I am a mathematician who is using category theory all the time in my work in algebraic geometry, so I am exactly the wrong audience for this write-up!
I think that talking about “bad definitions” and “confusing presentation” is needlessly confrontational. I would rather say that the traditional presentation of category theory is perfectly adapted to its original purpose, which is to organise and to clarify complicated structures (algebraic, topological, geometric, …) in pure mathematics. There the basic examples of categories are things like the category of groups, rings, vector spaces, topological spaces, manifolds, schemes, etc. and the notion of morphism, i.e. “structure-preserving map”, is completely natural.
As category theory is applied more broadly in computer science and the theory of networks and processes, it is great that new perspectives on the basic concepts are developed, but I think they should be thought of as complementary to the traditional view, which is extremely powerful in its domain of application.
the traditional presentation of category theory is perfectly adapted to its original purpose
I think this is too generous. The traditional way of conceptualizing a given math subject is usually just a minor modification of the original conceptualization. There’s a good reason for this, which is that updating the already known conceptualization across a community is a really hard coordination problem—but this also means that the presentation of subjects has very little optimization pressure towards being more usable.
I am a mathematician who is using category theory all the time in my work in algebraic geometry, so I am exactly the wrong audience for this write-up!
I think that talking about “bad definitions” and “confusing presentation” is needlessly confrontational. I would rather say that the traditional presentation of category theory is perfectly adapted to its original purpose, which is to organise and to clarify complicated structures (algebraic, topological, geometric, …) in pure mathematics. There the basic examples of categories are things like the category of groups, rings, vector spaces, topological spaces, manifolds, schemes, etc. and the notion of morphism, i.e. “structure-preserving map”, is completely natural.
As category theory is applied more broadly in computer science and the theory of networks and processes, it is great that new perspectives on the basic concepts are developed, but I think they should be thought of as complementary to the traditional view, which is extremely powerful in its domain of application.
I think this is too generous. The traditional way of conceptualizing a given math subject is usually just a minor modification of the original conceptualization. There’s a good reason for this, which is that updating the already known conceptualization across a community is a really hard coordination problem—but this also means that the presentation of subjects has very little optimization pressure towards being more usable.