The fundamental lemma of homological algebra discusses the extension of arrows to chain maps from a projective to an arbitrary resolution, and the uniqueness-up-to-homotopy of such an extension. Indeed, if $\mathsf{A}$ is an abelian category and $P_\bullet, Q_\bullet$ are respectively a projective and an arbitrary resolution of objects $A,B$, then the fundamental lemma says $$\mathsf{Hom}_\mathsf{A}(A,B)\cong [P_\bullet,Q_\bullet].$$

The acyclic model theorem seems to go in much the same spirit. The main difference seems to be the the acyclic model theorem has a notion of freeness, while the fundamental lemma employs projectivity. Indeed, the acyclic models theorem more-or-less says that natural transformations between the zeroth homology of a free functor taking values in $\mathsf{Ch}^+_\bullet(\mathsf A)$ to the zeroth homology of a parallel acyclic functor are in bijection with (chain) homotopy classes of natural transformations: $$\mathsf{Nat}(H_0F,H_0G)\cong [F,G].$$

Is the fundamental lemma of homological algebra a special case of (some version of) the acyclic model theorem? What is the big picture here?

  • $\begingroup$ Hopefully I have some time later to elaborate on this, but there is a short discussion on Wikipedia about this. It would be helpful, too, if you say which version of the fundamental lemma and the theorem on acyclic models you are using so any answer you get is actually helpful! en.wikipedia.org/wiki/… $\endgroup$ – Takumi Murayama Jun 13 '15 at 18:34
  • $\begingroup$ @TakumiMurayama I edited the question. $\endgroup$ – Arrow Jun 16 '15 at 12:37

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