### 713 Assignments for Fall 2015

1. p.133 #1 -- "get-acquainted special"

2. 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.

3. 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.

4. p.137 #4

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

5. p.140 #1

6. p.146 #4

7. 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.

8. p.153 #3

9. p.520 #12

10. p.162 #5

11. Prove Thm.3 p.166.

12. p.167 #6

BONUS 2 p.167 #7

13. p.170 #13.(b) and p.173 #4.(b)

14. p.178 #4

15. p.184 #2

16. p.192 #1

17. p.192 #4

18. p.325 #1

19. p.325 #6

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

20. p.328 #3

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

21. p.333 #2

22. Prove that the coproduct in CRng is the tensor product.

23. Determine the monics and epics in the category P of Assignment 1.

24. p.511 #2

BONUS 5 Find and verify all 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 .

The remaining assignments are due Wed 9 Dec by 4:30 pm

From the handout:

25. p.1 #1, 2

26. p.2 #1, 3

All rewrites of ungraded assignments
and any bonus problems
are due Mon 7 Dec by 4:30 pm.

Lists of assignments from Spring '13 and prior.