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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A

A

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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

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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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FF

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

212121

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 ,