CS 6117 (Spring 2020)  Category Theory for Computer Scientists
Overview
CS6117  Category Theory for Computer Scientists is an introduction to category theory, with a focus on material with established applications to computer science. The course will emphasize developing comfort with abstraction and instantiation while not assuming students have a strong background in mathematics.
Class Times: MWF 1:252:15
Location: 320 Hollister Hall
Instructor:
Ross Tate
Office: 434 Gates Hall
Office Hours: Friday 9:0010:00
Grading There will be a small homework assignment almost every week (no midterms or final):
Assignments: 100%
 There will be roughly 14 assignments, each worth equal credit.
 The two worst assignments will be ignored.
 Assignmets are due at the beginning of lecture on Wednesday.
 No late assignments.
S/U is determined based on attendance. To receive an S, all but two lectures you miss must be the first lecture you missed that calendar week.
Piazza: http://piazza.com/cornell/spring2020/cs6117
Related Reading
 Abstract and Concrete Categories  The Joy of Cats by Jiri Adamek, Horst Herrlich, and George E. Strecker (2004)
 Higher Operads, Higher Categories by Tom Leinster (2003)
 Categories for the Working Mathematician by Saunders Mac Lane (1971)
Schedule
Date  Topic  Reading  Assignments 

Week 1  
Jan 22  Categories, Part 1  Reference (updated throughout the semester) Preface 3.1(ignore c), 3.2, 3.3(2b,2d2f,4a,4c4f), 3.4 Categories 

Jan 24  Categories, Part 2  Available: Assignment 1 

Week 2  
Jan 27  Categories, Part 3  
Jan 29  Isomorphisms  3.8, 3.1012, 3.13(12,4,6), 3.1415 Isomorphisms 
Due: Assignment 1 Available: Assignment 2 
Jan 31  Monomorphisms and Epimorphisms Balanced Categories Subobjects and Quotient Objects 
7.32, 7.33(1,34,9), 7.34, 7.39, 7.40(12,5), 7.4142 7.4950 7.7781, 7.8485, 7.86(1) 

Week 3  
Feb 3  Equalizers and Coequalizers Regular Mono/epimorphisms 
7.51, 7.52(1), 7.5355, 7.68, 7.69(13), 7.70 7.5659, 7.71, 7.72(1,4), 7.73, 7.75(1), 7.76 
Available: Assignment 3 
Feb 5  Initial, Terminal, and Zero Objects Sections and Retractions Separators and Coseparators 
7.1, 7.2(12,5,89), 7.37.4, 7.5(13,56), 7.67.9 7.19, 7.20(47), 7.21, 7.24, 7.25(1,4), 7.2627, 7.3536, 7.4243 7.10, 7.11(1,45), 7.16, 7.18(1,3,6,8) 
Due: Assignment 2 
Feb 7  Snow Day  
Week 4  
Feb 10  No Lecture  
Feb 12  No Lecture  Due: Assignment 3 

Feb 14  Duality  3.57  Available: Assignment 4 
Week 5  
Feb 17  No Lecture  
Feb 19  Functors  3.1718, 3.20(14,68,10,13), 3.23, 3.50 Functors 

Feb 21  The Category of Categories  3.24, 3.25(1), 3.26(13), 3.30(1) 7.2(3), 7.18(7), 7.40(6), 7.72(5), 7S(b) 3.4043 

Week 6  
Feb 24  Break  Available: Assignment 5 

Feb 26  IsomorphismPreservation Faithful/Full Functors 
3.21,3.22(2) 3.27(23), 3.29(2,4), 3.3032, 7.37(2), 7.4445, 7.2829 
Due: Assignment 4 
Feb 28  Concrete Categories Concrete Functors Fibres/Fineness/Coarseness Transportability 
5.1, 5.2(12,4,7), 5.3 5.9, 5.10(13), 5.11(2,4), 5.12, 5.1415, 5.20 5.4(13), 5.5(1,3), 5.78, 5.1819 5.28, 5.30(12) 

Week 7  
Mar 2  Algebras Spaces Coalgebras 
5.37, 5.38(1,4), 5.39 5.4042 Coalgebras 

Mar 4  (Co)Structured Arrows (Co)Free Objects (Op)Cartesian Liftings 
8.15(12), 8.16(3), 8.17(12,6), 8.18(1), 8.19(1), 8.40(1) 8.22, 8.23(1a,3,56,11,1314,16), 8.2428, 8.40(2), 8.41(2b2c) 8.6(1), 8.8(1,5,8,1112), 8.9, 8.10(1), 8.11(23), 8.1314 
Due: Assignment 5 
Mar 6  (Mono)Sources Products (Epi)Sinks Coproducts 
10.16, 10.7(2), 10.810 10.19, 10.20(12,58), 10.2130, 10.31(1,4), 10.3437 10.6264 10.65(1,3,69) 
Available: Assignment 6 
Week 8  
Mar 9  Limits Pullbacks and Intersections Colimits Pushouts and Cointersections 
11.14, 11.67 11.826 11.27, 11.28(12,4), 11.29 11.28(3,5), 11.3031, 11.33 

Mar 11  Internal Logic Regular Categories 
Due: Assignment 6 

Mar 13  (Co)Recursion  Available: Assignment 7 

Week 9  Spring Break  
Week 10  Spring Break  
Week 11  Spring Break  
Week 12  
Apr 6  Natural Transformations, Part 1  6.1, 6.2(2,56)  
Apr 8  Natural Transformations, Part 2  6.6(1), 6.1315, 6.5, 6.6(45), 6.17  Due: Assignment 7 
Apr 10  Natural Transformations, Part 3  6.34, 6A  Available: Practice 8 
Week 13  
Apr 13  Reflective Subcategories Coreflective Subcategories 
4.16, 4.17(A1,B3,B7,C10C12), 4.1823 4.25, 4.26(A1,B4), 4.27 

Apr 15  Transpositions  Transpositions and Adjunctions  
Apr 17  Adjunctions  Available: Practice 9 

Week 14  
Apr 20  Monads and Comonads  20.13 Monads and Comonads 

Apr 22  Monad Algebras  20.4, 20.5(13), 20.711, 20.12(13,58,10) Monad Algebras 
Available: Practice 10 
Apr 24  Effects  20B(ac) Kleisli Categories 

Week 15  
Apr 27  Multicategories  Multicategories  Available: Practice 11 
Apr 29  Tensors and Exponentials  Tensors and Exponentials  
May 1  Monoidal Categories  Monoidal For Fun: Physics, Topology, Logic and Computation: A Rosetta Stone 

Week 16  
May 4  Enriched Categories  Enriched Categories  
May 6  Toposes, Part 1  
May 8  Toposes, Part 2  
Week 16  
May 11  Realizability Models  
May 13  End of Classes  
May 15  End of Classes 
Homework Policies
Cornell University has a Code of Academic Integrity, with which you should be familiar. Violations of this code are treated very seriously by Cornell and can have longterm repercussions. In this course, you are encouraged to discuss the content of the course with other students, and you may also discuss homework problems with other students. However, you must do your own work, write up assignments yourself, and if you discuss a problem with another student, you are expected to document this fact in your writeup. It is a violation of the code to copy work, including programs, from other students; it is also a violation to use solutions to homework problems from previous iterations of the same course. Note that Cornell holds responsible for the code violation both the recipient and the donor of improper information.