Product HE II 112 112 112 x 141,1 112 contains Sue.IE 3 et Ise - PDF document

5 co products and categories Products products with rectus difference between products we'll motivate the co What is digg IR infinite dimensional space Question several things this could mean There are Product HE II 112 112 112 x 141,1 112 contains Sue.IE 3 et Ise lREitolRe ztolRej o contains 5e f finite sums Vi t t Via 3 O O O O 45,99 5 Note There is E lR canonical embedding a non empty family of If Xa a A Def then a is objects a prod t of the Xj of an fbject P and a pair P is pa Xa satisfying the following unipoperty Pa P morphisms morphisms far It Given any H and a c A object Xa paf I F H P s 1 faeA 3 a pair S W object S and of the XIs of is coat A an satisfying the following universalproperhy y Xx S morphisms la Xp H Given any It LEA and morphisms object s 1 flea fin F FLEA f S It

E X H It X coproduct 7 i product if F yep Ph JS F of the XIs is the smallest object that projects into poet each factor Xx is the smallest object into of the Xp The coproduct or seen which each factor Xx maps and La Xx usually Pa P 75 Xi let Xj REE IR Back to vector spaces for motivation H H in It I X X X x Xzx X Xr c X Xz 4 Pi Pz Lz what does not work Xr X X xx x fr Y ii t Xr X X xx C x X X X X E Lz L Pi Pc T too big too small if How to define U l l

0 Note that this construction works regardless of the number of factors uncountable etc finite countable and all X Ga We'll focus on the case where is a maps gap homomorphisms are x c A it is unique If exists for G Propst P a product Pa and each pg P is ont up to G isomorphism We have for each a A PI let P Pa'D be another product pi P P Ga P Ga and h i 9 F pla y i P p we get for each LEA If the diagrams stack we P Pa we also know P P but G G I 9 up i Pa 1 p w Pon f up p Pa Paf p fg Pa pal Similarly gf 1p so f and g fog Ip are By uniqueness inverse isomorphisms we'll show each pin is onto Next define Fp Ga For each Gs to be Gp

I c Gp trivial map X C Gpt G fp x XE Gp x e Gp G identity map For all BEA Fix H G special case for 1a H Ga Ga Ga Gp of pea T P P D 1 since 1a paf and so is pa is onto Thm5 nonempty family of groups then If Ga Left is a a A the product of exists Ga show that the Cartesian product is the product PI We'll P TI Ga let LEA write elements where Xa ya Xx as Xaya A pa P Define the projection map Ga as XL Xp Suppose fa H for all DEA G is homomorphism a f h Define f H Fach deA P Cheech paf f Echl H h Ga T pl ft i F Pa ffdh p A

and H Ga Consider Uniqueness Hp i Jyp Paf Pag i.e Then ffcxik pafcy pagcx fg.ly V Axe H fly _gCx D G fix Gn If we write 42,3 on A 42 n or denoted TIG This is the directproduct G xGzxG Left The homomorphism pi TIG is the projection of the G LEA product onto the directfactor Ga Suppose G Q E G Th 3 satisfy A G i Gi Gz 1 ii G n Gz iii G Gz G Then GE G xGc X Xz uniquely with Xi C G since GnGz 1 each X PE ketch Also Eli X Xi x KEG NG I X Xz Xz4 Define the projection maps pilx it check homon 2 Xi homomorphisms f Now for any It i 42 Gi f h fdh fch H G define f Check that pif fi

also satisfies pig fi Suppose g H G for each i Uniqueness for any xfH p f Then then we have pig Axl p fan paffex p glad pdgki glad Since products exist and are unique G is and a product GIG xG follows that it D Thm 5.3 naturally extends to Reina 2 factors Condition ii n for each i 1 G NG 1 Gin Gj becomes we say that G is the when the assumptions of Thin 5.3 hold internal direct product of its subgroups G Ga G Gn o Isiah consists of Defi A category E A class of objects E Oble A class of morphisms Hom e between objects with II 4 Identity morphism 1A A A for all A c Ob E Composition Fe HomelA B gofe Home Acc c Home B C Cii g ho goof hog of Associative Iii a directed multi graph Think of a category as vertices morphisms object edges

