ELEMENTS IN MATHEMATICS FOR INFORMATION SCIENCE NO.14 CATEGORY THEORY. Tatsuya Hagino

Transcription

1 1 ELEMENTS IN MTHEMTICS FOR INFORMTION SCIENCE NO.14 CTEGORY THEORY Tatsuya Haino

2 2 Set Theory Set Theory Foundation o Modern Mathematics a set a collection o elements with some property c x loical ormula about x Description o elements is important: i.e. when x Limit o set theory Russell's Paradox The collection o all sets is not a set. R = x x x} R R or R R

3 3 Cateory Theory lternative oundation o Mathematics Some call it bstract Nonsense. Describe thins with relationship with others. Set Theory x description o inside contents Cateory Theory description rom outside actions Uniy many concepts in one. Can see symmetry easily.

4 Cateory cateory C consists o: collection o objects: C =,, C, For objects and, a collection o arrows (or morphisms): hom C, = {,, h, } I hom C (, ), we may write it as: : is the domain o is the codomain (or rane) o 4 cateory C mush satisy the ollowin properties: For : and : C, : C For :, : C and : C D, h = (h ) For each object, there exists an identity arrow 1 : and or :, 1 = and 1 = C (h ) h D h C 1 1

5 5 Example o Cateory Set: the cateory o sets objects: sets arrows: unctions is the unction composition 1 is the identity unction o Grp: the cateory o roups 1 ) objects: roups (G,, e, arrows: homomorphisms is the unction composition 1 is the identity unction o 1 (G,, e, ): roup x y z = x y z x e = e x = x x x 1 = x 1 x = e homomorphism: : G H x y = x y CPO: the cateory o complete partial ordered sets objects: CPO arrows: continuous unctions is the unction composition. 1 D is the identity unction o D

6 6 Example o Cateory (cont.) Monoid M,, e as a cateory object: only one object arrows: M (i.e. elements in M) is 1 is e (M,, e): monoid x y z = x y z x e = e x = x Partially ordered set (D, ) as a cateory objects: D (i.e. elements in D) arrows: (i.e. at most one arrow rom x y) is ``i x y and y z, then x z'' 1 x is ``x x'' (D, ): partially ordered set x x i x y and y z, then x z i x y and y x, then x = y

7 7 Dual Cateory Dual Cateory C op o cateory C C op objects = C objects C op arrows: hom C op, = hom C (, ) Reverse the direction o arrows. C C C op C ny property which is true in cateory C is also true in its dual cateory C op. (C op ) op = C

8 8 Mono, Epi and Isomorphic Mono : is mono-morphism or any object D and any arrows : D and h: D, i = h, then = h. Epi h D : is epi-morphism or any object D and any arrows : D and h: D, i = h, then = h. Isomorphic Object and is are isomorphic there are : and : such that = 1 and = 1. h D Mono in C op is epi in C. Epi in C op is mono in C. Isomorphic objects play the same role in C.

9 9 Initial and Final Objects Initial object I or any object, there is a unique arrow rom I to. I! Final object F or any object, there is a unique arrow rom to F.! F Theorem: The initial object, i it exists, is unique up to isomorphic. Dual Theorem: The inal object, i it exists, is unique up to isomorphic. Initial Object Final Object Set { } Grp { e } { e } CPO { } Partially Ordered Set 丅

10 10 Product and Co-Product is the product o and There are two arrows π 1 : and π 2 :. For any object C and arrows : C and : C, there exists a unique arrow h: C such that the ollowin diaram commutes: + is the co-product o and There are two arrows ι 1 : + and ι 2 : +. For any object C and arrows : C and : C, there exists a unique arrow h: + C such that the ollowin diaram commutes: π 1 π 2 ι 1 ι 2 + C!h C!h π 1 h = π 2 h = h ι 1 = h ι 2 =

11 11 Product and Co-Product Set Partially Ordered Set D, Product = { x, y x and y } π 1 x, y = x π 2 x, y = y For : C and : C, h z = z, z x y = x y π 1 : x y x π 2 : x y y For : z x and : z y, h: z x y Co-Product + = x, 1 x y, 2 y ι 1 x = x, 1 ι 2 y = y, 2 For : C and : C, h x, 1 = x and h y, 2 = y + = x y ι 1 : x x y ι 2 : y x y For : x z and : y z, h: x y z Theorem: I product exists, it is unique up to isomorphic. Dual Theorem: I co-product + exists, it is unique up to isomorphic.

12 More Topic o Cateory Theory 12 More diarams Equalizer and Co-Equalizer Pushout and Pullback Limit and Co-Limit Functions between cateories Functor and Natural Transormation djunction let and riht adjoint Product is the riht adjoint o diaonal unctor. Co-product is the let adjoint o diaonal unctor. Exponential (unction space, riht adjoint o product) Natural Number Object Cartesian Closed Cateory Topos

13 13 Summary Cateory Theory lternative oundation o Mathematics Cateory Objects and rrows Special objects Initial and inal objects Product and co-product