Categories of Functors
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Transcript of Categories of Functors
Categories of Functors
http://cis.k.hosei.ac.jp/~yukita/
2
Functors
. if )()()(
,1)1( satisfying
, ))(),((Hom ),(Hom:
, of , objects ofpair each for and,
, obj obj: functions of
consits to fromfunctor A .categories be and Let
. to from arrows ofset thedenotes ),(Hom
.in objects twobe , andcategory a be Let
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obj
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BABA Def.
AA
3
Ex. 1. Group Homomorphisms
.11 and )(
such that
,arr arr :function a is :functor athen
),invertible arrowevery with categoriesobject (one
groups are and If
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FF
BABA
BA
4
Ex. 2.
map. preservingorder an is
to fromfunctor a then posets, are and If BABA
5
Category Cat
functors. are arrows
whoseand categories are objects hosecategory w a is
. functor a is Then
.
functors beG F, and ,categories be and , ,Let
small
GF GF
GF
Cat
CA
CBA
CBA
6
Ex. 3. Functors are ConstructionsBasis Vector Space
FXX
X
FXX
FXX
VextSets :onconstructi thegives This
.in elements of nscombinatio
linear finite formal all ofset thebe to Take . basiswith
space vector aconstruct can we,set finite aGiven
7
Ex. 4. Functors are ConstructionsX Stacks of X
).()()(
)()(:)(
have should we,:Given
:
:onconstructi thegives This . of stacks all ofset the
),(set new aconstruct can we,set any Given
2121 nn xfxfxfxxx
oo
YStackXStackfStack
YXf
Stack
X
XStackX
SetsSets
8
Ex. 6. Functors are representations (or models) of categories
arrows.identity on for choice
no s there';10function a and ,1 and 0
sets, twogive tohave we,:functor a give To
.10category a be Let
F
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SetsA
A
9
Ex. 7. Functors are representations (or models) of categories
.)(
,in and each for such that :
npermutatio a , each toand , ofobject one the
toingcorrespond set a of consists :functor A
group. a be Let
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XXF
XF
A
AA
SetsA
A
10
Def. Faithful/Full Functor
.
call wesurjective are functions theseall instead, If, injective. is
),(Hom),(Hom:function the,,
objects ofpair each for if BAfunctor a call We
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full
faithful
F
FAFAAAFAA BA
11
Ex. 8. Giving a functor is specifying three sets and two maps in Sets.
0
1
2
Category A Sets
12
Remark
• In Ex. 8, three sets and two arrows constitute a single functor.
• Complex entities can be thus represented.
13
Ex. 9-11
.)(
,*
is,That .:function idempotentan with set a is
to fromfunctor . satisfying **: arrow
one and *object oneby generatedcategory thebe Let
.in objects of sequence a
is to},,,1,0{ fromfunctor A
.in objects ofpair ajust is to}1,0{ fromfunctor A
22
2
FeFeeFFe
FeXF
XXX
eee
n
SetsAA
A
B
BA
BBA
14
Ex. 13. Directed Graphs
functions. parallel twoand
, sets two
of consists to fromfunctor A
:category thebe Let
YX
SetsA
A1 0
X Yd0
d1
15
Ex. 13. continued
edcd
adcd
cdbd
cdbd
bdad
ecbaYX
dd
XY
: :
: :
: :
: :
: :
and },,,,{ and },,,,{Let
edges.an to verticesassign two and
edges. directed ofset a as and vertices,ofset a as ofThink
10
10
10
10
10
10
16
Ex. 13. continued
b
a c e
17
Ex. 14. Deterministic Automata
. ,(*) have We
.in letter each for : smendomorphi
an with set a as thingsame theis :functor A
.},,,{alphabet on monoid free heConsider t *
a
a
fFaXF
XXf
XF
cba
SetsA
A
18
The Functor Category from A to B
),( :
: of arrows
offamily a is ) a called (also to
from morphismA . to from functors be Let
A
B
BA Def.
