TY - JOUR
T1 - Pointed homotopy of maps between 2-crossed modules of commutative algebras
AU - Akça, I. Ilker
AU - Emir, Kadir
AU - Martins, João Faria
N1 - JFM was partially supported by CMA/FCT/UNL, under the project UID/MAT/00297/2013 and by FCT (Portugal) through the "Geometry and Mathematical Physics Project", FCT EXCL/MATGEO/0222/2012. KE expresses his gratitude to CMA for the hospitality in 2014/15. We would like to thank Ronald Brown for useful comments and several corrections.
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
KW - 2-crossed module of commutative algebras
KW - Crossed module of commutative algebras
KW - Quadratic derivation
KW - Simplicial commutative algebra
UR - http://www.scopus.com/inward/record.url?scp=85046671254&partnerID=8YFLogxK
U2 - 10.4310/HHA.2016.V18.N1.A6
DO - 10.4310/HHA.2016.V18.N1.A6
M3 - Article
AN - SCOPUS:85046671254
SN - 1532-0073
VL - 18
SP - 99
EP - 128
JO - Homology, Homotopy and Applications
JF - Homology, Homotopy and Applications
IS - 1
ER -