Mathematics 584: Category TheoryInstructor: Jonathan Smith, 496 Carver, 4-8172 (voice mail) e-mail: jdhsmithATmathDOTiastateDOTedu (substitute punctuation) Office Hours: Mon. 11am, 2:10 pm; Wed. 11am, 2:10pm, 5pm (subject to change) Grading: based on four graded homework assignments (60%), and final (40%). Textbook: S. Mac Lane, Categories for the Working Mathematician, 2nd. ed., Springer, ISBN 0-387-98403-8
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Study plan: reserve 1 - 2 hours for homework between each pair of classes. Successful performance in the class depends critically on completion of the homework assignments. Syllabus: Chapters I - IV Homework Assignments8/28 for 9/4: Show that id : Ob(C) —Mor(C) is 1-1.8/30 for 9/4: Let ≤ be a pre-order on a set S. Define a relation E on S by x E y if and only if x≤y and y≤x.Show that E is an equivalence relation on S. 9/6 for 9/13: Section 1.3: 2, 3(a)(b), 4, 5. 9/9 for 9/13: Section 1.4: 1, 2, 3. First graded homework due 9/13: Three questions from the above two assignments (Sections 1.3, 1.4). 9/23 for 9/27: (a) Give an example of a poset with a bottom element, but no top element. (b) Give an example of an infinite poset in which each subset has a bottom element, but no infinite subset has a top element. Prove that the example does have these properties. 9/25 for 9/27: (c) Let T1 and T2 both be terminal objects of a given category C. Give a careful proof that T1 and T2 are isomorphic. Also Section 1.5: 3. 9/27 for 9/30: Section 1.5: 2 Second graded homework due 10/7: Three questions from Section 1.5: 4, 5, 6, 7, 8. 10/11 for 10/14: Section 3.4: 1 (first part only, not Top), 6 10/16 for 10/18: Section 2.6: 2 Third graded homework due 10/21: Three questions from Section 3.4: 4, 5, 8, 7, 9. 10/28 for 10/30: Let (F,G,f) be an adjunction. Verify that the counit e is a natural transformation from FG to the identity functor on the domain category A of G. 11/1 for 11/4: Let (F,G,f) be an adjunction. Verify that for each object X of B and A of A, the component at X and A of the natural isomorphism f-1 is the function B(X,GA) —A(FX,A) ; g |—eAoFg .11/4 for 11/6: In the context of Theorem 2 of Section IV.1 of Mac Lane (p. 83 in the 2nd edition, p. 81 in the 1st), write out a careful proof that the data in (iv) determine an adjunction. Fourth graded homework due 11/20: Three questions from Section 4.2: 4, 5, 9, 12; Section 4.5: 3. Take-home final in portable document format (due 12/11). |