Mathematics 584: Category Theory

Instructor: 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

Library Reserve:

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 Assignments

8/28 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).