Notes about Category Theory

Build

A compiled pdf encompassing the whole work is already provided in this repo. If you want to build it by yourself, there is a Makefile for the purpouse.

$ make

For I want to make my work easier, there are also the pdfs of the single chapters with the cross-references handled by the package xr. A distinguished script is for that task and it sleeps in this repo as well.

$ ./make-articles.sh

TODO-s

Ch 0 | Preamble

• Write a short intro for NBG? (Could a short intro for NBG even exist?)

Ch 1 | Basic vocabulary

• Free categories can be readily introduced in this chapter and return to them when we deal with limits od adjunctions. This may be the occasion in which some sort graphs appear, in fact categories are some kind of graphs. Maybe this may become an exercise for the section Other exercises...

• Reorganise the section of foundations: it is a quite poorly mantained part. The idea is that in some sense things are to be stated for later scopes.

• The sections about iso-, mono- and epimorphisms need a rewrite. Put all in a single section? As for monos and epis, they can be introduced when it comes to talk about the hom functor. Consider that.

• On the section "Basic constructions", prepare more examples for (co)comma categories: there are field extensions and covering spaces, for example. But let us start with simple examples. Have we mentioned subcategories before?

• The section about the functor hom is just a stub. I am thinking of talking about monos and epis for the first time here.

• More examples about natural transformations. Talk about natural isomorphisms.

• Rethink the section Equivalent categories. A section about representable functors and and after that one dedicated to Yoneda Lemma follow. Cool examples here involving natural transformations, the first examples about universal properties.

• Prepare a Other exercises, as it is done for the chapter of limits. Some initial ideas: relevant subcategories (like Mon(a), Epi(a), ...) and properties (first contact with universal properties?), currying functors (this will involve natural transformations), some basic stuff about 2-cats [they will not subject of these notes, though], some words about string diagrams? and maybe exercises in the spirit "from representability to limits and adjointness"? Insert exercises about facts about equivalences too.

Ch 2 | Limits and colimits

• The section Definition needs some refinement: Currently there is just a definition and some incooplete example and a proposition. However, I think there are some examples that can be made without anticipating the coming attractions. One example is the limit of a sequence of objects and morphisms "... -> * -> * -> * -> ...", that might be used later for pullbacks and pushouts.

• Write about the evaluation homomorphism for polynomials in any commutative ring. Maybe this is some example which is finer to be encountered in the section about representable functors.

• There is a certain example about the universal covering space. This will require some time, but it could be nicely delineated and some nice bibliography entry could be added for the topic.

• Extend comments following some examples: invest more time in explaining what is the point of the things we are doing. Focus one the computational aspect of the things too.

Ch 3 | Adjointness

• More examples and more names.

About the TeX behind

• Use tcolorbox. At the current state, the are some nuisances, like underfull \vbox through the book.