Products and Sums yukita
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![Page 1: Products and Sums yukita](https://reader030.fdocuments.net/reader030/viewer/2022032415/56649f0c5503460f94c20606/html5/thumbnails/1.jpg)
Products and Sums
http://cis.k.hosei.ac.jp/~yukita/
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2
Products
• To express the notion of function with several variables
• We need to talk about products of objects.
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3
Ex. 1. Add and Multiply
ZYXf
yxyx
multiply
yxyx
add
:
form theof arrowan need We
),(
:
),(
:
RRR
RRR
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4
The one point set and the empty set
. is possiblityonly The
. ofsubset a is : above, case In the
.),(;!; hereproperty w with the
ofsubset a ,definitionby is, :function A
.:!by thisdenote also We
. to fromfunction oneexactly is thereset any Given [2]
{*}.:!by thisdenote We
{*}. to fromfunction oneexactly is thereset any Given [1]
f
XXf
fyxYyXxYX
YXf
X
XX
X
XX
Remark
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5
Prop. Property [2] characterizes the empty set.
.isomorphic are and that showswhich
,:1 and :1 have also We
.1 bemust which function oneexactly and
:function oneexactly is theresays ofproperty same The
.1 bemust which function oneexactly and
:function oneexactly is thereSo
.function oneexactly is there,set any For
[2].property thehasset that another be Let
Z
ZZ
Z
ZZ
Z
XZX
Z
Z
Z
Proof.
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6
Prop. Property [1] characterizes the one point set.
.isomorphic are *}{ and that showswhich
,{*}*}{:1 and :1 have also We
.1 bemust which {*}*}{function oneexactly and
{*}:function oneexactly is theresays *}{ ofproperty same The
.1 bemust which function oneexactly and
{*}:function oneexactly is thereSo
.function oneexactly is there,set any For
[1].property thehasset that another be Let
{*}
{*}
Z
ZZ
Z
ZZ
Z
ZXX
Z
Z
Z
Proof.
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7
What is the use of this kind of argument?
• We respect specification by arrows.
• Properties [1] and [2] are specifications.
• Corresponding implementations are the one point set and the empty set.
• There are many cases where specification determines implementation up to isomorphism.
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8
Def. Initial and terminal objects
.1 arrow unique a is therein object any for if
called is (one) 1object An
.0 arrow unique a is therein object any for if
called is (zero) 0object an category aIn
XX
terminal
XX
initial
A
A
A
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9
Ex. 2. Elements of a set
.) of or (or
of 1 arrows call wein terminalis 1 If
. ofelement an toscorrespond {*}function Each
Aconstantspoints
AelementsA
XX
A Def.
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10
Ex. 3. The power set 2X
not. do categories small objects;
terminaland initial have to tendcategories large general,In
arrow. oneexactly with
monoid theisobject terminala have tomonoidonly The
.2in terminalis itself subset The
Note.
4. Ex.
XX
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11
.conversely and
,in object initialan is then in object terminala is If
dual. are terminaland initial of sdefinition The op AA
Remark.
AA
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12
Products
.categoriesother in products of
concept apply thecan that weso arrowsonly using and to
it relatingby product cartesian thezecharacteri try toWe
ofcategory in the
},|),{(
by defined is and ofproduct cartesian The
YX
YX
YyXxyxYX
YX
Sets.
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13
YX YX 2p1p
x y
1
Cartesian product of X and Y
. and of , us gives , with composingthen
, of 1: an and 1object terminala have weIf
21 YXyxelementspp
YXYXelement
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14.then
,,
such that : arrowan is there
such that :, arrows Given two s)(uniquenes
.,
such that : arrowan is there
,:,: arrowsGiven )(existence
:assplit becan The
.,
such that : arrow unique a is there
,:,:
arrows and object any given if, and of a called is
arrows with twoobject an ,category aIn
2211
21
21
21
pppp
YXZ
YXZ
ypxp
YXZ
YZyXZx
conditionty universali
ypxp
YXZ
YZyXZx
ZYXproduct
YYXX
YXpp
Note.
ADef.
