# Is "Categories and Sheaves" a good followup to Aluffi's "Algebra: Chapter 0"?

I'm about to finish Aluffi's "algebra: chapter 0" and am a bit confused as to what should be my next move. I've been planning to read Tom Dieck's Algebraic Topology for some time now. I glimpsed at it several times and the style of writing very much matches my own taste.

The thing is, although I have some grasp of categories from Aluffi, my grounding in category theory isn't as solid as I would like. And since the book by Dieck uses categories extensively I'd like to be a bit more fluent in the language before approaching it.

Towards this purpose (and since i'm interested in both homological and commutative algebra regardless) I had the idea of finding a book that starts with a formal treatment of the basics of category theory and moves to more advanced/specialized concepts in a moderate pace. That way i could start reading until i'm completely comfortable with the language, then pick up Dieck's "algebraic topology" and read the two books simultaneously. After surfing the web a bit i found the following title.

It looks like the book for me. The problem is i didn't find any reviews about it so i'm not so sure.

Can anyone recommend a book that could fill the roles i described?

Might be relevant that I prefer to read books cover to cover than to pick up different things from different sources.

My background (rough description):

• Differential geometry (Guilliam and Pollack + in the middle of Jefferey lee's book)
• Algebra (Aluffi + Herstein).
• Topology (Munkres)
• Analysis (baby+big Rudin, currently reading "Functional Analysis" by Rudin)
• Categories and sheaves is quite difficult if you are not already familiar with categories, I think. Jul 28, 2015 at 12:03

0) The excellent mathematician you evoke has as family name (=surname) tom Dieck and as first name Tammo: tom is part of his surname and has nothing to do with Tom, the endearing form of Thomas.

1) Your idea of "finding a book that starts with a formal treatment of the basics of category theory and moves to more advanced/specialized concepts in a moderate pace" is exactly backward: you should learn some mathematical subject that attracts you first and learn the relevant categorical facts as the need makes itself felt.
Else you will learn an unmotivated and dry formalism with no clue as to its usefulness.

2) I can't recommend strongly enough to keep away from Kashiwara-Shapira's book.
It is an extremely technical and advanced monograph written by and addressed to experts.
Kashiwara is a world renowned specialist in algebraic analysis, a domain created by his advisor Mikio Sato (who introduced hyperfunctions, a sort of alternative to Schwartz's distributions). Kashiwara has arguably been the dominant figure in $\mathcal D$-module theory before he turned to other subjects like the microlocal theory of sheaves which he created with Schapira.
In summary: be very aware that this monograph is addressed to quite advanced readers, as its inclusion in Springer's prestigious series Grundlehren der mathematischen Wissenschaften already shows.

3) The book by tom Dieck is an excellent choice: it covers basic algebraic topology and has a very healthy attitude toward category theory.
Namely, when the author needs a concept he introduces it, explains it, and uses it in the particular context where it arises.
A typical example is on page 60, where tom Dieck introduces the concept of $2$-category while studying the homotopy groupoid $\Pi(X,Y)$.
The only caveat is the length of the book, but since the book is well organized you don't have to read it from cover: just select the morsels you find most appetizing.

• Thanks for the help. This answer does cheer me up a little. Although i'm about to finish Aluffi's book and had some experience with diagram chasing, functors and limits, category theory still looks like this huge freighting language, and the endless list of abstract concepts i didn't yet assimilate into my language (like monads, algebras, groupoids, represantability and the list goes on) is really daunting. I take it your advice is to learn these (concepts) when i absolutely need them, and not just for the sake of learning "the language" (?) Jul 28, 2015 at 12:26
• Dear @Saal: no,no, tom Dieck is not verbose. He just wants to cover a broad territory in algebraic topology. Another choice is Rotman's Introduction to Algebraic Topology, which is more elementary and very user friendly. The same Rotman wrote a nice introduction to homological algebra, using more category theory, which you might also like. Jul 28, 2015 at 13:56
• Dear Saal, it makes me very happy that I could be of help. And, yes, algebraic topology is a wonderful subject. It is there that I first saw the power of functorial thinking: applying suitable functors like $\pi_1$ or $H_i$ solves some problems so easily that one has the feeling one is cheating. But, hey, in love and math all is fair :-) Jul 28, 2015 at 14:52
• @SaalHardali Concerning your apprehension of "this huge freighting language", I used to be freightened as well by those big words. But often, when I forced myself to learn about one of those abstract concepts because I needed it, I ended up thinking: "Oh well, that's all it is...". Once you tackle a little with a categorical notion and once you had some nice examples in mind, you often understand it very well. In short, the slogan is: it is far too general to be difficult.
– Pece
Jul 28, 2015 at 16:54
• @Pece: nice slogan, that! Jul 28, 2015 at 18:59