## Mathematics 584: Category Theory

Instructor: Jonathan Smith, 496 Carver, 4-8172 (voice mail)

e-mail: jdhsmithATmathDOTiastateDOTedu (substitute punctuation)

Office Hours: Mon. 2:10 pm, 5:30pm; Wed. 2:10pm, 4:10pm, 5pm (subject to change)

Textbook: S. Mac Lane, Categories for the Working Mathematician, 2nd. ed., Springer, ISBN 0-387-98403-8

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

Three questions from the following:
1. Let  S  be an object of the category  Set  of sets. Show that  S  is a terminal object of  Set  if and only if each object of the category  Set  is a copower of  S .
2. In a category  C , an epimorphism  u : B --> E  is regular if it is the coequalizer of a pair of arrows  f, g : A --> B . Show that in the category  Set  of sets, each epimorphism is regular.
3. Consider the categories  Gp  of groups and  Mon  of monoids. Let  F  be the forgetful functor from  Gp  to  Mon  that forgets inversion. Show that  F  has a right adjoint  G  whose object part sends a monoid to its group of units (invertible elements). What is the counit of this adjunction?
4. Show that the category of finite-dimensional real vector spaces is equivalent to the category  MatrR  of all rectangular real matrices.
5. For each finite-dimensional vector space  V , let  FV  denote the vector space of linear functionals on  V . For a linear transformation  f : V1 --> V2 , let  Ff : FV2 --> FV1  take a functional  l  on  V2  to the composite functional  l o f  on  V1 . Prove that there is no natural transformation  t  whose component at  V  is an isomorphism  tV : V --> FV .
11/15 for 11/29: Section 4.3: 4(a).

Three questions from Section 4.2: 5, 9, 10, 12; Section 4.5: 3.

10/27 for 11/1: Given an adjunction
( F : X --> A , G : A --> X , h , e ) ,
prove that  eFx o Fhx  =  idFx  for each object  x  of  X .
10/25 for 10/27: Section 4.2: 4.
10/18 for 10/22: Section 5.1: 6, 7.

Three questions from Section 2.4: 6; Section 3.4: 5, 6, 8; Section 3.6: 1.

10/1 for 10/4: Section 2.4: 2, 4.
9/27 for 10/1: Section 3.6: 4.
9/22 for 9/27: Sections 2.3: 1, 2; 3.5: 1.

Three questions from Section 1.5: 1, 4, 5, 6, 7.

9/13 for 9/15: Section 1.4: 2, 3.
9/8 for 9/13: Section 1.5: 8. Also:
Let T1 and T2 both be terminal objects of a given category C. Prove that T1 and T2 are isomorphic.