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Differential homological algebra in its initial formulation is due to Eilenberg and Moore, who first published the homological version of the Eilenberg–Moore spectral sequence in 1965 (and the cohomological version never), sufficiently long after its date of discovery that other accounts had been published in the meantime.

Paul Baum's 1963 thesis gives an account, but Moore had spoken on this theory at least as early as the 1959–60 Seminaire Henri Cartan, and in that appearance, says he and Eilenberg had worked it out and that it had appeared in notes multigraphié from the 1957–8 Princeton seminar on algebraic topology, which thus may be the earliest written source.

Are these still extant?

If so, where can they be found?

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    $\begingroup$ see mathoverflow.net/q/375776/11260 $\endgroup$ Nov 6, 2020 at 9:25
  • $\begingroup$ The term 'differential' is redundant. Homological algebra is always differential. $\endgroup$ Dec 13, 2020 at 7:08
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    $\begingroup$ Thank you for your answer, but differential homological algebra is the name of a different subject developed apparently in the mid-or-late fifties by Eilenberg and Moore which takes as its object of study differential graded modules over differential graded algebras. This is distinct from the classical Cartan–Eilenberg development of homological algebra in which the modules in a complex do not themselves carry differentials. $\endgroup$
    – jdc
    Dec 14, 2020 at 15:10

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The origins of differential homological algebra

According to Weibel, who has written one of the standard texts on homological algebra:

Homological algebra had its origins in the 19th century, via the work of Riemann (1857) and Betti (1871) on 'homology numbers' and the rigorous development of homology numbers by Poincare in 1895. An observation of Emmy Noether in 1925 shifted attention to homology groups of a space ... [later] Cartan and Eilenbergs book crystallised and completely redirected the field ...

I'd also add, which is often not noted, that the term 'differential' for the morphisms between the chain modules actually harks back to calculus. After all, the differential d, satisfies d^2=0, which is the signature relation for the fluxions of Newton and the infinitesimals of Liebniz.

It's also worth noting, in line with this connection, that the differential calculus on a manifold (and which physicists would recognise as vector analysis) assembles itself into the de Rham complex with the de Rham differential.

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