Seminar: Winter 2016

Seminar - Winter 2016

Category Theory

John Baez

Here are the notes from a basic course on category theory. Unlike the Fall 2015 seminar, this tries to be a systematic introduction to the subject. A good followup to this course is my Fall 2018 course.

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:

Lectures notes typeset by Michael Nelson and Marco Aurelio García Portillo.

You can get the LaTeX files created by Nelson and García Portillo here. You can email me about mistakes or fix them yourself and send me a fixed version, which I'll check. Their typeset version was based on these handwritten versions:

Christina Osborne's handwritten notes.

Samuel Britton's handwritten notes.

Or, you can look at individual weeks:

Week 1 (Jan. 5 and 7) - The definition of a category. Some familiar categories: \(\mathrm{Set}\), \(\mathrm{Gp}\), \(\mathrm{Vect}_k\), \(\mathrm{Ring}\), \(\mathrm{Top}\), \(\mathrm{Met}\) and \(\mathrm{Meas}\). 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.

Christina Osborne's notes.

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.
- Christina Osborne's notes.
Week 3 (Jan. 19 and 21) - Equalizers and coequalizers, and what they look like in \(\mathrm{Set}\) and other familiar categories. Pullbacks and pushouts, and what they look like in \(\mathrm{Set}\). Composing pullback squares.
- Christina Osborne's notes.
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 \(G\)-set is a functor from a group \(G\) to \(\mathrm{Set}\). What is a natural transformation between such functors?
- Christina Osborne's notes.
Week 5 (Feb. 2 and 4) - A \(G\)-set is a functor from a group \(G\) to \(\mathrm{Set}\), and a natural transformation between such functors is a map of \(G\)-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 \(\mathrm{Grp}\) to \(\mathrm{Set}\). The left adjoint of the forgetful functor from \(\mathrm{Vect}_k\) to \(\mathrm{Set}\). The forgetful functor from \(\mathrm{Top}\) to \(\mathrm{Set}\) has both a left and right adjoint. If a category \(C\) has binary products, the diagonal functor from \(C\) to \(C \times C\) has a right adjoint. If it has binary coproducts, the diagonal functor has a left adjoint.
- Christina Osborne's notes.
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.
- Christina Osborne's notes.
Week 7 (Feb. 16 and 18) - Adjunctions. The naturality of the isomorphism \( \mathrm{hom}(Fc,d) \cong \mathrm{hom}(c,Ud)\) 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.
- Christina Osborne's notes.
Week 8 (Feb. 23) - Internalization. The concept of a group in a cartesian category. Any pair of objects \(X, Y\) in a cartesian closed category has an 'internal' hom, the object \(Y^X\), as well as the usual 'external' hom, the set \(\mathrm{hom}(X,Y)\). Evaluation and coevaluation. Internal composition. In a category with a terminal object, we can define the set of elements of any object.
- Christina Osborne's notes.
Week 8 (Feb. 25) - Guest lecture by Christina Osborne on symmetric monoidal categories.
- Christina Osborne's notes.
Week 9 (Mar. 1 and 3) - For any category \(C\) with a terminal object, elements define a functor \(\mathrm{elt} : C \to \mathrm{Set}\). If \(C\) is cartesian, this functor preserves finite products. If \(C\) is cartesian closed, \(\mathrm{elt}(Y^X) \cong \hom(X,Y)\), so it converts the internal hom into the external hom. The 'name' of a morphism. Subobjects. The subobject classifier in \(\mathrm{Set}\). 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.
- Christina Osborne's notes.
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 \(\mathrm{Sub}(X)\), whose elements are subobjects of an object \(X\) in a topos. The correspondence between set theory and logic: given a set \(X\), subsets of \(X\) correspond to predicates defined for elements of \(X\), intersection corresponds to 'and', union corresponds to 'or', the set \(X\) itself corresponds to 'true', and the empty set corresponds to 'false'. The intersection of subsets is their product in \(\mathrm{Sub}(X)\), their union is their coproduct in \(\mathrm{Sub}(X)\), the set \(X\) is the terminal object in \(\mathrm{Sub}(X)\), 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 \(X\) in any topos, \(\mathrm{Sub}(X)\) is a Heyting algebra. If we think of these elements of \(\mathrm{Sub}(X)\) as predicates, the exponential is 'implication'.
Where does topos theory go from here?
- Christina Osborne's notes.