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How can one define the chain homotopy in non-abelian category? (The category I have in mind is the category of chain complexes of monoids.)

  • The notion of chain homotopy is induced by an "algebraic" interval, namely the chain complex corresponding to the simplicial complex $\Delta^1$. I don't see how to build that chain complex without using subtraction. – Zhen Lin Apr 18 '14 at 16:26
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    Replace it with simplicial homotopy of maps between simplicial objects. (There's no reason, from the perspective of homotopical algebra, to work in the category of chain complexes of monoids, as the Dold-Kan theorem fails in this setting.) – Qiaochu Yuan Apr 18 '14 at 17:26
  • For commutative simplicial monoids one may add up separately odd-numbered and even-numbered face operators to obtain a globular structure (something like chain complex with parallel pairs of boundary operators). See e. g. this paper.

    I don't know if something similar can be done in the noncommutative case...

     – მამუკა ჯიბლაძე Apr 18 '14 at 18:59
  • This mathoverflow thread might be useful: http://mathoverflow.net/questions/430/homological-algebra-for-commutative-monoids – David White Apr 18 '14 at 19:25

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