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