Two-fold homotopy of 2-crossed module maps of commutative algebras

Abstract

We address the homotopy theory of 2-crossed modules of commutative algebras. In particular, we define the concept of a two-fold homotopy between a pair of one-fold homotopies connecting 2-crossed module maps $\mathcal{A}\to \mathcal{A}\prime$. We also prove that if the domain 2-crossed module $\mathcal{A}$ is free up to order one (i.e., if the bottom algebra is a polynomial algebra) then we have a 2-groupoid of 2-crossed module maps $\mathcal{A}\to \mathcal{A}\prime$ and their homotopies and two-fold homotopies.

Keywords:

• 2-crossed module of commutative algebras
• crossed module of commutative algebras
• quadratic derivation
• simplicial commutative algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 55U10 (principal)
• 18D05
• 18D20
• 55Q15 (secondary)

Acknowledgements

Kadir Emir was collaborating member of CMA/FCT/UNL, 2014/15.

Additional information

Funding

This work was supported by FCT/Portugal Strategic Project [grant number: UID/MAT/00297/2013].