morphism feltonelA B DEI A is a if fg fg gz omorphism g mon if gf off epimorphism g g and gf 1A if 7g CHomeCB A with fg 1 isomorphism In this case we say that A and B are equivalent collection of LEA Xx Given can define Renard objects we a see the beginning of these notes their product and coproduct see HW for details Exampled uct Objects Product Coprod Category Morphisms Cartesian pod Set sets Disjoint union function Disjoint union Cartesian pad Contin maps Top Top spaces Free product Direct product Gnp Groups Homomorphism A b Abelian groups Direct product Direct sum Homomorphisms linear maps K vector spaces Direct product Vectk Direct sun Reina Ay post defines a category Products and coproduct defined via universalmappingproperties are i e by the existence of certain uniquely determined morphisms can be generalized This

D is urinal if for each An object I c Ob E Lef or initial J pie HomelI Ci Ci c Oble if for each or terminal An object TeOb e is couriversat G c Ob e 4 c Home Ci J T An initial and terminal object is a aroobject Conniversalftermin Universallintia Exampled Category Ex set 0 Cary x x Top I 1 Grp Thm5 Any two universal objects are equivalent let and J PI I be universal sketch 1 fog I J Ty t g I f zig JJ and gf I f J Thus fg II are equivalent 1 D can show that any two conniversal objects are we similarly HW equivalent are the universal object For suitably chosen categories co products

Ai l ie I be a family of in C let Example objects new category D as follows Define a where fi c Home B A fi lie I Pairs B Objects fi Elements he HomelB D Ai B Morphisms g oh fi s t tie I hi g D In this category the conniversal terminal object is Cheech pili e I Ai If A exists in e are unique up to equivalence Products and coproduct Cer when they exist shortasideitopology analysis 4 geometry without the metric can be thought of as Topology can be continuously deformed to the if Two spaces homeomor phic one are other and vice versa For example I spheres are tori plural of torus are g Think about fundamental differences b w the sphere and 6 rug 4 color theorem 7 color theorem e g vs some geodesics wait intersect any 2 great circus intersect twice us D

Devey topological space X an associated fundamentalgroye has Ty x closed loops up to continuous deformation consisting of O _bqcd d a badlab IT 5 IT S Ed tf fa blab ba Ttt L 2 2 idi ii Top There Key functor Gr between categories is a that preserves the structure of morphisms F f continuous map is Y X a T f T fit is the induced homomorphism y t x This also preserves products and coproduct category C and let fit HomelA Ai Defi let A be objects in A Az a for i 1,2 A pushet or fiber coproduct for CA Ai Az fi fz is with the following commutative diagram property f A A for any object Ceobfe and E iii iii iii in iii it h 19 a for h g h i I 2 i a

another pushont for CA A A f E A Pros is Suppose g Then B'EB Eaf gi Pfi Hw Examples let A A Az Eset A A in Az A fi Ai Putout fief 191 A A uh g with gfA Ig Az Thiele AUA identified Top_ E or Set D dD S D d Dr consider 2 disjoint disks 4 They have boundary circle dDi S Og top The pushont of S1 D Dz 4 he hemisphere w are inclusion maps Li S where Di Dr In is the 2 sphere 5 hemisphere glue 2 disks along their boundary circle Thi we get the map that if what do he S D is Question around dD twice s wrap O

then the e G If f pushont is the freepro are injective due is the quotient G G fG G and withamalgamation f where he H fdh N f G H this is just the free product If H 1 Note I E g the GIG coproduct gz Application Seifert Van Kampen theorem algebraic topology UnV be path connected top and A UuV Sketchofmain ideas ht X The fudamer IgopI of X is spaces IT IX Xo Ttu µ ITN xD Xo a Example X T Horus UN b e A UN disk TIME L f vueo.name estoawedgeeo annulus two circles S vs z yQ with fundamental group Fr 91 La b 1 Tlv Xo I 2 2 v K Siefert Vankampen says this pushart 7 s a Inst fig it Top functor Grp also preserve g pushouts O

which is trivial in DD I 1 9 loop f 1 g 1 Ti TilDixo A Xo Hit 16 It risks ix it xd p.fi sIi which is trivial a b 1 151 1 2 2 La blaba in T