In this study, given two crossed complexes 𝒞{\mathcal{C}} and 𝒟{\mathcal{D}} of associative R -algebras and a crossed complex morphism f:𝒞→𝒟{f\colon\mathcal{C}\to\mathcal{D}}, we construct a homotopy as a pair (H,f){(H,f\/)}, where H=(Hn){H=(H_{n})} is a sequence of R -linear maps Hn:Cn→Dn+1{H_{n}\colon C_{n}\to D_{n+1}}. Then we show that for a fixed pair 𝒞{\mathcal{C}} and 𝒟{\mathcal{D}} of crossed complexes of associative R -algebras, the family of all homotopies between crossed complex morphisms from 𝒞{\mathcal{C}} to 𝒟{\mathcal{D}} has a groupoid structure with crossed complex morphisms as objects and homotopies as morphisms.

中文翻译：

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关联R-代数的交叉复态的同伦

在这项研究中，给出了关联R-代数的两个交叉复形𝒞{\ mathcal {C}}和𝒟{\ mathcal {D}}和交叉复态态f：𝒞→𝒟{f \ colon \ mathcal {C} \到\ mathcal {D}}，我们将同伦构造为一对（H，f）{（H，f \ /）}，其中H =（Hn）{H =（H_ {n}）}是一个序列R线性映射Hn：Cn→Dn + 1 {H_ {n} \冒号C_ {n} \到D_ {n + 1}}。然后我们证明，对于固定的对R-代数的交叉复形对𝒞{\ mathcal {C}}和𝒟{\ mathcal {D}} ，, {\ mathcal {C }}至𝒟{\ mathcal {D}}具有一个类群结构，其中交叉复杂的态射为对象，同态为射态。