Abstract Algebra MATH 647
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This 1 page Class Notes was uploaded by Henderson Lind II on Tuesday September 8, 2015. The Class Notes belongs to MATH 647 at University of Oregon taught by Staff in Fall. Since its upload, it has received 52 views. For similar materials see /class/187187/math-647-university-of-oregon in Mathematics (M) at University of Oregon.
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Date Created: 09/08/15
1 to g Algebra 6477 Homework 17 solutions L7 1 harnX7 Io Y7 yo is a subset of h0mX7 Y in the category of sets So in particular it is a set In addition the identity is a perfectly good map of pointed sets since 1X 10 10 We just really need to check if f X710 A Km and g z yo A Z720 is a map of pointed sets then so is g o But this is clear since 9 0 fro yfro 9yo 20 i L7 2 Suppose 9 also satis e f 09 l and g o f 14 We get 990190f09 yofog 1og 95 It is worth observing that we just proved something stronger than what was required Suppose f is a morphism with a left inverse 9 meaning go f l and a right inverse 9 meaning f o g 1 Then 9 g and is a proper twosided inverset ll 39 7 Let A in C be an object whose underlying set has more than one element Let X be a set with more than one element Let 11 12 be distinct ele ments of X Pick two distinct elements of 041042 6 A and let j z X A A be a set map so that jzl a1 and jzg a If i z A F is a free object on X there is a unique map f F A A so that f oi j Since jzl jzg it follows that f So 239 sends distinct elements of X to distinct elements of F i Give an example of a concrete category and a morphism f z A A B in that category that is an isomorphism on the underlying set7 but not an isomor phism in the category Here is a very simple example Take the category to be partially ordered sets Take a partially ordered set with two elements X ab and the only relation being a S a and b S 124 This satisfies the axioms for a partial order though it isn t very interesting Now take Y 17y with I S y and of course I S I y S Take f z X A Y so that fa z and y f is a set bijection and is a map of partially ordered sets But f which exists as a map of sets is not a map of partially ordered sets So f has no inverse in the category of P40 sets
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