Vol. 19 (2018), No. 1, pp. 37–47 DOI: 10.18514/MMN.2018.1695
ON THE BARR EXACTNESS PROPERTY OF BXMOD=R
H. G ¨ULS ¨UN AKAY AND U. EGE ARSLAN Received 29 May, 2015
Abstract. In this work, it is shown that the categoryBXMod=Rof braided crossed modules over a fixed commutative algebraRis an exact category in the sense of Barr.
2010Mathematics Subject Classification: 18A30; 18E10
Keywords: braided crossed module, exact category, regular category, coequaliser
1. INTRODUCTION
Exact categories for additive categories were introduced by Quillen [9]. Barr defined exact categories for non-additive categories [1]. He introduced exact cat- egories in order to define a good notion of non-abelian cohomology.
We will mention regular categories closely related by exact categories. There exist in the literature many different definitions of regular categories, which are all equi- valent under the assumptions that finite limits and coequalisers exist. We will recall the following definition from [7], but a weaker version of it is given in [1].
A categoryC is called regular if it satisfies the following three properties:
i: C is finitely complete,
ii: Iff WX !Y is a morphism inC, and Z
p1
p0 //X
f
X
f //Y is a pullback.thenZ
p0
p1
X is called the kernel pair off /, the coequaliser of p0; p1exists;
The first author was partially supported by T ¨UB˙ITAK (The Scientific and Technological Research Council Of Turkey).
c 2018 Miskolc University Press
iii: Iff WX !Y is a morphism inC, and W
g
//X
f
Z //Y
is a pullback, and iff is a regular epimorphism, thengis a regular epimorph- ism as well.
If a regular category additionally has the property that every equivalence relation is effective that is every equivalence relation is a kernel pair, then it is called a Barr exact category.
Examples of exact categories are:
.1/The categorySet of sets.
.2/The category of non empty sets.
.3/Any abelian category.
.4/Every partially ordered set considered as a category.
.5/For any small categoryC, the functor category.Cop; Set /.
Brown and Gilbert introduced in [3] the notion of braided regular crossed module of groupoids and groups as an algebraic model for homotopy 3-types equivalent to Conduch´e’s2-crossed module. The reduced case of braided regular crossed module is called a braided crossed module of groups. Braided crossed modules in the category of commutative algebras were defined by Ulualan in [10].
The purpose of this paper is to answer the question whether the category of braided crossed modules of commutative algebras is exact. We prove that the category of braided crossed modules of commutative algebras is an exact category.
Throughout this paper k will be a fixed commutative ring with 0¤1. All k- algebras will be commutative and associative.
2. BRAIDED CROSSED MODULES
In order to give the notion of braided crossed modules of commutative algebras, we will recall the concept of crossed modules of commutative algebras. Crossed modules of groups originate in algebraic topology and more particularly in homotopy theory. Mac Lane and Whitehead showed in [6] that crossed modules of groups modelled homotopy 2-types (3-types in their notation). The commutative algebra case of crossed modules is contained in the paper of Lichtenbaum and Schlessinger [5] and also in the work of Gerstenhaber [4] under different names. Some categorical results and Koszul complex link are also given by Porter [8].
Definition 1 ([8]). A crossed module of commutative algebras, .C; R; @/, is an R-algebra,C, together with an R-algebra morphism@WC !R such that for all
c; c02C
@ .c/c0Dcc0
whereRis ak-algebra. A morphism of crossed modules from.C; R; @/to.C0; R0; @0/ is a pair ofk-algebra morphisms,WC !C0and WR !R0such that
.i / @0D @ and .i i / .rc/D .r/ .c/
for allr2Randc2C. We thus get the categoryXModof crossed modules.
There is, for a fixed algebraR, a subcategoryXMod=Rof the category of crossed modules, which has as objects those crossed modules withRas the “base”, i.e., all .C; R; @/ for this fixedR, and having as morphisms from.C; R; @1/ to.C0; R; @2/ just those.f1; f0/inXModin whichf0WR !Ris the identity homomorphism on R.
