Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. XXII I , 153-163 (1977)

On Lattice-Like Properties of Orthoquaternary Categories (*)

G I U S E P P E CONTE F E I ~ D I N A N D O MOI~A (**)

Introduction.

I t is well known t h a t canonical isomorphisms be tween subquot ients are, generally, not composable (see [10] pg. 53).

For a ca tegory C having qua te rna ry symmet r i za t ion C w (a sort of (( ca- tegory of relat ions )~ for C, see [4, 11]) this componibi l i ty is equivalent to the o r thodoxy of C w (see [5, 6]).

This pape r is devoted to point a class of categories for which necessary and sufficient conditions in order they have an or thodox qua te rna ry sym- metr iza t ion are expressible b y means of lattice-like propert ies of subobjects , in analogy with [7] 1.10 for exac t categories.

I n the sequential ly following pape r [2] it is shown how this par t icular s t a t ement of the previous conditions is useful in order to define a sort of dis t r ibut ive expans ion (see [2, 8]), which results to have an or thodox qua- t e rna ry symmetr iza t ion , for a large class of categories (including Groups, Pointed Set). So oppor tune classes of thei r subquot ients can be studied as subobjects of an or thodox category.

I n the first pa rag raph we recall t h a t for a b ica tegory (C, I , P) the existence and o r thodoxy of C w is equivalent to the axioms CW2) 3) 4), OQ and OQ*.

In the second pa rag raph axioms D1) 2) 3) are in t roduced: they , assuring the existence of finite intersections and of certain finite unions of subobjects and assuring t h a t lqoether i somorphism theorems hold, allow us to find, in the th i rd paragraph , latt ice-like proper t ies equivalent to o r thoqua te rna r i ty and to s ta te t h e m b y means of axioms Dd)5) .

I n the four th pa rag raph we resume the main results and briefly discuss axiom D3).

(*) Written while the authors were members of C.N.R. sec. G.N.S.A.G.A. (**) Address of the authors: Istituto di Matematica, vi~ L. B. Atbcrti 4,

16132 Genova.

154 G I U S : E P P : E C O N T E - F ] E I ~ D I N A N D O M O R A

Conventions and notations.

In w167 2 and 3 after introducing an axiom (D1)-5)) we suppose tha t C

verifies it. The word t< bicatcgory >> is used ia the sense of [9]; if ( C , I , P ) is a

bicategory we picture a map in I or P, respectively, by ~ and -->>.

Antilexicographic notat ion is used for the composition of maps; 0 de- notes the zero object and zero map.

We denote the commutat ive square

by [ml~ m2~ ms~ m4].

By a cubic diagram T we mean a commutat ive diagTam of the form (1)

(1)

J

m 5

m 3

V~ ~ ~ V~

m 2

V )

m

V ~. 6

,/9~ m l O

V s ~

m 7

S V

3

m 8

) V

J m l z

Y 7

ON LATTICE-LIKE PROPERTIES OF 0RTHOQUATERNARY CATEGORIES

or of the form (2)

155

(2)

Ill 5

M 6

V 2

V6

1112

1113

mll

V3

m 7

mlo v - . >~ V

5 7

m 8

)) V 8

m12

In the following, speaking of a cubic diagram, we refer to (1) or (2) with

the maps and the vertices numbered as in the picture and the same orienta-

tion. So we can identify a cubic diagram with an ordered set of maps and

objects naming only the ones we are interested to consider. In each equivalence class of I a representunt is chosen to become a

subobject of its codomain; dually for quotients.

By abuse of notat ion we often denote a subobject B ~ A by B_cA and a normal subobject by B < A; in this case A/B is the codomain of

cok(B ~ A).

w 1. - P r e l i m i n a r y results .

1.1 Let (C, I , P) be a bieategory having an SW-symmetr iza t ion, de-

noted C w, i.e. a bicategory satisfying the following axioms (see [11]):

CW2) The subcategory I has pull-backs (1).

CW2*) The subcategory P has push-outs (1).

(1) It is easy to verify that pull-backs in I and push-outs in P remain such in C.

156 G I U S E P P i E C O N T : E - F : E I : t D I N & N D O ~ O R A

CW3) Any pair of maps (re, p), m in I a n d p in t ) , with the same codomain can be filled in a pul l -back [p' , m', m, p] with m ' in I and p ' in P.

CW4) I f T is a cubic d iagram of the fo rm (2) in C, its left face is a pul l-back ~nd its uppe r face is a push-out , then its r ight face is a pul l -back and its lower one is a push-out .

