This course is part of the Mastermath programme.

Lecturer is Jaap van Oosten. Assistant for the exercise class is Mark Kamsma (mark@markkamsma.nl).

This course will be given in weeks 6-20 (February-June 2018).

The course is on Mondays, 10:15-12:00 (lecture) and 12:15-13:00 (exercises), in Amsterdam Science Park, room A1.04.

The course will be concluded by a written exam, for which you must obtain at least the grade 5. There are, during the course, 6 hand-in exercises, which may count for 30% of your final grade (it is not compulsory, although strongly recommended, to hand in these exercises). That is, your final grade will be calculated as follows: let H be your grade for the hand-in exercises and W your grade for the written exam. **Provided W is at least 5**, your final grade will be the maximum of W and the number (3H + 7W)/10.

The hand-in exercises have to be handed in at the exercise class in the week they are mentioned below; they can also be sent by email to the assistant, before noon on the same day. Exercises, to be handed in in week x+1, will appear on this page ultimately the Wednesday of week x.

Here are the hand-in exercises with solutions for the old ones.

Literature: Lecture Notes.

This course will treat some applications of Category Theory to Logic. Although this is only a minor part of the course, people unfamiliar with Logic may wish to consult sections 2.1-2.3 of the basic logic course Sets, Models and Proofs, for the notions of a *language* and *structures for a language*.

Suggestions for improvement of the lecture notes are very welcome!

Mark Kamsma prefers this tool. He gives the following instructions:

Draw your diagram using that tool, and then use the button with the braces { } to copy the LaTeX code for your diagram. You can then paste the code in a tikzcd environment, or more concrete:

\[

\begin{tikzcd}

Paste your code here

\end{tikzcd}

\]

Do not forget to include \usepackage{tikz-cd} in your LaTeX document to use the tikzcd environment.

- Week 6: Chapter 1: Categories and Functors, examples, special objects and arrows.

Exercises: 1,4,5,7,9. - Week 7: Chapter 2: Natural Transformations

exercises: 20,22,27,32,33. - Week 8: Chapter 3: Limits and Colimits

Exercises:34,35,36,38,44.**Hand-in exercise: 1** - Week 9: Chapter 4: Regular Categories and a first glimpse at Categorical Logic.

Exercises: 65,66,67,70. - Week 10: More on Regular Categories and Regular Logic.

Exercises: 75,79,88,89,92.**Hand-in exercise: 2** - Week 11: Adjunctions.

Exercises: 95,98,99,103. - Week 12: Monads and Algebras.

Exercises: 111,113,117,118,123.**Hand-in exercise: 3** - Week 13: Cartesian Closed Categories and Natural Numbers Objects.

Exercises: 125,126,130,132. - Week 14:
**Easter Monday. No lecture**. - Week 15: Presheaf categories, cartesian closed structure, every object a colimit of representable presheaves, subobject classifier.

Exercises: 144,145,146,149 and: Calculate the subobject classifier in the category of presheaves over a monoid M.**Hand-in exercise: 4** - Week 16: Continuation of theory of presheaves; sections 9.1 and 9.2.

Exercises: 152,153,154,157. - Week 17: Logic of presheaves; definition of Grothendieck topology, separated presheaves and sheaves.

Exercises: 158,160,161,162.**Hand-in exercise: 5** - Week 18: rest of 10.1, 10.2

Exercises: 164,165,167,168. - Week 19: 10.3 and 10.4.

Exercises: 176,177.**Hand-in exercise: 6. This exercise may be sent in until May 9, midnight.** - Week 20: Further perspectives: morphisms of toposes.

Left-over exercises; questions. - Week 25:
**Exam, Monday June 18, 10:00--13:00, SP C0.05.****At the exam, the use of lecture notes, books, or any kind of device which can store information, is not allowed.**

Here is the exam, with solutions. - Week 27:
**Retake Exam, Monday July 2, 10:00--13:00, SP B0.204.**

Here is the exam of 2014, with solutions.

Here is the retake exam of 2014, without solutions.

Here is the exam of 2016, with solutions

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