We consider the décalage construction \({{\,\mathrm{Dec}\,}}\) and its right adjoint \(T\). These functors are induced on the category of simplicial objects valued in any bicomplete category \({\mathcal {C}}\) by the ordinal sum. We identify \(T{{\,\mathrm{Dec}\,}}X\) with the path object \(X^{\Delta [1]}\) for any simplicial object X. We then use this formula to produce an explicit retracting homotopy for the unit \(X\rightarrow T{{\,\mathrm{Dec}\,}}X\) of the adjunction \(({{\,\mathrm{Dec}\,}},T)\). When \({\mathcal {C}}\) is a category of objects of an algebraic nature, we then show that the unit is a weak equivalence of simplicial objects in \({\mathcal {C}}\).

中文翻译：

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总附加费的单位

我们考虑décalage构造\（{{\，\ mathrm {Dec} \，}} \）及其右伴随\（T \）。这些函子是按序数和在任何双完全类别\（{\ mathcal {C}} \\）中赋值的简单对象类别上产生的。对于任何简单对象X，我们用路径对象\（X ^ {\ Delta [1]} \）来标识\（T {{\，\ mathrm {Dec} \，}} X \）。然后，我们使用该公式以产生用于该单元的明确退避同伦\（X \ RIGHTARROWŤ{{\，\ mathrm {十二月} \，}} X \）的红利的\（（{{\，\ mathrm {十二月} \，}}，T）\）。当\（{\ mathcal {C}} \）是一个具有代数性质的对象的类别，然后我们证明该单位是\（{{mathcal {C}} \\）中的单纯对象的弱等价形式。