LAUSR

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