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

I. Ilker Akça, Kadir Emir, João Faria Martins

Research output: Contribution to journalArticle

3 Citations (Scopus)
3 Downloads (Pure)

Abstract

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
Volume18
Issue number1
DOIs
Publication statusPublished - 2016

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Crossed Module
Commutative Algebra
Homotopy
Homotopy Theory
Groupoid
Equivalence relation
Morphisms
Strictly

Keywords

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

Cite this

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abstract = "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.",
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Pointed homotopy of maps between 2-crossed modules of commutative algebras. / Akça, I. Ilker; Emir, Kadir; Martins, João Faria.

In: Homology, Homotopy and Applications, Vol. 18, No. 1, 2016, p. 99-128.

Research output: Contribution to journalArticle

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AU - Emir, Kadir

AU - Martins, João Faria

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