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I have stumbled across statements like

  • Homological algebra is linearized homotopical algebra.

  • Chain complexes are linearizations of simplicial complexes.

The Dold-Kan correspondence was often mentioned as well. Unfortunately, I don't understand it, and would have never imagined there was such a thing at all...

So what do the above statements actually mean? In what sense is the word linearization being used?

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    $\begingroup$ By the Dold-Kan correspondence, chain complexes are simplicial abelian groups. These are precisely the objects you get when you linearize simplicial sets (by which I mean, in this case, taking levelwise free abelian groups). This is a simplicial model of the process of taking singular chains on a space, which computes its homology. $\endgroup$ Jan 16, 2015 at 23:04
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    $\begingroup$ If you work with simplicial sets, rather than simplicial complexes, the idea is clear, as the comments suggest. Given a simplicial set $X$ we can take the free abelian group functor $\mathbb Z[X]$, which yields a simplicial abelian group. From this simplicial abelian group you can define a chain complex with the same sequence of abelian groups and alternating sums of face operators as differentials $d=\sum_{i=0}^n(-1)^id_i$. $\endgroup$ Jan 17, 2015 at 0:12
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    $\begingroup$ @QiaochuYuan why does taking levelwise free abelian groups deserve the name linearization? Is it just because freeness allows to "extend by linearity"? $\endgroup$
    – Exterior
    Jan 17, 2015 at 9:34
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    $\begingroup$ @Exterior: in my simple mind anything where you start with something, then wave a magic wand and end up with something else where you can perform addition, should be called "linearization". I don't think ALL linearizations should be shadows of some unique uberuniversal categorical machine, but they all usually simplify the problem at hand (eg taking homology of a space remembers only a good chunk of it, and only up to homotopy). $\endgroup$ Jan 18, 2015 at 12:40

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