### 713 Assignments for Spring 2013

1. Consider the set P of all positive integers. Define an arrow from to if and only if divides . Show that this provides the structure of a (non-concrete) category on the positive integers. -- "get-acquainted special"

2. p.133 #1

3. p.137 #4

4. Prove that a group is a (non-concrete) category with one object in which all arrows are invertible, and that a morphism of groups is a functor of such categories.

BONUS 1 p.133 #3 and p.137 #1

5. p.140 #1

6. p.146 #4

7. p.153 #3

8. Prove that a lattice is a category in which any two objects have at most one arrow between them, and that has products and coproducts.

9. p.162 #5

10. Prove Thm.3 p.166.

11. p.167 #6

BONUS 2 p.167 #7

12. p.178 #4

13. p.184 #2

14. p.192 #1

15. p.192 #4

16. p.325 #1

17. p.325 #6

BONUS 3 Prove Prop.18 p.324 for f.g. free/projective modules.

18. p.328 #3

BONUS 4 Same for f.g. free/projective modules.

19. p.333 #2

20. p.334 #6

21. Determine the monics and epics in the category P of Assignment 0.

22. p.500 #4

23. p.511 #2

24. Show that the isomorphisms in (35) p.181 and (32) p.321 are natural in ModK.

BONUS 5 Find and verify all other natural isomorphisms in ModK from Chapters V and IX.8--12.

BONUS 6 Regard a group as in Assignment 2. Prove that an automorphism is inner if and only if it is naturally equivalent to 1G .

25. p.516 #4--5

26. p.520 #4

27. p.520 #5.(b)

28. p.520 #12

From the handout:

29. p.1 #1

30. p.1 #2 -- due Fri 3 May

The remaining assignments are due Wed 15 May by 4:50 pm

31. p.2 #1, 3

32. p.3 #3, 4, 5