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I just started nonabelian homological algebra.
I may be stupid, but I do not quite understand a sentence in homological algebra saying that “Homological algebra thus studies, in particular, the homology of chain complexes in abelian categories – therefore the name.” as a conclusion to the introduction about what the derived functors do. I do not understand, the homology of a complex is just an element of the formalism, it is not an example of what the previous text does – a derived functor. It is rather an example of a homological functor from a triangulated category to an abelian category.
On the other hand the higher categorical and nonabelian analogues like homological algebra for crossed complexes is described in the article as a nonlinear analogue.
But while the syzygies of Hilbert are often considered as a birthmark of homological algebra and give relations among linear relations, the nonlinear version which is earlier and due Cayley, is the computation of nonlinear relations among polynomials, that is nonlinear dependence, and than one is given by the theory of resultants. Cayley has computed a resultant of two polynomials as a determinant of the corresponding Koszul complex, so the linear homological algebra was usuful in this work. The generalizations for more polynomials are still not understood well enough.
I want to quote interesting Tim’s remarks from a bit obsolete query now erased in homological algebra:
Tim: I would support a different wording as well. The above does not make clear that, say, group (co)homology or Lie algebra (co)homology might be considered as subclasses of ’homological algebra’. It also does not really include the crossed homological algebra that Ronnie has developed. There is also the point, which sometime I will make more precise, that homological algebra is a linearised version of homotopical algebra and that there are crossed, quadratic and so on versions intermediate between the two.
I am also slightly worried by too much emphasis on ’stable’ as this can be a ’weasel word’ like ’progress’!
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