Definition 2([10]). A braided crossed module of commutative algebras@WC!R is a crossed module with the braiding functionf ; g WRR !C satisfying the following axioms:
BCM1/ @fr; r0g Drr0; BCM 2/f@c; @c0g Dcc0;
BCM 3/f@c; rg Drcandfr; @cg Drc;
BCM 4/frr0; r00g fr; r0r00g D0;
for allr; r0; r002Randc; c02C.
We denote such a braided crossed module of commutative algebras byfC; R; @g. IffC; R; @1gandfC0; R0; @2gare braided crossed modules, a morphism
.f1; f0/W fC; R; @1g !˚
C0; R0; @2 ;
of braided crossed modules is given by a morphism of crossed modules such that f ; g.f0f0/Df1f ; g:
We thus get the categoryBXModof braided crossed modules of commutative al- gebras.
In the case of a morphism .f1; f0/ between braided crossed modules with the same baseR, i.e. where f0 is the identity on Rwith f1f ; g D f ; g, then we say thatf1is a morphism of braided crossedR-modules,@1WC !Ris a braided crossedR-module and we usefC; @1ginstead offC; R; @1g. This gives a subcategory BXMod=RofBXMod. Our results are obtained for this subcategory.
Several well known examples of crossed modules give rise to braided crossed mod- ules as follows.
Examples of braided crossed modules
.1/ Any identity map ofk-algebras@WX !X is a braided crossed module with fx; yg Dxy.
.2/ If C is ak-algebra andC2 is an ideal generated byfc1c2jc1; c22Cg. Then
@WC2 !C is a braided crossed module withfc1; c2g Dc1c2, forc1; c22C.
.3/AnyR-moduleM can be considered as anR-algebra with zero multiplication and hence the zero morphism0WM !Ris a braided crossed module withfr; r0g D0.
.4/LetfC; R; @1gandfC0; R0; @2gbe two braided crossed modules, then
˚CC0; RR0; @ is a braided crossed module.
3. THEBARR EXACTNESS PROPERTY OF BRAIDED CROSSED MODULES
Our aim now is to obtain thatBXMod=Ris an exact category. For this purpose, we have to prove some statements in this section.
Proposition 1. InBXMod=Revery pair of morphisms with common domain and codomain has an equaliser.
Proof. Letf; gW fC; @g ! fD; ıgbe two morphisms of braided crossedR-modules.
Let E denotes the set E D fc2C jf .c/Dg .c/g: It can be easily checked that fE; "ghas the structure of a braided crossedR-module, and the inclusionuW fE; "g ! fC; @gis a morphism of braided crossedR-modules and clearlyf uDgu.
Suppose that there exist a braided crossed R-module, fE0; "0g and a morphism u0W fE0; "0g ! fC; @g of braided crossed R-modules such that f u0 Dgu0: Then for all x2 E0; f .u0.x//Dg .u0.x// ; and hence u0.x/2E: Thus, we have ˛ W fE0; "0g ! fE; "gby˛ .x/Du0.x/from which we get
"˛ .x/D"u0.x/D@u0.x/D"0.x/ ;
˛ .rx/Du0.rx/Dru0.x/Dr˛ .x/ ;
for allx2E0; r2R. Sinceu,u0are braided crossedR-module morphisms, it is clear that˛fr; r0g D fr; r0gfor allr; r02R.
Let˛0WE0 !Ebe a morphism of braided crossedR-module such thatu˛0Du0: Since˛ .x/Du0.x/Du˛0.x/D˛0.x/for allx2E0, we have that˛is the unique morphism which makes the diagram
fE; "g u //fC; @g
f //
g //fD; ıg
fE0; "0g
˛
OO
u0
;;w
ww ww ww ww ww ww w
commutative. Henceuis the equaliser of.f; g/ ;as required.
