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.

在这项工作中，给定两个交叉的模块\（\ mathcal {M =} \ left（\ mu：\ mathrm {M} \ rightarrow \ mathrm {A} \ right）\）和\（{\ mathcal {N}} = \左（\ ETA：\ mathrm {N} \ RIGHTARROW \ mathrm {B} \右）\）的- [R -algebroids和交叉模块态射\（F：{\ mathcal {M}} \ RIGHTARROW {\ mathcal {N }} \），我们引入f-导数作为地图\（H_ {1}的有序对\（H = \ left（H_ {1}，H_ {0} \ right）\）：\ mathrm {Mor} \ left（\ mathrm {A} \ right）\ rightarrow \ mathrm {Mor} \ left（\ mathrm {N} \ right）\）和\（H_ {0}：\ mathrm {A} _ {0} \ rightarrow \ mathrm {Mor} \ left（\ mathrm {B} \ right）\）满足某些公理并显示f和H确定交叉模态态（\：g：{\ mathcal {M}} \ rightarrow \ mathcal {N} \）。然后将这样的一对\（\ left（H，f \ right）\）称为f到g的同伦，我们证明存在一个类群结构，其对象与\（{\ mathcal {M}} \ ）到\（{\ mathcal {N}} \），并且态射是交叉模块态射之间的同伦。此外，给定两个交叉的模块态射\\（f，g：{\ mathcal {M}} \ rightarrow {\ mathcal {N}} \），我们引入fg -map作为map \（\ varLambda：\ mathrm {A } _ {0} \ rightarrow \ mathrm {Mor} \ left（\ mathrm {N} \ right）\）服从某些条件，然后证明\（\ varLambda \）为从f到g的每个同伦\（\ left（H，f \ right）\）确定同伦\（\ left（H ^ {\ prime}，f \ right）\）从f到g。此外，将这样的一对\（\ left（\ varLambda，\ left（H，f \ right）\ right）\）从\（\ left（H，f \ right）\）到\（ \ left（H ^ {\ prime}，f \ right）\）我们证明了交叉模态从\（{\ mathcal {M}} \）到\（{\ mathcal {N}} \\ ）它们之间的同型异位体通过2倍同型异位体升级为2-groupoid结构。此外，为了查看所提及的所有一般结构的简化版本，我们在预备阶段研究了关联R-代数的交叉模块的同伦性。