## Professor Andre Scedrov

Professor Scedrov's Office: DRL 4E6.
Professor Scedrov's Office Hours: Online by appointment.

## Prerequisites

Math 5700 / Lgic 3100 or permission of the instructor.

## Topics Covered Include

Categories, Basic Constructions in Categories, Functors, Natural Transformations, Universality, Limits and Colimits, Adjoints, The Curry-Howard Correspondence, Linearity, Monads and Comonads, Algebras and Coalgebras.

Dependent Type Theory, Dependent Function Types, Inductive Types, Identity Types, Universes.

## Basic Course Information

There will be two take-home midterms, the first one due in Canvas on Thursday, February 22, 2024 and the second one due in Canvas on Monday, April 29, 2024. Each midterm will be worth 33% of the grade and will have at least a two-week lead time.

The take-home final exam will be due Monday, May 13, 2024 during the final exam period and will be worth 34% of the grade. The take-home final exam will also have at least a two-week lead time.

## Midterm #1 due in pdf in Canvas by 3pm on Thursday, February 22, 2024

Please bear in mind that everyone is expected to submit written solutions individually. That is, even if in some cases your work may be a result of group discussions, each person is responsible to write up the solutions in their own words by themselves. Please take the time to show all your work and provide a detailed explanation of your reasoning in your own words.

• Exercise 1.1.6.1bdefg on p. 17 of the first textbook.
• Exercise 1 on p. 27 of the first textbook.
• Exercise 2 on p. 27 of the first textbook.
• Prove that a coequalizer is an epimorphism.
• Show that pushouts can be constructed in terms of coproducts and coequalizers. Prove that the required universal property holds.
• Given two pushout squares placed side by side and sharing one morphism, show that they form a larger pushout square when ignoring the inner shared morphism. Prove that the required universal property holds. That is,
if maps f: C --> A, g: C --> B, and h: B --> D are given, and
the pushout of f and g is given by i: A --> P and j: B --> P, and
the pushout of j and h is given by k: P --> Q and m : D --> Q,
then the pushout of f and g;h is given by i;k : A --> Q and m : D --> Q.
• Exercise 49bd on p. 34 of the first textbook.
• Exercise 50 all on p. 34 of the first textbook.
• Exercise 1.3.5.1 on p. 35 of the first textbook.
• Exercise 1.3.5.2 on p. 35 of the first textbook.
Please put MATH 5710 Midterm1 and your name in the name of the pdf file.

This is the complete set of problems for Midterm #1 due in pdf in Canvas by 3 pm on Thursday, February 22, 2024.

## Midterm #2 due in pdf in Canvas by 3pm on Monday, April 29, 2024

Please bear in mind that everyone is expected to submit written solutions individually. That is, even if in some cases your work may be a result of group discussions, each person is responsible to write up the solutions in their own words by themselves. Please take the time to show all your work and provide a detailed explanation of your reasoning in your own words.

• Exercise 56 on p. 40 of the first textbook.
• Exercise 1.4.4.2 on pp. 41-42 of the first textbook.
• Exercise 1.4.4.4 on p. 42 of the first textbook.
• Exercise 1.5.5.2abc on pp. 55-56 of the first textbook.
• Exercise 1.5.5.4abc on p. 56 of the first textbook.
• Exercise 87 on p. 61 of the first textbook.
• Exercise 93 on p. 64 of the first textbook.
• Exercise 106 on p. 72 of the first textbook.
• Exercise 1.6.7.1 the first two bullets on p. 73 of the first textbook.
• Exercise 1.6.7.2 the first two bullets on p. 73 of the first textbook.
Please put MATH 5710 Midterm2 and your full name in the name of the pdf file.

This is the complete set of problems for Midterm #2 due in pdf in Canvas by 3 pm on Monday, April 29, 2024.

## Final Exam due in pdf in Canvas by 3 pm on Monday, May 13, 2024

Please bear in mind that everyone is expected to submit written solutions individually. That is, even if in some cases your work may be a result of group discussions, each person is responsible to write up the solutions in their own words by themselves. Please take the time to show all your work and provide a detailed explanation of your reasoning in your own words.

• Exercise 125 on p. 87 of the first textbook.
• Exercise 131 on p. 92 of the first textbook.
• Exercise 2.4ab on pp. 18-19 of the second textbook.
• Exercise 4.2abc on pp. 35-36 of the second textbook.
• Exercise 5.5ab only on pp. 49-50 of the second textbook.
Please put MATH 5710 Final and your full name in the name of the pdf file. Please submit only one pdf and indicate clearly which part is the final and which is the resubmission of some midterm problems.

This is the complete set of problems for Final Exam due in pdf in Canvas by 3 pm on Monday, May 13, 2024.