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15
YX YX 2p1p
x y
Z
Universality (existence)
commute. diagram following themakes that : arrowan have we
, diagramany Given
YXZ
YZX yx
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16
YX YX 2p1p
Z
Universality (uniqueness)
).,(by denote We
. have then we, and
such that :, arrows any twoGiven
2211
yx
pppp
YXZ
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17
. and such that :
arrowan is e that thermeans ofproperty defining The
. toisomorphic is that
show Weproduct.another is Suppose
2211
21
qpqpYXQ
YX
YXQ
YQX qq
Proof.
Prop. The product of two objects in a category is unique up to isomorphism.
YX YX 2p1p
1q 2q
Q
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18
. and such that :
arrowan is e that thermeans ofproperty defining The
2211 pqpqQYX
Q
)(continued
YX YX 2p1p
1q 2q
Q
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19
YX
Q
2q1q
1q
Q1
2q
Q
.1 have we
property, uniqueness By the
Q
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20
YX
YX
2p1p
1p
YX1
2p
YX
.1 have we
property, uniqueness By the
YX
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21
Note
.1 and 1 such that : and
: morphisms twoare thereif isomorphic be to
said are and category. a of objects are and Let
BA gffgABg
BAf
BABA
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22
Ex. 6. Category 2X
product. of uniqueness theguaranteesproperty This
arrow. onemost at is thereobjects ofpair any Between
.
isproduct Their . of subsets are and Let
.in contained is means category In this
product. a has 2category in the objects ofpair Each
VVUU
XVU
BABA
X
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23
Preordered Category
• The product of two objects, if it exists, is their intersection.
• In other words, the greatest lower bound of the two objects.
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24
Ex. 7. The monoid with one object A and two arrows 1A and , satisfying 2=, does not have products.
. arrow single and
, arrows of pairsbetween bijection is thereand
again. be it would existed, If
AAAA
AAAA
AAAAA
AAA
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25
Ex. 8. The Diagonal Function
function. or thecalled
),,(
:
function, a is thereset aGiven
copyfunction diagonal
xxx
XXX
X
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26
Def. Diagonal in an Arbitrary Category with Products
:commute diagram following themaking arrow unique theis X
.XX XX 2p1p
X1X X1
X
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27
parallel.in and functions two thesay, toSo
)).(),((),(
:
:by denotedfunction a is there
:,: functions given two ,In
2121
2121
2211
gf
xgxfxx
YYXXgf
YXgYXf
Sets9. Ex.
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28
Def. Parallel Functions in an Arbitrary Category with Products
commute. diagram following themaking arrow unique theis arrow the
,: ,: arrows Given two 2211
gf
YXgYXf
2Y1Y21 YY
2Yp1Yp
f gf g
21 XX 1X 2X2Xp
1Xp
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29
Ex. 10. The Twist Function
),(),(
:
function a is there and sets given two ,In
,
xyyx
XYYXtwist
YX
YX
Sets
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30
Def. The Twist Function in an Aribitrary Category with Products
:commute diagram following themakes that arrow unique theis Then
. of sprojection thebe , and of sprojection thebe ,Let
,
2121
YXtwist
XYqqYXpp
.XY XY
2p 1p
1q
twist
2q
YX
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31
Object oriented viewpublic class ObjCatA{
}
public class ProdCatA extends ObjCatA{
ObjCatA x, y;
public ProdCatA(ObjCatA x, ObjCatyA y){
this.x = x; this.y = y;
}
public ArrowCatA<z,this> factArrows
(ObjA z, ArrowCatA<z,x> f, ArrowCatA<z,y> g){
return /* the unique arrow that satisfies
the property in the last slide */
}
}
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32
Ex.11. Category Circ
.0*
:
and
,1*
:
namely , from functions twoare There
sets. esebetween th functions all :Arrows
}|),,,{(
,{*}, where,,,, :Objects
}.1,0{Let
10
10
10
21
10210
BBfalse
BBtrue
BB
BxxxxB
BBBBBB
B
inn
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33
Negation, And, Or
1(1,1)
1(1,0)
1(0,1)
0(0,0)
: ,
1(1,1)
0(1,0)
0(0,1)
0(0,0)
:
01
10 :
1212
11
BBorBBand
BB
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34
Claim. Category Circ has products.
