Probably Random

A blog by Paul Siegel

Category Theory References

April 28, 2017

This post contains a list of references for learning the basics of category theory and its applications to computer science. It will grow over time as our interests and goals become clearer.

General category theory references

There are numerous introductory references for category theory, but one must be careful: many of them are oriented towards topology, algebraic geometry, and homological algebra. These subjects are probably the best possible environment for learning category theory, but the prerequisites can be overwhelming. So the references below are intended to be more generic - if possible, some will hopefully include applications to computer science.

  1. Categories for the Working Mathematician - Mac Lane: This is a bit dated, but is still the standard reference for the basics.

  2. Basic Category Theory - Leinster: Great exposition, but examples are quite mathy. Still, this is the best modern reference that I have found.

  3. Category Theory for Computing Science - Bar and Wells: The exposition is quite good, and lots of the examples are aimed at computer scientists.

  4. nLab: This is a community-drivin wiki for category theory. I generally find that Wikipedia articles are easier to understand because nLab authors generally try to use the most general language possible, but there is content here that is hard to find elsewhere.

The category of graphs

Perhaps the simplest example of a category - other than the category of sets - which is rich enough to illustrate the basic definitions is the category of graphs (with graph homomorphisms). So far I have not been able to find a single satisfactory reference on the subject, but here are some good starting points.

  1. Category Theory for Scientists - MIT course: Introduction to category theory, using the category of graphs as an example.

  2. Graph Homomorphisms - Wikipedia: Not much category theory, but this at least carefully defines and motivates the idea of a graph homomorphism.

  3. Hom complexes and homotopy theory in the category of graphs - Dochtermann: This paper is probably too hard, but the early sections look readable and the references look good.