Category Theory in Geometry Abigail Timmel Mentor: Thomas - - PowerPoint PPT Presentation
Category Theory in Geometry Abigail Timmel Mentor: Thomas Brazelton Categories Category: a collection of objects and morphisms between objects Every object c has an identity morphism I c For morphisms f : c d and g : d
Mentor: Thomas Brazelton
Category: a collection of objects and morphisms between objects β’ Every object c has an identity morphism Ic β’ For morphisms f : c d and g : d e, π π there is a composite morphism gf : c e π Examples: β’ Sets & functions β’ Groups & group homomorphisms β’ T
An isomorphism is a morphism f : c d with g : d c so that fg = I π π
d
and gf = Ic c d f g Examples β’ Set: bijections β’ Group: group isomorphisms β’ T
Functor: a map F : C D between categories taking π
β’ Preserves identity morphisms β’ Preserves function composition C c Examples: β’ Forgetful: Group Set sends groups to sets of elements π β’ C(c, - ): C Set sends x to set of morphisms c x and π π morphisms x y to C(c,x) C(c,y) by postcomposition π π β’ Constant: C c sends every object in C to c, every π morphism to the identity on c
Diagram F : J C: π β’ An indexing category J of a certain shape β’ A functor F assigning objects and morphism in C to that shape c cβ d d β f g h i
Natural transformation F β G of functors F, G : C D: π β’ A collection of morphisms called components Ξ±c : Fc Gc π β’ For all f : c cβ, the diagram π commutes If the components are isomorphisms, we have a natural isomorphism F β G Fc Fc β Gc Gc β F f Gf
Ξ±c Ξ±c
β
Cone over a diagram F : J C: π β’ A natural transformation between the constant functor c : J c and π the diagram F : J C π β’ The components Ξ»j are called legs c F(1) F(2) F(3) F(4) F(5)
Ξ»1 Ξ»2 Ξ»3 Ξ»4 Ξ»5
A functor F : C Set is π representable if there is an object c in C so that C(c, - ) β F β’ Recall C(c, - ) takes an object cβ to the set of morphisms c cβ π The functor F encodes a universal property
A limit is a universal cone: β’ There is a natural isomorphism C( - , lim F) β Cone( - , F) β’ Morphisms c Lim F π are in bijection with cones with summit c
c F(1) F(2) F(3) F(4) F(5) Lim F
Product X Y XxY
Diagram shape
Product of spaces Z
Spaces
Pullback
Diagram shape
f-1(x)
Fiber of x=i(*)
Y X f i
Spaces
Category Theory is everywhere β’ Mathematical objects and their functions belong to categories β’ Maps between difgerent types of objects/functions are functors β’ Universal properties such as limits describe constructions like products and fjbers
βCategory Theory in Contextβ by Emily Riehl