Commentationes Mathematicae Universitatis Carolinae
Jiří Sichler
Category of commutative groupoids is binding
Commentationes Mathematicae Universitatis Carolinae, Vol. 8 (1967), No. 4, 753--755
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Commentationes Mathematicae Univers i t a t i s Carolinae
8 , 4 (1967)
CATEGORY OP COMMUTATIV1 GROCPOID& IS BINDIHG
J i ř í SICHLER, P r aha
Let
C/L ( 1 , 1 )
denote the category of a l l
(universal)
algebras with two unary operátions and t h e l r homomorphisms.
Similarly,
t
(Z )
meane the category of a l l commutative
groupoids together with t h e i r homomorphisms*
A category
&
%
i s c a l l e d representable i n a category
if» there e x i s t s an i s o f unctor
subcategory of
onto a f u l l
«£ .
A category i n which
l e d binding.
<$ : 3C -> *&
Ol ("1,1 ) i s representable i s c a i -
(Cf.tl]).
By methode very s i m i l a r t o those ušed i n [2J i t i s p o s s i b l e t o prove
Theorem 1 .
Ot C 1 , 4 )
i s representable i n
*£(!).
One of the c o r o l l s r i e a i s tlue folLowíngTheorem 2. On every s e t
such that Ci) co (x^y.
y
in
X , Cii) t h e algebra
t y mapping of
of
) ~ co (<y.9 x )
( X*, co )
f o r any
and
has only the i d e n t i -
A -
(X > Cf , yr )
be an object
OL C"M), Put Z = X u í ^ (X), #2 o o , x3(XU
X n i z^ (X )> &±(X),
« ( Z *y co ) , o>
x
X as an endomorphisnu
Proof of Theorem 1 . Let
where
X there i s a binary operát ion
z,$ (X)
} ~ 0
being the binary operát i on on
follows:
- ¥53 ~
.
>
Put <f» CA)«
Z def ined as
(Wa write
m^ inetead ef z± (A )
eyte, 2r3)=ř ^ftr 3 , -z ) = Z^
.)
for every x inZ
,z
4* z^
a> O , ^ ")ma>(Xi, *)**<$<* > f * étery x 6 X
^ C * , ^ ^ ^ ^ , * ) * * ^ * ) *or every * * X
^3
<L> Č I X , ^ ) s
ř « every *x 6 X , <y. e X
CO (Zi , ar^ ) * * 3
O (Z2,Z2)
t*r i + á
o* <i ~ £ ** A
» ^f
Lat A ' * CK') Cf' ,1?' ) ba another objeet ©f ( ^ C V ^ ) ,
#C/4') * f X ' u { * f r X ' ) ,
| - A -> A ' ba a morphiaia of
Í $ 6 4 ) - > $fA'>
* 3 f X ' ) j 5 &')
J^CK'),
Oí (1, 1 ) . Define $ < Y ) ;
by
$Cf ) ( * ) « «f Cx )
for every * in X
*<f)te*CX»» * j O < ' )
Clearly,
tion, $
of
$ ř#)
for
i . ^ , ^ 3
ia jaorphiam in <í?(2)
.
and, in addi-
ia a one-to-one functor.
It remains to prove that Ita image ia a f u l l subcategory
*€ (2) .
In the aequel, co and co*
poaition.
Také
g, : $ (A)-*
Provided
s
, let
}
£^3
=r 2 ^
^
§ (A' )
9. ( ^ 3 ) € X '
ť
£ *3 '
) « xf%
s
will ba deeignated by juxta-
?
- a Borphiam of
we háve <fr(Z2 ) * 9, f ^ ^ 3 ) =
^3 > further
-
*ř C 2 ) .
9. f « 3 ) «
9 (z% z^ ) *
» C O I l t r a d i c t ÍOH.
Similar computation ean be uaed for the proof of g~ (z^ ) #
+ ** i * » 4f 2
- ^ o * only
9»ť£ t ) s
laat fact yielda immediataly
U
tyC*')
= zf%
X3
g. CZ% > * z'%
ia poaaible. The
and <^(Z1 ) ~ z^.
for aome x in X f then
- ¥f 4 -
9. Cz 3 ) *
« a <\x*x )*;*£;*£ « ^
£ ;&'
- «• c o n t r a d i c t i o n , e i m i l a r l y ^ G x ) ^
i s obtained. F i n a l l y , sup po se
g, č y Cx >) « 9. CvX a^ )KX^X^
q,(x
* Z^ . ke
) ** ^
Cf(x
)e A
. It is
;this
is
a contradictory t o the preceding statement#
Hence
q, (X)
£ X',
£<a^ ) »
Define a mapping -f •. X - * X '
^
by -f f*x )& g^(^ ) . I t i s
and we obtain the samé re s u i t f o r W
a morphiea i n
/
Oí C í1 1),
for i * 1,2,3.
and ty
• Hnus
f
is
$ Cf > * fy •
Proof of Theorem 2» a) Xf
X
i s i n f i n i t e the s t a t e -
ment i s an easy consequence of [3],C2J and the conatruction
given i n the proof of Theorem ! •
b) Let
X be f i n i t e ,
X « { J C 1 9 - • • , «X^ \ . The bina-
ry symmetric operation w i l l be defined by c*>f%X^,*>Q ) « iX^
for
-i *¥
fr
or -i * ^ * TL
co (z*i^ x ^ > s
»x i 4 . ^
for i .. 4 , . . . , ^
- 4 *
A l i t t l e computation concludes the proof*
I am indebted t o A.Pultr for the suggestion of the problém*
R e f e r e n c e
[1]
s
Z. HEDELÍN, A. PULTE and V. TRNKOVÁ: Concerning a c a t e g o r i a l approach t o t o p o l o g i c a l and algebraic
theorie3«Froceeding8 of the Second Prague Topol o g i c a l Symposium 1966,Praha 1967,176-181,
t23
Z. HEDRLÍH, A. PULTR: On f u l l embeddlngs of eategories
of a l g e b r a e . I l l i n o i s J . o f Math.10(1966),392-406.
[33
P . VOPfiHKA, A. PULTR, Z. HSORLÍHS A r i g i d r e l a t i o n e x i s t s
011 any set.Comment.Math.Univ»Carolinae,vol*6
(1965),149-155.
(Received November 13*196?)
-fff5
-