Proposition 2. BXMod=Rhas finite products.
Proof. LetfC; @gandfD; ıgbe braided crossedR-modules. The productfCuD; g is the pullback over the terminal objectfR; iRg;
fCuD; g
ı0
@0 //fC; @g
@
fD; ıg ı //fR; iRg
whereCuDD f.c; d /j@ .c/Dı .d /gandWCuD !Ris defined byD@@0D ıı0:Then by induction,BXMod=Rhas finite products.
Proposition 3. BXMod=Ris finitely complete.
Proof. Follows from Propositions1and2.
Proposition 4. InBXMod=Revery morphism has a kernel pair, and the kernel pair has a coequaliser.
Proof. LetfA; @gandfB; ˇgbe two braided crossedR-modules. Letf W fA; @g ! fB; ˇgbe a morphism of braided crossedR-modules. Then.A; f / is a crossedB- module, whereBacts onAviaˇand the homomorphism˛WABA !Bdefined by˛ .a; a0/Df .a/Df .a0/is a crossedB-module, where
ABAD˚ a; a0
jf .a/Df a0 is aB-algebra:
˛ b a; a0
Df .ba/Dbf .a/Db˛ a; a0
;
˛ a; a0
a1; a01
D f .a/a1; f a0 a01
D a; a0
a1; a01
; for all.a; a0/ ; a1; a10
2ABA; b2B:
We can define@0WABA !Rby@0.a; a0/D@ .a/D@ .a0/, sinceˇ˛Dˇf D@.
It is easily checked that.ABA; R; @0/is a crossed module:
@0 r a; a0
D@ .ra/Dr@ .a/Dr@0 a; a0
;
@0 a; a0
a1; a01
D@ .a/ a1; a01 D @ .a/a1; @ .a/a01 D @ .a/a1; @ a0
a01 D aa1; a0a01
D a; a0
a1; a10
;
for all.a; a0/ ; a1; a10
2ABA,r2R.
Below we will show that fABA; @0g is a braided crossed R-module with the braiding map
f ; g WRR !ABA defined byfr; r0g D.fr; r0g;fr; r0g/,
BCM1/
@0˚
r; r0 D@˚
r; r0 Drr0; BCM2/
˚@0 a; a0
; @0 b; b0 D f@a; @bg;˚
@a0; @b0 D ab; a0b0
D a; a0 b; b0
; BCM3/
˚@0 a; a0
; r D f@a; rg;˚
@a0; r D ra; ra0 Dr a; a0
˚r; @0 a; a0 D fr; @ag;˚ r; @a0 D ra; ra0 Dr a; a0
; BCM4/
˚rr0; r00 ˚
r; r0r00 D ˚
rr0; r00 ;˚
rr0; r00 ˚
r; r0r00 ;˚ r; r0r00 D ˚
rr0; r00 ˚
r; r0r00 ;˚
rr0; r00 ˚ r; r0r00 D.0; 0/ ;
for all.a; a0/ ; .b; b0/2ABA,r; r0; r002R.
The following diagram
ABA
p1
//
p2
//
@0
A f //
@
B
ˇ
R R R
commutes and the morphisms p1 and p2 above are morphisms of braided crossed R-modules. This construction satisfies universal property: LetfE; ıgbe a braided
crossedR-module andp10; p02W fE; ıg ! fA; @gbe any morphisms of braided crossed R-modules withfp10 Dfp20;then there exist a unique morphism
hW fE; ıg ! fABA; @0g
given byh .e/D p10.e/ ; p02.e/
;for alle2E;which makes the diagram
fE; ıg
p20
p10
((
h
%%
fABA; @0g
p2
p1
//fA; @g
f
fA; @g f //fB; ˇg
commutative.
Then.p1; p2/is the kernel pair of the morphismf.
Now we will show that the pair.p1; p2/has a coequaliser. LetI be an ideal ofA generated by all the elements of the formp1.x/ p2.x/ ;for allxD.a; a0/2ABA.
