Pointed homotopy of maps between 2-crossed modules of commutative algebras

I. Ilker Akça, Kadir Emir, João Faria Martins

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4 Citations (Scopus)
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We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy relation, and prove that it yields an equivalence relation in very unrestricted cases (freeness up to order one of the domain 2-crossed module). This latter condition strictly includes the case when the domain is cofibrant. Furthermore, we prove that this notion of homotopy yields a groupoid with objects being the 2-crossed module maps between two fixed 2-crossed modules (with free up to order one domain), the morphisms being the homotopies between 2-crossed module maps.

Original languageEnglish
Pages (from-to)99-128
Number of pages30
JournalHomology, Homotopy and Applications
Issue number1
Publication statusPublished - 2016


  • 2-crossed module of commutative algebras
  • Crossed module of commutative algebras
  • Quadratic derivation
  • Simplicial commutative algebra


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