products. ofproperty check themust We
),,(),,(),,(
follows. as and ofproduct thedefine We
111
21
nmmnmm
npnmnmpm
nm
xxxxxx
BBBBB
BB
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35
.nBmB nmB 2p1p
f g
X
property. ith thisfunction w oneonly theis
thatand , and check thateasily can We
)).(),(),(,),(()(Then
)).(),(()( and ))(,),(()(Let
21
11
11
gpfp
xgxgxfxfx
xgxgxgxfxfxf
nm
nm
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36
not
BB :
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37
&
BB2 :&
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38
BB2 :or
or
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39
wires.up splits : 2BB
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40
side.by side components twoputs : 22 BB gf
f
g
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41
series.in components twoputs : BB fg
f g
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42
wires. theof twos twist:twist 2 2BB
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43
Boolean Gates f(x,y,z)
&
or
not
not
not
not
&
&
&
x
y
z
),,( zyxf
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44
),,()&&,&&(
)&&,&&(),&,,&(
),&,,&(),,,,,(
),,,,,(),,,,,(
),,,,,(),,(
or
2&&
41&1&6
6116
6333 3
zyxfzyxzyx
zyxzyxzyxzyx
zyxzyxzyxzyx
zyxzyxzyxzyx
zyxzyxzyx
BB
BB
BB
BB
BBBB
2
4
BB
BB
B
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45
3)11()1&1(&&)(&or
follows. ascircuit aby dimplemente becan
BBBBB f
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46
Summary
• Using wires, we can implement products.
• Every function BmBn can be implemented using not, &, or, true, false, using products and composition.
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47
commute. diagram
following themakesthat
arrow unique a is there
),( :
arrows offamily a and object any Given
property. following thehaswhich
),( :
sprojection offamily a with togetherobject an isfamily theof
product The . of objects offamily finite a be )(Let
KlXZf
Z
KlXXp
X
X
ll
lKk
kl
Kkk
Kkk
ADef.
.lXKk
kXlp
lf
Z
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48
Note. The product of the empty family
The product is a terminal object. Since the family is empty, the only requirement is that, given Z, there is a unique arrow from Z to the product.
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49
. and such that :
arrowan is e that thermeans ofproperty defining The
. toisomorphic is that
show Weproduct.another is Suppose
llllKl
l
Kll
Kll
lq
qpqpXQ
X
XQ
XQ l
Proof.
Prop. The product of a family of objects in a category is unique up to isomorphism.
lX Kl
lXlp
lq
Q
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50
. such that :
arrowan is e that thermeans ofproperty defining The
llKl
l pqQX
Q
)(continued
X Kl
lXlp
lq
Q
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51
lX
Q
lq
lq
Q1
Q
.1 have we
property, uniqueness By the
Q
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52
lX
lp
lp
Kl
lX1
Kl
lX
.1 have we
property, uniqueness By the
Kl
lX
Kl
lX
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53
Prop. 4.2. If products of all pairs of objects exist in A and a terminal object exists then products of finite families exit.
. then and unique have wediagram, in the as ,,,Given
.,,Let
.)(family theofproduct theis )( that show will We
3212211 321
}3,2,1{321
hgfZ
pppppppp
XXXX
XXXXXXX
kk
Proof.
321 XXX 21 XXp
1Xp
2Xp
f
gh
3Xp21 XX
2X
Z
.3X
1X
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54
.)(product the
ofproperty uniqueness by the Hence
. and onto projectionwith
composites same thehave and Hence,
.product theofproperty uniqueness by the Hence
. and onto sprojection with composites same thehave and Then,
.
and ,
,
equations. threefollowing thesatisfying arrowanother is
Suppose unique. is such that show We
321
321
21
21
33
22
11
2121
2121
3
2212
1211
XXX
XXX
XXpp
XXpp
hppp
gppppp
fppppp
XXXX
XXXX
X
XXXX
XXXX
inued).Proof(cont
321 XXX 21 XXp
1Xp
2Xp
f
2121
XXXX pp
gh
3Xp
21 XX
2X
Z
.3X
1X
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55
e)associativ(Strictly
indentity. theis misomorphis thecategory In
. toisomorphic areBoth
.isomorphic are )( and )(
321
321321
Circ14. Ex.
Proof.
Cor.
XXX
XXXXXX