We call it a quaternary bicategory and we have (see [11]) t ha t C w is a factorizing regular involut ion category; [6] proves t h a t canonical isomor- phisms between subquot ients of C are composable iff C w is o r thodox (i.e. composi t ion of idempoten t endomorphisms is idempotent) .

So we s tate:

1.2 :DEFINITION. A qua te rna ry bicategory (C, I, P) is said an ortho- quaternary bicategory iff C w is an orthodox category.

The same arguments as in [7] 1.9 make it possible to prove the following

1.3 THEOnE)[. Let (C~ I , P) be a qua te rna ry bicategory. The following are equivalent

a) (C, I , P) is au o r thoqua te rna ry b ica tegory;

b) (C, I , P) verifies the following two axioms:

OQ) I f T is a cubic d iagram of the fo rm (1) and its upper face is a pul l -back such is the lower one.

OQ*) I f T is a cubic d iagram of the fo rm (2) and its upper face is a push-ou t such is the lower one.

w

2.1 I n the following C denotes a category with 0 in which any m a p / has a canonical factor izat ion ] = m p, where m is monic and p is conormal .

Such a factor izat ion coincides with the monic-ep imorphism one in t roduced in [1]. C has then an obvious b icategory s t ructure assuming as I and P, respect ively, the class of monics and tha t of ep imorphisms ( = conormal maps).

D1) ( = CW2)) C has ]inite intersection o/ subobjects.

2.2 P~oPosITION. Given b: B >-> A, a: A ~ A ' , ker(a): C ~ A, let [ b ' , / , k e r ( a ) , b ] be a pul l -back. We h a v e ] = k e r ( a b ) and, if a b = m p is the canonical factorizat ion, then p = cok(]).

ON L A T T I C E - L I K E P ROP E RT IE S OF ORTHOQUATERNARY CATEGORIES 157

I n pa r t i cu l a r C < A implies C n B < B.

P~ooF. B y [ t ] 6.2 a nd the c o n o r m a l i t y of ep imorphisms .

D2) ( = CW3)) Any pair of maps (m, p), m in I and p in P, with the same codomain can be filled in a pull-back [p', m', m, p] with m' in I and p' in P.

2.3 t~EMARK. C has kernels.

2.4 PI~OPOSITIOIN'. Given b: B ~ C, c: C >-> A, wi th c and c b no rma l

maps , we h a v e t h a t m ' : C/B >-> A / B is t he kernel of p : A / B -~ A/C. This implies also A/C = (A/B)/ (C/B) (1-st I~oether i somorphism) .

PROOF. Consider the d i a g ra m consis t ing of the squares [1B, b, c b, c], [c, cok(b), cok(c b), m ' ] and [cok(e), cok(e b), l~lc, p].

Le t [g,/, eok(c b), ker(p)] be a pu l l -back and call n the un ique m a p such

t h a t k e r ( p ) n ~ - m ' , n is monic. B y [4] 6.2 we h a v e g =- ker(cok(c)) and so the un ique m a p h such ~hat

c-- - -g h and n c o k ( b ) = ] h is an i somorph i sm; this implies t h a t n cok(b)

is an e p i m o r p h i s m and so n is an i somorphism.

2.5 PI~OPOSITION. I f [m, q, p, n] is a pu l l -back wi th m, n in 1 and p, q in P, t h e n m ker(q) --~ ker(p).

PI~OOF. Call A and B, respect ive ly , Ker (p) and Dom(n) and call g the

un ique monie m a p such t h a t ke r (p )g - -~ m ker(q). As n0As = p ker(p) we can define a as the un ique m a p such t h a t

m a ---- ker(p) and q a z 0AB, and b as the un ique m a p such t h a t ker(q) b = a,

a and b are b o t h monic . B y ker(p) g b ~ taker (q) b ----- m a = ker(p) we have g b --~ 1 A so g is an

i somorphism.

2.6 PI~OPO SITIO~-.

a)

b)

c)

The fol lowing assert ions are equ iva l en t :

the converse of 2.5 holds ;

(Axiom A7 of [3]) Given b:B->>A, b':B'-+>A, ker (b ) :C>+B, ker(b ' ) : C >-> B ' and a m a p m: B'- ->B such t h a t mker (b ' ) ~ ker(b) and b ' = b m, if m is mon ic t h e n it is an i somorph i sm;

in the c o m m u t a t i v e diagTam 2.6.1 let [m', p ' , p , n] be a pul l -back. Then X = B U C. So the re exists the un ion be tween a subob jec t

and a no rma l one.

