Bayle Shanks's website: notes-math-categoryTheory

Category theory is the study of paths through graphs, and the equational theory of paths in particular graphs (e.g. in some graphs you may wish to assert that path "A" is the same as the composition of path "B" with path "C") with the following complications:

you can have more than one path between the same two nodes
- for example there might be three different one-step paths from x to y
paths are transitively closed under concatenation
- for example if there is a path from x to y and a path from y to z, then these can be concatenated into a path from x to z
each path is reified into its own edge in the graph
- for example if there is a path from x to z, then there will be an edge from x to z representing this path
each node has a distinguished self-loop, that is, an edge from itself to itself, although it may or may not also have other self-loops. The self loops are identities on paths.
- for example, any node x will have a distinguished edge from x to x

The edges in category theory are called 'morphisms'.

This abstraction is useful for representing things like function composition (let the nodes in the graph be the domain and codomain of the function, let the edges in the graph be functions. Function composition is path concatenation; identity functions are self-loops) and homo- and iso- morphisms between structures, and automorphisms. It allows a number of facts about these application areas to be generalized to a more abstract setting. Often times it takes a property that you might typically express using a statement about elements of sets and shows you how to express it in terms of function composition, leaving a set as an opaque object and not referring to its elements. However there are also many categories whose morphisms are not homomorphisms, for example the category of the <= relation on natural numbers.

If you like thinking about homomorphisms, and about graphs, then you will probably like category theory.

https://news.ycombinator.com/item?id=7715277 claims that an interpretation of the Yoneda lemma is "the collection of all ways to relate an object to other objects is isomorphic to the object itself", that is, objects "are what they are because of how they relate to other things".

notes-math-categoryTheory

Links