My grad students and I have been using a lot of category theory in our work on networks in engineering, chemistry and biology. So I decided to teach an introductory course on category theory, and it was surprisingly popular: 25 grad students registered for it! Clearly there’s a lot of interest in the subject, and we don’t regularly teach a course on it at U. C. Riverside.
Here are the notes from my course. In my earlier fall Fall 2015 seminar, I had tried to explain how category theory unifies mathematics, without getting into lots of technical details. This time I tried to give a systematic introduction to the subject, leading up to a little taste of topos theory. But many proofs were offloaded to another more informal seminar, for which notes are not available.
When I started teaching this course, I imagined that the notes might someday grow into a book I’d always dreamt of: an introduction to category theory that includes lots of examples, talks to the reader in a friendly way, and explains what’s ‘really going on’. However, while teaching the course, I noticed that Emily Riehl has written a book like this, probably better than I ever could. Even better, her book is free online and will be published by Dover:
• Emily Riehl, Category Theory in Context, 2014.
So, I don’t feel much urge to write that book anymore. But there might still be room for a more quirky book on category theory that only I could write. It would probably need to include not only this ‘standard’ material but more on applications.
If you discover any errors in the notes please email me, and I’ll add them to the list of errors.
You can get all 10 weeks of notes in a single file here:
Or, you can look at individual weeks:
Week 1 (Jan. 5 and 7) – The definition of a category. Some familiar categories. Various kinds of categories, including monoids, groupoids, groups, preorders, equivalence relations and posets. The definition of a functor. Doing mathematics inside a category: isomorphisms, monomorphisms and epimorphisms.
Week 2 (Jan. 12 and 14) – Doing mathematics inside a category: an isomorphism is a monomorphism and epimorphism, but not necessarily conversely. Products. Any object isomorphic to a product can also be a product. Products are unique up to isomorphism. Coproducts. What products and coproducts are like in various familiar categories. General limits and colimits. Examples: products and coproducts, equalizers and coequalizers, pullbacks and pushouts, terminal and initial objects.
Week 3 (Jan. 19 and 21) – Equalizers and coequalizers, and what they look like in and other familiar categories. Pullbacks and pushouts, and what they look like in Composing pullback squares.
Week 4 (Jan. 26 and 28) – Doing mathematics between categories. Faithful, full, and essentially surjective functors. Forgetful functors: what it means for a functor to forget nothing, forget properties, forget structure or forget stuff. Transformations between functors. Natural transformations. Functor categories. Natural isomorphisms. In a category with binary products, the product becomes a functor, and the commutative and associative laws hold up to natural isomorphism. Cartesian categories. In a cartesian category, the left and right unit laws also hold up to natural isomorphism. A -set is a functor from a group to What is a natural transformation between such functors?
Week 5 (Feb. 2 and 4) – A -set is a functor from a group to and a natural transformation between such functors is a map of -sets. Equivalences of categories. Adjoint functors: the rough idea. The hom-functor. Adjoint functors: the definition. Examples: the left adjoint of the forgetful functor from to The left adjoint of the forgetful functor from to The forgetful functor from to has both a left and right adjoint. If a category has binary products, the diagonal functor from to has a right adjoint. If it has binary coproducts, the diagonal functor has a left adjoint.
Week 6 (Feb. 9 and 11) – Diagrams in a category as functors. Cones as natural transformations. The process of taking limits as a right adjoint. The process of taking colimits as a left adjoint. Left adjoints preserve colimits; right adjoints preserve limits. Examples: the ‘free group’ functor from sets to groups preserve coproducts, while the forgetful functor from groups to sets preserves products. The composite of left adjoints is a left adjoint; the composite of right adjoints is a right adjoint. The unit and counit of a pair of adjoint functors.
Week 7 (Feb. 16 and 18) – Adjunctions. The naturality of the isomorphism in an adjunction. Given an adjunction, we can recover this isomorphism and its inverse from the unit and counit. Toward topos theory: cartesian closed categories and subobject classifiers. The definition of cartesian closed category, or ‘ccc’. Examples of cartesian closed categories. In a cartesian closed category with coproducts, the product distributes over the coproduct, and exponentiation distributes over the product.
Week 8 (Feb. 23) – Internalization. The concept of a group in a cartesian category. Any pair of objects in a cartesian closed category has an ‘internal’ hom, the object as well as the usual ‘external’ hom, the set Evaluation and coevaluation. Internal composition. In a category with a terminal object, we can define the set of elements of any object.
Week 8 (Feb. 25) – Guest lecture by Christina Osborne on symmetric monoidal categories.
Week 9 (Mar. 1 and 3) – For any category with a terminal object, elements define a functor If is cartesian, this functor preserves finite products. If is cartesian closed, so it converts the internal hom into the external hom. The ‘name’ of a morphism. Subobjects. The subobject classifier in The general definition of subobject classifier in any category with finite limits. The definition of a topos. Examples of topoi, including the topos of graphs.
Week 10 (Mar. 8 and 10) – The subobject classifier in the topos of graphs. Any topos has finite colimits. Any morphism in a topos has an epi-mono factorization, which is unique up to a unique isomorphism. The image of a morphism in topos. The poset whose elements are subobjects of an object in a topos. The correspondence between set theory and logic: given a set subsets of correspond to predicates defined for elements of intersection corresponds to ‘and’, union corresponds to ‘or’, the set itself corresponds to ‘true’, and the empty set corresponds to ‘false’. The intersection of subsets of is their product in their union is their coproduct in the set is the terminal object in and the empty set is the initial object. A lattice is a poset with finite limits and finite colimits, and a Heyting algebra is a lattice that is also cartesian closed. For any object in any topos, is a Heyting algebra. If we think of these elements of as predicates, the exponential is ‘implication’.
Where does topos theory go from here?
All the notes are handwritten, by Christina Osborne and Samuel Britton. I’d consider paying someone to TeX them up! But as you see, there are a lot of diagrams…. so you should only try it if you want to learn category theory while practicing your TikZ. And if you try it, you should let us know – it would be silly to have more than one person doing the same job, while chopping the job into parts might work well.