Algebraic Topology via Categories

[Reedeegi]

Does anyone know of a modern book on algebriac topology developed in a purely categorical framework? I've been recommended Eilenberg and Steenrod (which I may end up getting regardless), but I'm looking for more recent developments in both material and pedagogy.

 

[MathematicalPhysicist]

I am not sure if there any modern textbooks which are purely categorical, but I guess Spanier's textbook is as close as it comes of being categorical in nature.

But be warned it's tersier than any other textbook in the field.
I stopped reading after the first chapter, some of it because I didn't have time to do the exercises and really assimilate the material.

 

[Landau]

I don't know what you mean by a "purely" categorical framework. But Peter May has written a modern book with lot of category theoretic language: click (pdf).

 

[mathwonk]

well this frustrating browser just erased my post.

basically i said May's free! book looks great, and thanks to Landau.

I suggest reading May's introduction and his guide to further reading, and then you will have an expert's answer to essentially your question, and much more.

also i recommended starting by learning the fundamental group, if you are a beginner.

The reason people are puzzled by your question as posed is that algebraic topology is by definition a study of functors from topology to algebra, so almost any treatment falls under this heading, except maybe a really old one like Hocking and Young.

I.e. essentially any treatment that constructs an algebraic object out of a topological space and then immediately also constructs the induced algebraic homomorphism coming from a continuous map of the top. spaces, and checks that compositions go to compositions, and identities go to identities, hence necessarily isomorphisms go to isomorphisms, is taking a categorical approach.

But maybe you are way beyond this and interested in spectra, cohomology operations, axiomatic homotopy, derived categories, and so on. Anyway, May discusses everything from the perspective of an expert.

 

[atyy]

Wow, mathwonk is back. That's great, even if it's only for a while!

 

[mathwonk]

to the physicist who quit reading spanier after chapter one, let me suggest that chapter one may be the most terse and unreadable chapter, and that chapters 2,3,4 may be much better and easier.