Hello,
Learning some Alg.geometry and Sheaf theory, I got used to the notion that cohomology arises naturally as a derived functor of some sort.


This has led me thinking, singular cohomology, from algebraic topology, was never defined (In all books i've checked) as a derived functor, but just by giving cycles and boundaries.
I could not figure out by myself any reasonable functor whose derived functors yield singular cohomology, So I pose this question out here.


I hope this might shed some more insight on what singular cohomology actually measures.


Thanks



Singular Homology/Cohomology as a derived functor?
On seeing the question, I knew that I had seen something like this. Rinehart had a TAMS paper:
1972 called SINGULAR HOMOLOGY AS A DERIVED FUNCTOR. His interpretation of derived functor is, of course, that of his era but it does give a setting that may be adapted to a more modern treatment. Following up on David's reply, it is also possible to use Quillen's treatment of cohomology from the original Homotopical Algebra SL notes, together with ideas on monadic cohomology (the Triples volume of SLN paper by Applegate and Tierney, plus ideas from Mike Barr) to get another take on the question.... but without the various volumes in front of me I will not attempt to do that here and now. :-)


