (cid:12)bration whose (cid:12)bres are pointed categories, i.e.
categories with azero object (see Bourn, 1996,[17]). Thisbook willgiveevidence
that the (cid:12)bration of points emphasizes the importance of split epimorphisms in
the context of algebraic theories, but also that this (cid:12)bration of points has a very
strong classi(cid:12)cation power: see on page 390 the table summarizing these classi(cid:12)-
cation properties.
ThisbookliesthusnaturallyattheconfluentofCategoryTheoryandUniversal
Algebra, even if the (cid:12)rst aspect is predominant, since both authors have a deep
categorical background. The book will show in particular that, in the context of
General Algebra, there exists an intrinsic conceptual notion of central morphism
with a strong discriminating power. This notion is structurally connected, but
generally not equivalent,to the theory of commutators.
General Algebra begins, in a way, with the datum of a binary operation. In
a similar way, our (cid:12)rst chapter elementarily begins with the study of magmas,
i.e. sets endowed with a binary operation having a unit element. Although an
extremely poor algebraic notion, the structure of magma determines a category,
denoted Mag, which satis(cid:12)es a property which deserves attention. Indeed, the
equality(x;y)=(x;1)(cid:1)(1;y)intheproductX(cid:2)Y oftwomagmasX andY implies
thateverysubmagmaZ ofX(cid:2)Y containingX andY (ormorepreciselyX(cid:2)1and
1(cid:2)Y) is actuallyequal to X(cid:2)Y. In other words, for each pair (X;Y) of objects
in Mag, the pair of canonical inclusions (l ;r ) is jointlystrongly epimorphic:
X Y
l r
Xqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq X qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqX (cid:2)Y qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq Y qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqY:
We call unital a category satisfying this property. It is straightforward, from the
previous observation on magmas that the categories Mon, ComM, Gp, Ab, Rg of
respectively monoids,commutativemonoids,groups,abeliangroups,ringsarealso
unital. More generally,an algebraicvariety turns out to be unitalif and only ifit
is Jonsson-Tarski (see [66]).
We show next that, inside a unital category E, there is an additive core which
will appear to have a powerful classi(cid:12)cation potential. More precisely we de(cid:12)ne
PREFACE ix
the right ideal Z(E) of central morphisms. The set Z(X;Y) of central morphisms
between theobjectsX andY formsacommutativemonoidwhichactscanonically
on the set E(X;Y) of all maps in E between X and Y. From this ideal Z(E), we
extract the right ideal (cid:6)(E) of symmetrizable morphisms. The set (cid:6)(X;Y) of
symmetrizable morphisms between X and Y becomes now an abelian group. Of
course the identity mapid is central (respectively, symmetrizable) if and only if
X
the object X is commutative (respectively, abelian).
Themaintooltode(cid:12)ne centralmaps,namelythe notionofcooperating pairof
morphisms with the same codomain,goes back to Huq (1968,[55]). But Huq was
working in a much heavier context than ours, so that on one hand the examples
Mag, Mon and ComM were excluded and, on the other hand, the discriminatory
power could not appear.
When the unital category E is also regular, then all the information is con-
centrated in the notion of cooperating pair of subobjects. When furthermore E
is (cid:12)nitely cocomplete, it is possible to associate with every pair (X ;X ) of sub-
0 1
objects of X a map which universally makes them cooperate. This map measure
the obstruction for the pair (X ;X ) of subobjects to cooperate and gives rise to
0 1
a (cid:12)rst approach of the notion of commutator.
The notion of strongly unital category is de(cid:12)ned as well: in this case we have
always (cid:6)(E) = Z(E), i.e. every central map is symmetrizable. A category is
strongly unital when for each object X, the followingpair of canonical inclusions
(lX;sX0 ) is jointlystrongly epimorphic, withs0;X: Xqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqX (cid:2)X the diagonal:
l sX
Xqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq X qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqX (cid:2)Xqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqX:
An algebraicvarietyis stronglyunitalifand onlyifthe theory admitsexactlyone
constant 0 and contains a ternary operation p satisfying the axioms:
p(x;x;y)=y; p(x;0;0)=x:
This is a weak occurrence of the Mal’cev axiom, extensively studied in our chap-
ter 2.
A (cid:12)rst aspect of the discriminatory power of the additive core of a unital
category is givenby the followingtable, where Ω(E) is the idealofzero maps,and
the intersection of the line L and the column C indicates the class of categories
which satisfy the property L=C.
= E Ω(E)
Z(E) linear categories antilinear categories
(cid:6)(E) additive categories antiadditive categories
Paradigmatic examples of linear, additive, antilinear and antiadditive categories
are respectively giventhe categoriesComMofcommutativemonoid,Ab ofabelian
groups, PrHe of preHeyting algebras and IMag of idempotent magmas.