TITLE: Singular Homology/Cohomology as a derived functor?
QUESTION [22 upvotes]: 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

REPLY [10 votes]: Here's an elaboration of David Roberts' answer, a perspective that can be found in Quillen's 'Homotopical Algebra'. Quillen thought of homology as "derived abelianisation", and the abelianisation of a space (by which I mean simplicial set) can only be the free simplicial abelian group on that space, seeing as it's left adjoint to the forgetful functor $sAb \to sSet$.
Composing the free simplicial abelian group functor with the Dold-Kan correspondence - an equivalence of categories between simplicial abelian groups and chain complexes - gives a functor $sSet \to Ch$, which happens to coincide (more or less) with the singular complex functor. If we endow $Ch$ with the injective model structure this is a left Quillen functor. By Quillen's philosophy, homology of spaces should be the total left derived functor of this functor. But the word 'derived' is sort of vacuous here, because all simplicial sets are cofibrant, and this negates the need to explicitly use any model category language. This is why we think of the singular complex as the "homology object" of a space.