Abstract
These notes were originally developed as lecture notes for a category theory
course. They should be well-suited to anyone that wants to learn category
theory from scratch and has a scientific mind. There is no need to know
advanced mathematics, nor any of the disciplines where category theory is
traditionally applied, such as algebraic geometry or theoretical computer
science. The only knowledge that is assumed from the reader is linear algebra.
All concepts are explained by giving concrete examples from different,
non-specialized areas of mathematics (such as basic group theory, graph theory,
and probability). Not every example is helpful for every reader, but hopefully
every reader can find at least one helpful example per concept. The reader is
encouraged to read all the examples, this way they may even learn something new
about a different field.
Particular emphasis is given to the Yoneda lemma and its significance, with
both intuitive explanations, detailed proofs, and specific examples. Another
common theme in these notes is the relationship between categories and directed
multigraphs, which is treated in detail. From the applied point of view, this
shows why categorical thinking can help whenever some process is taking place
on a graph. Form the pure math point of view, this can be seen as the
1-dimensional first step into the theory of simplicial sets. Finally, monads
and comonads are treated on an equal footing, differently to most literature in
which comonads are often overlooked as "just the dual to monads". Theorems,
interpretations and concrete examples are given for monads as well as for
comonads.
Description
[1912.10642] Notes on Category Theory with examples from basic mathematics
Links and resources
Tags
community