We will define the braided crossed R-moduleıWA=I !Rby ı .aCI /D@ .a/ ; fr; r0g D fr; r0g CI fora2A; r; r02R. SinceI Ker@,ıis well defined. By the definition offA=I; ıg, the morphismqW fA; @g ! fA=I; ıgis the induced projection and it is a morphism of braided crossedR-modules, i.e., the diagram
ABA
p1 //
p2
//
@0
""
EE EE EE EE EE EE
EE A q //
@
A=I
ı
}}|||||||||||||
R
commutes. Suppose there exist a braided crossedR-modulefA0; ˛0gand a morphism of braided crossedR-modulesq0W fA; @g ! fA0; ˛0gsuch thatq0p1Dq0p2;then there exists a unique morphism ' W fA=I; ˛g ! fA0; ˛0g defined by ' .aCI /D q0.a/, satisfying 'q Dq0; i.e. q is the universal among all the morphisms q0W
fA; @g ! fA0; ˛0g for any braided crossed R-module fA0; ˛0g: So we get the fol- lowing commutative diagram:
ABA
p1 //
p2
//
@0
""
EE EE EE EE EE EE
EE A q //
@
q0
BB BB BB BB B
!!B
B A=I
ı|||||||||
}}|||
'
R A0
ı0
oo
Thenqis the coequaliser of the pair.p1; p2/ :
Therefore in BXMod=R, the kernel pair of every morphism exists and has a co-
equaliser.
Proposition 5. InBXMod=Revery regular epimorphisms are stable under pull- back.
Proof. Let˚ W fC; g ! fD; gbe a regular epimorphism inBXMod=R;which means that C !D is a regular epimorphism of k-algebras. Let W fG; g ! fD; gbe any morphism inBXMod=R:The pullback of˚alongis
˚W fCDG; g ! fG; ggiven by˚.c; g/Dg
whereCDGD f.c; g/j˚ .c/D .g/gandWCDG !Ris a braided crossed R-module defined by .c; g/D .c/D .g/ withfr; r0g D.fr; r0g;fr; r0g/, for all r; r02R; c2C; g2G:Since in the category ofk-algebras the regular epimorphisms are characterised as the surjective homomorphisms, these are closed in this way under pullback. Thus˚is a surjective homomorphism.
fCDG; g
˚
//fC; g
˚
fG; g //fD; g
We claim that the surjective morphism˚is a regular epimorphism, that is˚is the coequaliser of a pair of morphisms. Define
ED˚
.x; y/2.CDG/.CDG/j˚.x/D˚.y/
and let fE; ˛g
p
q fCDG; gbe the first and second projections. Since G is iso- morphic to the quotient ofCDG,˚is the coequaliser ofp andq. Thus we get that if˚is regular epimorphism, so is˚, as required.
Theorem 1. BXMod=Ris regular.
Proof. The proof is a direct consequence of Propositions3,4and5.
Now we recall the definition of equivalence relation from [2].
Definition 3. LetAbe a category with finite limits. IfAis an object, a subobject d0; d1
WE !AAis called an equivalence relation if it is
ER1:Reflexive: There is an arrowrWA !Esuch thatd0rDd1rDidAI
ER2:Symmetric: There is an arrowsWE !Esuch thatd0sDd1andd1sDd0I ER3:Transitive: If
T
q2
q1 //E
d0
E d1
//A
is a pullback, there is an arrowtWT !E such thatd1tDd1q1andd0tDd0q2:
Proposition 6. Every equivalence relation
fE; @g uv //// fA; ˛g
in the category of braided crossedR-modules is effective.
