Having recently been critical of not a few books here(!), let me mention a rather good one for a change. I’ve had on my desk for a while a copy of Topology: A Categorical Approach by Tai-Danae Bradley, Tyler Bryson and John Terilla (MIT 2020). But I have only just got round to reading it, making a first pass through with considerable enjoyment and enlightenment.
The cover says that the book “reintroduces basic point-set topology from a more modern, categorical perspective”, and that frank “reintroduces” rather matters: a reader who hasn’t already encountered at least some elementary topology would have a pretty hard time seeing what is going on. But actually I’d say more. A reader who is innocent of entry-level category theory will surely have quite a hard time too. For example, in the chapter of ‘Prelminaries’ we get from the definition of a category on p. 3 to the Yoneda Lemma on p. 12! To be sure, the usual definitions we need are laid out clearly enough in between; but I do suspect that no one for whom all these ideas are genuinely new is going to get much real understanding from so rushed an introduction.
But now take, however, a reader who already knows a bit of topology and who has read Awodey’s Category Theory (for example). Then they should find this book very illuminating — both deepening their understanding of topology but also rounding out their perhaps rather abstract view of category theory by providing a generous helping of illustrations of categorial ideas doing real work (particularly in the last three chapters). Moreover, this is all attractively written, very nicely organized, and (not least!) pleasingly short at under 150 pages before the end matter.
In short, then: warmly recommended. And all credit too to the authors and to MIT Press for making the book available open-access. So I need say no more here: take a look for yourself!