I think I am not the only person to feel frustrated with category theory – or perhaps more with the textbooks. The ones I have come across – like Saunders MacLane and Emily Riehl – are certainly insightful and quite well-written, but I feel that I am drowning in exotic examples that I can't follow. This may be because I'm poorly educated or a bit dim, of course, but I feel that since category theory is such a powerful tool, and increasingly important, efforts ought to be made to make it accessible to people without a background in advanced, higher mathematics. As an example of the kind of difficulties I keep running up against (from Emily Riehl, "Category Theory in Context"):
The isomorphisms $A \cong T A \oplus (A/T A)$ are not natural in $A
\in Ab_{fg}$
Here, $A$ is Abelian, $TA$ is "the torsion subgroup of $A$" and $Ab_{fg}$ the category of freely generated, abelian groups. I understand Abelian group, and I can just about remember that I've heard about freely generated ones, but in the hassle of having to understand the premises of the example, I completely lose sight of the thing I'm supposed to learn here: how natural transformations work.
Are there no resources that treat the theory, while relying only on relatively elementary mathematics (naive set theory, elementary group theory, – vector spaces, – topology etc)?
EDIT: I have left the original last section in, since it is being referred to in the comments, but here is perhaps a better final section.
I am of course aware of the many, freely available texts about category theory – I've recently come across Algebra and Topology by Pierre Schapira, which looks promising. But what I have in mind is – if someone wanted to write "the ultimate introduction", targeted at, say the entry-level at university, is that even possible in a meaningful way? It ought to be – after all, we do teach naive set theory very early, and vector spaces and abstract algebra loom large as soon as you enter a maths department, I believe. The actual category theory isn't really that much harder, I think – what makes it hard is the examples.
Best Answer
Other than Simmons, a good suggestion from the comments, Leinster's Basic Category Theory aims to be more, well, basic than either example you give. Lawvere and Schanuel's Conceptual Mathematics has even, I think, been used with some success with high-school age students. You might also look into David Spivak and Brendan Fong's recent Seven Sketches in Compositionality, which aims to introduce category theory to the non-mathematician with applications in mind. John Baez has been leading an online discussion forum full of relative novices working through the latter.