Proof. LetA=E be the set of all equivalence classes Œa with respect to E;i.e., ŒaD fb2Aj.a; b/2Eg: A=E has the structure of anR-algebra. ˛WA=E !R induced by˛ is well defined since˛uD˛v:We can form the braided crossed R- module˛with braiding mapfr; r0g DŒfr; r0gand get the following diagram,
E u //
v //
@
@
@@
@@
@@
@@
@@
@ A q //
˛
A=E
˛
}}{{{{{{{{{{{{{
R
of morphisms of braided crossed R-modules, whereq is the projection ontoA=E. By the definition of an equivalence relation on A, we have.u .x/ ; v .x//2E; for x2E. Since
EARAD˚ a; a0
j˛ .a/D˛ a0 ;
we get˛u .x/D˛v .x/, thus.0; u .x/ v .x//2E;thereforequDqv. Suppose that there exist a braided crossedR-modulefD; !gwithu0; v0W fD; !g ! fA; ˛gsuch thatqu0Dqv0;soŒu0.d /DŒv0.d / ; i.e. .u0.d / ; v0.d //2E;and therefore there
exists a unique morphismW fD; !g ! fE; @g;such that the diagram fD; !g
v0
u0
''
##
fE; @g
v
u //fA; ˛g
q
fA; ˛g q //fA=E; ˛g commutes.
Thus.u; v/is the kernel pair of a morphismqW fA; ˛g ! fA=E; ˛gin the cat-
egory of braided crossedR-modules.
Theorem 2. The categoryBXMod=Rof braided crossedR-modules is a Barr exact category.
Proof. The conditions of exact category forBXMod=Rare satisfied by Theorem1
and Proposition6, which completes the proof.
REFERENCES
[1] M. Barr, P. Grillet, and D. Van Osdol,Exact categories and categories of sheaves, ser. Lecture notes in mathematics. Springer-Verlag, 1971.
[2] M. Barr and C. Wells, “Toposes, triples and theories.”Repr. Theory Appl. Categ., vol. 2005, no. 12, pp. 1–288, 2005.
[3] R. Brown and N. Gilbert, “Algebraic models of 3-types and automorphism structures for crossed modules.”Proc. Lond. Math. Soc. (3), vol. 59, no. 1, pp. 51–73, 1989.
[4] M. Gerstenhaber, “On the deformation of rings and algebras,”Annals of Mathematics, vol. 79, no. 1, pp. 59–103, 1964.
[5] S. Lichtenbaum and M. Schlessinger, “The cotangent complex of a morphism,”Transactions of the American Mathematical Society, vol. 128, no. 1, pp. 41–70, 1967.
[6] S. MacLane and J. H. C. Whitehead, “On the 3-type of a complex,”Proceedings of the National Academy of Sciences of the United States of America, vol. 36, no. 1, pp. 41–48, 1950.
[7] M. Pedicchio and W. Tholen,Categorical Foundations: Special Topics in Order, Topology, Al- gebra, and Sheaf Theory, ser. Categorical Foundations: Special Topics in Order, Topology, Al- gebra, and Sheaf Theory. Cambridge University Press, 2004.
[8] T. Porter, “Some categorical results in the theory of crossed modules in commutative algebras,”
Journal of Algebra, vol. 109, no. 2, pp. 415 – 429, 1987.
[9] D. Quillen,Higher algebraic K-theory: I. Berlin, Heidelberg: Springer Berlin Heidelberg, 1973, pp. 85–147.
[10] E. Ulualan, “De˘gis¸meli Cebirler ¨Uzerinde Kuadratik Mod¨uller,” Ph.D. dissertation, ESOGU FBE, Turkey, 2004.
Authors’ addresses
H. G ¨uls ¨un Akay
Eskis¸ehir Osmangazi University, Department of Mathematics and Computer Science, 26480 Eskis¸ehir, Turkey
E-mail address:hgulsun@ogu.edu.tr
U. Ege Arslan
Eskis¸ehir Osmangazi University, Department of Mathematics and Computer Science, 26480 Eskis¸ehir, Turkey
E-mail address: uege@ogu.edu.tr
