The unit of the total d\'ecalage adjunction

TL;DR

This paper investigates the de9calage adjunction in simplicial objects, explicitly describes the unit's homotopy, and proves it is a weak equivalence in algebraic categories.

Contribution

It explicitly identifies the right adjoint of the de9calage functor with path objects and proves the unit is a weak equivalence in algebraic contexts.

Findings

Explicit formula for the right adjoint as a path object

Construction of a retracting homotopy for the unit

Proof that the unit is a weak equivalence in algebraic categories

Abstract

We consider the d\'ecalage construction $\operatorname{Dec}$ 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 $T \operatorname{Dec}X$ 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 $X\to T\operatorname{Dec}X$ of the adjunction $(\operatorname{Dec},T)$. 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 $\mathcal{C}$.