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 β’ β’ For morphisms f : c π d and g : d π e, there is a composite morphism gf : c π e Examples: β’ Sets & functions β’ Groups & group homomorphisms β’ T opological spaces & continuous functions
Categories An isomorphism is a morphism f f : c π d with g : d π c so that fg = I d c d g and gf = I c Examples β’ Set: bijections β’ Group: group isomorphisms β’ T op: homeomorphisms
Functors Functor: a map F : C D between categories taking π C objects to objects and morphisms to morphisms β’ Preserves identity morphisms β’ Preserves function composition Examples: β’ Forgetful: Group π Set sends groups to sets of elements β’ C(c, - ): C π Set sends x to set of morphisms c π x and morphisms x π y to C(c,x) π C(c,y) by postcomposition c β’ Constant: C π c sends every object in C to c, every morphism to the identity on c
Diagrams Diagram F : J π C: i c d β’ An indexing category J of a certain shape f h β’ A functor F assigning objects and morphism in C to that shape cβ d g β
Natural Transformations Ξ± c Natural transformation F β G of Fc Gc functors F, G : C π D: β’ A collection of morphisms called F Gf components Ξ± c : Fc π Gc f β’ For all f : c π cβ, the diagram Fc Gc commutes Ξ± c β β If the components are isomorphisms, β we have a natural isomorphism F β G
Cones Cone over a diagram F : J π C: c β’ A natural transformation between the constant functor c : J π c and the diagram F : J π C Ξ» 1 Ξ» 4 Ξ» 5 Ξ» 2 Ξ» 3 The components Ξ» j are called legs β’ F(1) F(2) F(3) F(4) F(5)
Universal Properties A functor F : C π Set is 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 π cβ The functor F encodes a universal property of c
Limits A limit is a universal cone: c β’ There is a natural isomorphism C( - , lim F) β Cone( - , F) Lim F β’ Morphisms c Lim F are in π bijection with cones with summit c over F F(1) F(2) F(3) F(4) F(5)
Limits in Geometry Z Product X x Y X Y Spaces Diagram shape Product of spaces
Limits in Geometry Pullback f -1 ( x ) Y f X * Spaces i Diagram shape Fiber of x=i(*)
Conclusion 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
Reference βCategory Theory in Contextβ by Emily Riehl
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