Homotopies of Crossed Modules of R-Algebroids

Abstract

In this work, given two crossed modules $\mathcal {M=}\left( \mu :\mathrm {M} \rightarrow \mathrm {A}\right)$ and ${\mathcal {N}}=\left( \eta :\mathrm {N} \rightarrow \mathrm {B}\right)$ of R-algebroids and a crossed module morphism $f:{\mathcal {M}}\rightarrow {\mathcal {N}}$, we introduce an f-derivation as an ordered pair $H=\left( H_{1},H_{0}\right)$ of maps $H_{1}: \mathrm {Mor}\left( \mathrm {A}\right) \rightarrow \mathrm {Mor}\left( \mathrm {N }\right)$ and $H_{0}:\mathrm {A}_{0}\rightarrow \mathrm {Mor}\left( \mathrm {B} \right)$ which are subject to satisfy certain axioms and show that f and H determine a crossed module morphism $g:{\mathcal {M}}\rightarrow \mathcal { N}$. Then calling such a pair $\left( H,f\right)$ a homotopy from f to g we prove that there exists a groupoid structure of which objects are crossed module morphisms from ${\mathcal {M}}$ to ${\mathcal {N}}$ and morphisms are homotopies between crossed module morphisms. Moreover, given two crossed module morphisms $f,g:{\mathcal {M}}\rightarrow {\mathcal {N}}$, we introduce an fg-map as a map $\varLambda :\mathrm {A}_{0}\rightarrow \mathrm {Mor}\left( \mathrm {N}\right)$ subject to some conditions and then show that $\varLambda$ determines for each homotopy $\left( H,f\right)$ from f to g a homotopy $\left( H^{\prime },f\right)$ from f to g. Furthermore, calling such a pair $\left( \varLambda ,\left( H,f\right) \right)$ a 2-fold homotopy from $\left( H,f\right)$ to $\left( H^{\prime },f\right)$ we prove that the groupoid structure constructed by crossed module morphisms from ${\mathcal {M}}$ to ${\mathcal {N}}$ and homotopies between them is upgraded by 2-fold homotopies to a 2-groupoid structure. Besides, in order to see reduced versions of all general constructions mentioned, we examine homotopies of crossed modules of associative R-algebras, as a pre-stage.