AGAFA
GF
F, G
A
tiontransforma natural
2FA
1FA
2A
Ff Gf
1GA
.2GA
1A
19
Ex. 15. A natural transformation from F to G is a commutative squares in B.
.10: arrowan of consists to from functor a
;10: arrowan of consists to from functor A
.10 :category aConsider
GGGG
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BA
BA
A
1F
0F
1
F G
0G
.1G
0
20
Composites of Natural Transformations
.HBFB GBBB
Ff Gf Hf
GAFA HAAA
21
Def. Functor Catetories
ations. transformnatural ofn compositio isn compositio where
;: ations transformnatural are arrows the
; functors are objects the
:follows as defined is
,by denoted category,functor the, and categorisGiven
BA
BA
BBA A
GF
22
The category of directed graphs Grphs=SetsA
codomain. thecalled is
and domain, thecalled is
functions. with two
,, sets ofpair a is
. to fromfunctor a
is of object An
:category thebe Let
1
0
10
d
d
XXX
X
SetsA
Sets
A
A
1 0
X0
d0
d1
X1
23
A morphism in SetsA is a pair (0,
1).
0X ,0Y0
0d 0d
1Y1X1
0X .0Y0
1d 1d
1Y1X1
24
Def. Regular Languages
. arrow special awith together of elements the
being arrows with and , *object one graph with theis
).object end the( and )object beginning the(
objects heddistinguis twograph with finite a is where
: morphismgraph a isgrammar regular A
S
KJ
X
SX
25
Remark
.under togoy which the}{ of elements
with the of edges thelabeling and sketchingby
drepresente becan : morphismgraph A
.under togo arrows then whererestrictio no is There
. ofobject one the togomust ofobject Every
XX
SX
SX
26
Ex. 21. A regular grammar
*
.KJ
b
a
b
a
27
Note. A morphism of graphs
.)(:
then,in path a is
Suppose follows. as defined is ly;respective
and graphs on the categories free theare and where
:
functor a induces : graphs of morphismA
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X
xxx
YXYX
YX
YX
n
28
Def. Regular Languages
grammar. by the
generated thecalled is s' omitting
thenand applying and to from paths all taking
bygrammar regular a from obtained ofsubset The *
language regularKJ
29
Ex. 28. The Union Operation
1K1J
,KJ
2J
2K
U
V
30
Ex. 30. The Star Operation
,KJ
1K1JU
thenfor grammar a represents If
.in wordsofnumber any ingconcatenatby
obtainedset the, is so then languageregular a is If
11
**
UKJ
U
UU
U
.for grammar a is *U
31
Ex. 31. The Concatenation Operation
.for grammar a is
thenly,respective
, and grammarsrepresent and If
},|{
2211
2211
UV
KKJKJJ
VUKJKJ
VvUuuvUV
VU
VU
32
Ex. 32. Pumping a regular language
. claims which path, in the loop a
be should thereprinciple, holepigeon By the path. in the objectssuch )( are There
:factor first on the landspath that ofPart .in path a of image theis
.let and graph in objects of
number thebe let and , defininggrammar a is : Suppose
language.regular anot is Then,
}.,2,1,0|{ and },{Let
|1
||3
|2
|1
Ubaa
nkl
xxxxx
aXbaa
nkX
nUSX
U
kbaaUba
mkk
la
laaaa
kkk
kk
Proof.
1x a
1ix
.lx2x
1x
a
jx
a… …
…
33
Theorem. (Kleene)
. to from paths same the
keeping alwaysbut objects, removingby graph modify thely successive willWe
. to from edges labelled ofset thedenote Let
object. end oneonly is e that therassumemay We
s.' eliminatecan We
part. if"only " theprove Weshown.already ispart if"" The
. and ,, operations ofnumber finite a applying
by of subsetssignleton from obtained becan it ifonly and if is, that ;expression
regular aby described becan it ifonly and if languageregular a is ofsubset A
,
*
*
KJ
X
yxU
VUUVU
yx
Proof. a of Sketch
34
Repeat the procedure until we get to the following graph.
JJU , KJ KKU ,
KJU ,
JKU ,