It has so happened that I have come this far knowing nothing on the subject of algebraic topology (as in homology theories of topological spaces and their applications). I've decided to finally read up on that during the summer.

Seemingly, however, the authors of most books for beginners are hesitant to make use of nontrivial homological algebra and category theory, which, if I'm not mistaken, could be used to speed up and at the same time clarify the presentation. I, on the other hand, would dare say to be somewhat familiar with these disciplines. (I'm, to different degrees, acquainted with derived functors, spectral sequences, derived categories as well as sheaf cohomology and Lie algebra/group cohomology.)

Thus, what I'm looking for is an introduction to algebraic topology the author of which readily employs the above concepts when appropriate.