## Mathematics 617: Category Theory

Instructor: Jonathan Smith.

e-mail: jdhsmithATiastateDOTedu (substitute punctuation)

Office Hours: No physical hours. Please use e-mail for initial contact.

Grading: based on five graded homework assignments (final due Mon., 11/20).

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

The books by Adámek-Herrlich-Strecker (4MB) and Emily Riehl (1.5MB) are also available online.

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.

Communication devices must remain switched off during the class periods. For the safety of your classmates, please wear a face covering and maintain a six foot separation at all times.

CoViD-19 health and safety policies

Additional Mathematics course policies

Syllabus: Basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, cartesian-closed and enriched categories.

### Assignments

Final graded homework due 11/20:
Three questions from the following list of five: Section II.5: 8; Section IV.2: 11; Section IV.6: 1; Section VI.2: 3; Section VIII.2: 1.

11/20: Read Mac Lane, Section IV.10, and Riehl, Definition 3.3.4 and Appendix E.4.
Lesson summary

11/18 for 11/20: Read Mac Lane, Section IX.4.
Lesson summary

11/16 for 11/18: Read Mac Lane, Section IV.9.
Lesson summary

11/13 for 11/16: Lesson summary

11/11 for 11/13: Read Mac Lane, Section XI.2.
Lesson summary

11/9 for 11/11: Lesson summary

11/6 for 11/9: Read Mac Lane, Section VI.5.
Lesson summary

11/4 for 11/6: Read Mac Lane, Section VI.2.
Lesson summary

Fifth graded homework due 11/4:
Three questions from the following list of five: Section II.5: 2, 6; Section VIII.2: 3, 4; Section XI.3: 1.

11/2 for 11/4: Complete reading of Mac Lane, Section VI.1.
Lesson summary

10/30 for 11/2: Read first page of Mac Lane, Section VI.1. Do Riehl, Exercise 1.7.ii.
Lesson summary

10/28 for 10/30: Read Mac Lane, Section XI.4.
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10/26 for 10/28: Read Mac Lane, Section II.5, XII.3, and Riehl, Section 1.7.
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10/23 for 10/26: Read Mac Lane, Section VIII.2 to top of p.197.
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10/21 for 10/23: Read Mac Lane, rest of Section VIII.1, and Section VIII.3 to middle of p.200.
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10/19 for 10/21: Read Mac Lane, Section VIII.1 to top of p.193.
Lesson summary

10/16 for 10/19: Read Riehl, Def'n. 4.3.7 to (4.3.13).
Lesson summary

Fourth graded homework due 10/16:
Three questions from the following list of five: Section IV.2: 11; Section IV.3: 5; Section IV.4: 1, 2; Section V.5: 1.

10/14 for 10/16: Read Mac Lane, Section I.8.
Lesson summary

10/12 for 10/14: Read Mac Lane, Section VII.1.
Lesson summary

10/9 for 10/12: Read Riehl, Proposition 3.5.9 and Mac Lane, middle of p. 132.
Lesson summary

10/7 for 10/9: Read Mac Lane, Section V.5.
Lesson summary

10/5 for 10/7: Read Riehl, Definition 3.3.1 and Remark 3.3.2.
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10/2 for 10/5: Read Mac Lane, pp.88-9 "Duality functors ..." and p.93 "... skeleton ...".
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9/30 for 10/2: Read Mac Lane, Section IV.4.
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Third graded homework due 9/30:
Three questions from the following list of five: Section III.5: 5; Section III.6: 4; Section IV.2: 4, 5, 12.

9/28 for 9/30: Read Mac Lane, Section IV.3 and Riehl, Definition 4.5.12 through Example 4.5.14(v).
Lesson summary

9/25 for 9/28: Read Mac Lane, Section III.2 and Riehl, Section 2.2.
Lesson summary

9/23 for 9/25: Read Mac Lane, Section II.6.
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9/21 for 9/23: Read Mac Lane, Section IV.5.
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9/18 for 9/21: Read Mac Lane, Section IV.2.
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9/16 for 9/18: Read Mac Lane, Section IV.1, first six pages. Do IV.1, Exercise 3.
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9/14 for 9/16: Read Mac Lane, Section III.6 (just "only if" direction of Proposition 1).
Lesson summary
Consider a group G in the category Set, as defined on p.75 of Mac Lane.
Show that G need not be a group.

Second graded homework due 9/14:
Three questions from the following list of five: Section I.4: 3, 5; Section III.4: 4, 5, 8.

9/11 for 9/14: Read Mac Lane, Section III.5.
Lesson summary

9/9 for 9/11: Read Mac Lane, Section II.3 and Riehl, Definition 1.3.11 through Definition 1.3.13.
Lesson summary

9/7 for 9/9: Complete reading of Mac Lane, Sections III.3, 4.
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9/4 for 9/7: Read Mac Lane, Section III.4 from Infinite products through Pullbacks and Section III.3 from Infinite Coproducts through Cokernel Pair. Do Exercise III.4.9.
Lesson summary

9/2 for 9/4: Read Mac Lane, Section III.4 Products and Section III.3 Coproducts.
Lesson summary

Suppose that the product of objects X and Y exists in a category.
Show that it is isomorphic to the product of Y and X.

8/31 for 9/2: Read Mac Lane, Section II.4 and Riehl, Definition 3.1.1.
Lesson summary
Do Exercise 3, and interpret the polynomial ring  R[X] as a graded abelian group.

8/28 for 8/31: Read Mac Lane, Sections II.1-2.
Lesson summary

1. Give an example of a monoid M that is not isomorphic to its opposite Mop.
2. Prove that there is no monoid isomorphism between M and Mop.

8/26 for 8/28: Read Mac Lane, Section I.4.
Lesson summary

First graded homework due 8/28:
Three questions from the following list of five: Section I.3: 2, 4; Section I.5: 2, 6, 7.

8/24 for 8/26: Read Mac Lane, Section I.3.
Lesson summary

8/21 for 8/24: Read Mac Lane, Section I.5.
Lesson summary
1. Prove the lemma in the lesson summary.
2. Show that any two terminal objects are isomorphic.
3. Show that the inverse of an invertible morphism is unique.
4. Give an example of a monomorphism in Set which is not split.

8/19 for 8/21: Read Mac Lane, Sections I.1, I.2.
Lesson summary
1. Consider the function sending each object of a category to the identity function at that object. Show that the function is injective.
2. Show that the identity morphism at an object of a category is uniquely determined by Mac Lane's condition (2) on page 8.

8/17 for 8/19: Give an example of a directed graph C that is not isomorphic to its dual Cop. Prove that there is no directed graph isomorphism between C and Cop.
Lesson summary