Department of Philosophy
Carnegie Mellon University
Pittsburgh, PA  15213

Office: Baker Hall 148

email :

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Categorical Logic course 80-514/814 (Spring 2021)


Category Theory

Fall 2020
Teaching assistant: Jacob Neumann

Zroom for the class

Meeting ID: 947 5367 7006
Accessible from Canvas 80413
Only adresses admitted
Contact me for password

Office hours

M. Anel: after each course, in the same zroom, or by appointment

J. Neumann: Wednesday 4:40-5:40 in this zroom, or by appointement


We will be using Piazza for class discussion. Please register and post your questions here:


Homework will be available on Gradescope each Thursday after the course. They have to be turned in the before the end of next Thursday or they will not be accepted.

Link for the 80413 class
Link for the 80713 class

Some written corrections

HW4  HW5  HW6  HW7  HW8  HW9 

Slides of the lectures


What is category theory?

Category theory is a field of Mathematics: topology, geometry, algebra, logic, computer sciences, physics and more are using methods of category theory.

Category theory is a synthesis for many structures: sets, preorders, posets, total orders, well-orders, equivalence relations, monoids, groups, groupoids, path algebras of graphs, and more are all examples of categories.

Category theory is a classification tool: the collections of sets, posets, monoids, groups, abelian groups, vector spaces, topological spaces, manifolds, chain complexes, and pretty much any kind of mathematical structure do form categories. Most importantly, the collection of categories themselves is a category.

Category theory is a language: morphisms, isomorphisms, endomorphisms, monomorphisms, epimorphisms, functors, diagrams, natural transformations, and many other notions help to organize the relationships between mathematical objects.

Category theory is a calculus: limits and colimits, monoidal structures and internal homs, universal properties, adjunctions, presheaves, and many other notions provide general tools to operate on the objects of a category.

And so much more: category theory is a geometry (any category behave like a space); category theory is a linear algebra (categories behave like vector spaces); category theory is a logic (any category has an internal logical language); category theory is a philosophy (categories are the Kantian Transcendental Æsthetics of mathematics); Category theory is an art de vivre (surtout avec un saucisson bien sec)...


Course description

The purpose of the course is to introduce the basic tools of category theory (functors, transformations, diagrams, universal properties, (co)limits, adjunctions...) in order to allow an autonomous access to the more advanced literature.

Although the course is given in the Philosophy department, this is a mathematics course and students are expected to do proofs. Familiarity with abstract algebra or logic is required.

Weekly homework will be given.

Contact me for any questions, particularly if you are unsure the course is for you.


Our main reference for the course will be

  • Steve Awodey, Category Theory, 2nd edition, Oxford University Press, 2010.
Students are encouraged to read the book in advance.

Other references:

  • David Spivak, Category Theory for the Sciences, MIT Press, 2014 (very good for people with no or few mathematical background)
  • Tom Leinster, Basic Category Theory, Cambridge University Press, 2014 (very accessible reading, minimal amount of math background required)
  • Emily Riehl, Category Theory in Context , Courier Dover Publications, 2017 (for students interested in maths)
  • Saunders Mac Lane, Categories for the Working Mathematician, 2nd edition, Springer, 1998 (for students interested in maths)
  • Samuel Eilenberg, Saunders Mac Lane, the original paper! (I recommend reading the introduction)

Other ressources:

  • The nLab: a wiki-lab on Mathematics and theoretical Physics from the point of view of (higher) category theory. They have pages on most concepts in our course and many beyond.
  • Category Theory: the entry at the Stanford Encyclopedia of Philosophy by Jean-Pierre Marquis discusses the history and philosophical significance of Category Theory.


  • Weekly homework (2/3 of final grade)
  • Final exam (1/3 of final grade)

80-413 vs. 80-713

The undergraduate and graduate sections of this course will be taught together. However, students enrolled in the graduate section will be expected to work a little bit harder. Each homework set will contain 1-2 starred problems that only the 80-713 students will be expected to solve. The tests may also have additional starred problems.


Students are encouraged to write their assignments in LaTeX. The Wikibook on LaTeX is a good place to start. We recommend using the tikz-cd package for creating commutative diagrams; see the manual for details. You will need an up-to-date installation of TeX Live to take advantage of the newest version of tikz-cd.


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