The unit of the total décalage adjunction

Abstract

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 $${\mathcal {C}}$$.