] 5 8 G I U S E P P E CONTE - F E R D I N A N D O MORA

C ~

(2.6.1)

ker(p)

PROOF. a) ~ b)

b) =:> c)

e) ::> b)

) A P "

m X n

/ / i f

/

" q B >;' "

Obvious.

[3] prop. 1.3.

Consider ugain 2.6.1, b y C = Ker(q) we have X : B u C : B, so the map i which fills in commuta t i ve ly is an isomorphism.

2.7 Therefore let us assume

D3) The equivalent assertions of 2.6 hold.

2.8 PROPOSITIO~ (2-rid ~ o e t h e r isomorphism). Given B _c A and C <~ A,

let b ' : B n C ~ B , b:B)-->BU C and c : C ~ B ~ ) C be the immersions. There exists an unique i somorphism i : B / B ~ C--~BU C/C such t h a t i eok(b') = b eok(c).

PROOF. Let us draw the d iagram 2.8.1

( 2 . 8 . 1 )

b ~

B~,C;

c' C >

/ a'

C>

c

cok(b') B ))B/B,,C

) B,.,C

~ A cok(a')

/ cok(c}

d

)) ~

); B v C l C

ON L A T T I C E - L I K E PROPEI~TIES OF ORTHOQUATERNAlZY CATEGOI~II~S 159

where [b', c', a, a ' ] is a pu l l -back (by D1)) ; d cok(b') = eok(a ' ) a (by 2.2); [q, d', d, eok(a ' ) ] is a pu l l -back (by D3) ) ; d' e = a'.

Now, b y 2.5, d ' ker(q) = ker (eok(a ' ) ) = a ' = d'c, hence ker(q) = c and

eok(c) = q, more precisely there exists a un ique i somorph i sm i such t h a t

i q = eok(c).

2.9 PROPOSITION. Let D_CA, B e D , C<a D. Then the ration in A of C

and B exists.

P~ooF. B y D3) the un ion of C a nd B exists in D. Le t us call it X. Le t D'_CA, C C_D', B C_D'. Then C a nd B are bo th con ta ined in D (~ D' .

D n D ' C _ D ; so X _ C D n D ' C _ D ' a n d X is the un ion of C and B in A.

w

3.1 By ax ioms D 1 ) 2 ) 3 ) there exist iu C finite in tersect ions of sub- objects , and un ions C w B, when C, B_CA and there exists D_CA with C<~ D and B c _ D .

We can now t rans la te the (( cubic )) ax ioms CW 4), OQ and OQ* by means

of lat t ice-l ike p roper t i es : a t this a im we consider these d is t r ibu t ive laws:

~u:(Bu M) r h ( C U M ) = (Brh C) U M

q ) n : ( B n M ) w ( B m N ) = B r h ( M u N )

for B , C - C A and M<~ A

for B O A ~md M, 2V-~ A

~u implies the m o d u l a r law:

. J I L : M t _ ) ( B ( 3 N - ) : B ( 3 ( M u N ) for B - C A ; M , N < ~ A ',rod M = B .

Now we need t he fol lowing:

3.2 LE~I~A. Given in C a cubic d i ag ram T of the fo rm (1), if its upper ,

rear and r ight faces are pull-backs, such is the lower one (D3) is unnecessary) .

PROOF. Le t [al, a2, ran, ml~] be a pu l l -back ; t h e n there exists a mon ic nl

such t h a t m S = aln~ and m~o = a2"nt. I f [a3, n2, m, , a2] and [a'~, .n~, m6, a~]

are pul l -backs , also [m, a3, n~, m12a.,, ms], [maa' 4, n:, renal , ms] are such. B y ral~a~ = mna~ we have an i somorph i sm i such t h a t n~ = n~i atld

msa'4i -= m4aa; p u t t i n g a~i - - a4 we ob ta in the square [a4, as, m~, real and the exis tence of an uilique m a p n~ such t h a t mln~ = a~ attd mona = a:,.

NOW a 2 n ~ = m T a a = m T m 2 n 3 : m~omsn a : a2~lm~na; so n. -- n~m~)~a and n I is an i somorph i sm as n. is an ep imorph ism.

