## Mathematics 584: Category Theory
## 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
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 T and _{1}T both be terminal objects of a given category _{2}C. Give a careful proof that T and _{1}T are isomorphic.
_{2}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
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). |