3.3 PlCOPOSIT~O%. C verifies OQ iff ff)u holds.

1 6 0 G I U S : E P P E C O N T ~ E - F E R D I N A N D O M O R A

1)l~~176 ifu =:> OQ) Le t T be a cubic diagram of the form (1) whose upper face is a pull-back. So the set of vertices of T is {C N B, C, B, A, X, Y, Z, A/M}. By pull-back constructions of the pairs (msm~ = mlom~, ms), (ran, ms) and (rex2, m~) and pull-back properties we obtain a cubic diagram T', of the form (1), whose set of vertices is { ( B n C ) U M , C u M , B u M , A, X, :Y, Z, A/M} by D3) (2.6c).

~low all vert ical faces of T' are pull-backs, the lower one is the same as in T and the upper one is a pull-back by i fu"

Applying 3.2 to T' we get t ha t also the lower face of T is a pull-back. OQ ~ if u) Given C, B_cA and M<_ A, b y a pull-back construct ion

and factorizations, we obtain the same cubic diagram T as before where the lower face is a pull-back by OQ.

Build as before T' , where now the lower face, the left one and the r ight one are pull-backs. This implies tha t also the upper face is a pull-back ([12] 2.6 lemma 3) and so ( B n C) U M : (BuM) n(CUM).

D4) The union o/ normal subobjects is a normal subobject.

3.4 As a consequence, the anti isomorphism Ker-Coker between the poset of normal subobjects and conormal quotients is an anti isomorphism of lattices. In part icular C has finite intersections of quotients.

3.5 PI~OP0SlTIOI~. C verifies OQ* iff i fn holds.

PROOF. ff)n ~ OQ*) Le t T be a cubic diagram of the form (2) whose upper face is a push-out. By means of 3.4 and 2.2 we can say t h a t its set of vertices is

{A, AIM , AIN, AIM u N, B, B/M f5 B, BIN ~ B, BIB n (M u iV)}.

B y i fn we have K e r ( m n m g ) = B C ~ ( M L ) i v ) = ( B n M ) L J ( B n N ) , therefore, by 3.4, Ires, mlo, m~l, m~] is a push-out.

OQ* ~ if)n) Let B _c A and M, IV < A, b y push-out construction and factorizations we obtain the same cubic diagram T as before whose upper face is a push-out. The lower one is again a push-out by OQ* and therefore Br~(MUiV)- - - - - (B V~ M) ~) (B ~ N).

3.6 PI~OPOSlTIO~. C verifies CW4) iff ~L holds.

PI~ooF. J~L ~ CW4) Le t T be a cubic diagram of the form (2) whose left face is a pull-back and whose upper one is a push-out. By means of 2.5, 3.4 and 2.2 we can say t h a t its set of vertices is

{A, A/M, A/IV, AIM u IV, B, B/M, B/IV n B, BIB (~ (M U IV)}.

As B c~ (M ~J N) = (B n IV) L) M, the lower face is a push-out.

(3.6.1)

ON LATTIC]~-LIK~ P R O P E R T I E S OF ORTHOQUAT]~RNARu CAT]EGORI~S 1 6 1

]qow we bui ld the commuta t ive d iagram 3.6.1; i and j are second ~ o e t h e r i somorphisms~ so there is an isomorphism i ' t ha t fills in commuta t i ve ly ; apply ing twice 2.4 we find another i somorphism j ' t h a t commutes wi th the other maps.

HuN>

J )

' lMu N

)

IN

IN

J >

>

In

NnB >

uN"

J >HuN

MuN} >

> Mu

1HuN

>Bu

J

S

)B

l ~ B

>-) A/HuN

7.. )) A/N

~, )"

) ) MuN/N

N,

>> BuN/

))MuN/N

~-), BuN/MuN

I ~.> B/Bn[MuN] I , I I I )) B/Nn8

T o w b y D3) the r ight face of T is a pul l -back. CW4) ~ Jt(~) Let B ~ A ; M, 2~<~ A and M_CB~ b y a push-out con-

s truct ion and factorizat ions we obta in the same cubic d iagram T as before whose left face is a pul l -back b y D3) and whose upper one is a push-out b y construction.

CW4) implies t h a t the lower face is a push-out and so

B ~ ( M U N ) : ( B ( ~ N ) U M .

3.7 We can now state the ~xiom:

D5) For any object A and subobjects B, C c_ A ; M~ • ~ A, ~D u and ~D n hold.

] 6 2 G I U S : E [ ' P E C O N T E - F E I ~ D I N A N D O M O I ~ &

w

4.1 I n this paragTaph we will s ta te the connect ion between latt ice-l ike proper t ies of subobjects and o r thoqua te rna r i ty axioms ill the following:

4.2 PROVOSITIO~. Le t C be a category with 0, monic-epimorphism factor izat ion and conorma] epimorphisms, ver i fying D3).

The following assertions are equivalent :

a) C verifies D1), D2), D4), Dh).

b) C is an o r thoqua te rua ry bicategory.

PROOF: D1) is CW2); D2) is CW3).

a) => b): CW2*) is 3.4; CW4) is prop. 3.6; OQ and OQ* props. 3.3 and 3.5.

b) ~ a) : let us proof D4). Then D5) will be equivalent to OQ and OQ*.

Let B, C <~ A. We get, b y a pushout construct ion and factorizations, the cubic d iagram T of the fo rm (2) whose vert ices are {A, A/B, A/C, A/B W C, B, O, BIB n C, 0}. I t s uppe r face is a pushout b y const ruct ion; its left one a pu l lback (B is normal in A).

CW4) implies t ha t the r ight face is a pul lback. So m s ~ ker m~. By [4] 6.2 then, if [g, ], m~, ms] is a pul lback, g ~ ker(m~m2). B u t b y

D3) g: B ~) C ~ A, and so finite unions of normal subobjects are normal .

4.3 I:~E]~hAI~K. I f nq is the factor izat ion of pro, where m is a normal monic ~nd p an epimorphism, t hen the above proof assures t ha t n is normal .

4.4 REMARK. Axiom D3) is not a consequence of o r thoqua te rna r i ty , us i t is shown b y the ca tegory in [3] pug. 199. I t is, however, the essential tool (see, for instance, props. 2.6 and 3.3) for the t rans la t ion of the (~ cubic )~ axioms into latt ice-like propert ies.

Pervenuto in Redazione il 4 aprile 1977.

RIASSUNT0

Si individua un~ classe di c~tegorie per le quali l'esistenza e l'ortodossi~ della simmetrizzazione qua term~ri~ b esprimibile mediante propriet'~ di tipo reticolare di sottooggetti, enunciate nell~ formu di assiomi D1)-5).

Cib garantisce I~ componibilit~ degli isomorfismi canonici tra subquozienti e la validit~ dei teoremi di isomorfismo di Noether.

ON LATTICE-LIKE PROPERTIES OF ORTttOQUATERNARY CATEGORIES 163

SUMMARY

We point a class of categories for which necessary and sufficient conditions in order they have an orthodox quaternary symmetrization are expressible by means of Iat~ice-Iike properties of subobjects, these ones given by axioms D1)-5).

This implies tha t canonical isomorphisms between subquotients are composablc and that Noether isomorphism theorems hold.

BIBLIOGRAPHY

[1] P. ARDUINI, Monomorphisms and epimorphisms in abstract categories, Rend. Sere. Mat. Universit~ di Padova, 42 (1969), pp. 135-166.

[2] A. B]~LCASTRO - M. MARGIOCCO, The normodistributive expansion o/ a category, Ann. Univ. Ferrara, Sez. VII, 23 (1977), pp. 143-151.

[3] M. S. BURGIN, Categories with involution and correspondences in categories, Trans. Moscow Math. Soe., 22 (1970), pp. 181-257.

[4] M. GRANDIS, Symetrisations de categories et ]actorisations quaternaires, Atti Accad. Naz. Lincei Mem. C1. Sci. Fis. Nat., to appear.

[5] M. GRANDIS, Canonical preorder and congruence in orthodox semigroups and categories (Orthodox categories, 1), Boll. Un. Mat. Ital., 13 B (1976), pp. 634-650.

[6] M. G]%~NDIS, Induction in orthodox involution categories (Orthodox categories, 3), Ann. Mat. Pura Appl., to appear.

[7] M. G~ANDIS, Quaternary categories having orthodox symmetrization (Orthodox symmetrizations, 1), to appear.

[8] M. GRANDIS, Exact categories and distributive lattice (Orthodox symmetriza- tions, 2), to appear.

[9] J. ISBELL, Some remarks concerning categories and subspaces, Can. J. of Math., 9 (1957), pp. 563-577.

[10] S. MAc LANE, Homology, Springer, Berlin, 1963. [11] FULVlO MONA, Quaternary symmetrizations o/ bicategories, in preparation. [12] B. PAREIGIS, Categories and ]unctors, Academic Press, 1970.