$\newcommand\sSets{\text{sSets}}\newcommand\sing{\text{sing}}\DeclareMathOperator\Hom{Hom}\newcommand\Top{\text{Top}}$I am looking for a reference describing the notions of homotopy and contractibility in terms of endofunctors of a simplicial category induced by endofunctors of $\Delta$.

Is there a definition of contractibilty in terms of the endofunctor of a simplicial category induced by the décalage endomorphism of $[+1]:\Delta\to \Delta $ shifting dimension by adding a new "always fixed" minimal element to a linear order $$n\mapsto n+1, \ \ f:n\to m \longmapsto f':n+1\to m+1, f'(0):=0; f'(i+1):=f(i) \forall 0\leq i\leq n. $$

The following naïve considerations suggest that a map of sufficiently nice topological space is contractible iff the corresponding map of singular complexes factors through the image with shifted dimension, see the claim below.

Instead of the endomorphism $[+1]:\Delta\to\Delta$ one might want to use $[\times 2]:\Delta\to\Delta$ "doubling" each linear order; $$0\le 1\le \dotsb \le n \longmapsto 0\le 1 \le \dotsb \le n \le 0' \le 1'\le \dotsb\le n'$$ $$n\mapsto 2n+1, \ \ f:n\to m \,\longmapsto\, f':2n+1\to 2m+1, f'(i+n+1)=f'(i)=f(i)\forall 0\leq i\leq n.$$

Claim. Let $F$ and $X$ denote “nice” topological spaces.

A map $h_0:F\to X$ is contractible, i.e. it factors through the cone of $F$ as $F\xrightarrow{x\mapsto (x,0)} F\times [0,1]/F\times \{1\} \xrightarrow h X$, iff in $\sSets$ the map $\sing F_\bullet \to \sing X_\bullet$ of singular complexes factors via $\sing X_\bullet\circ[+1]$, i.e. the map $\sing F_\bullet \to \sing X_\bullet$ factors as $$\sing F_\bullet \to \sing X_\bullet\circ[+1]\to \sing X_\bullet.$$ Here $\sing X_\bullet\circ[+1]\to \sing X_\bullet$ is the expected map “forgetting the first coordinate”.

Recall that the singular complex is defined using simplices \begin{gather*} \sing F_\bullet(n):=\Hom_{\Top}( \Delta^n, F) \\ \sing X_\bullet(n):=\Hom_{\Top}( \Delta^n, X) \\ \sing X_\bullet\circ[+1](n)=\Hom_{\Top}( \Delta^n\times [0,1]/{\Delta^n\times\{1\}}, X) \end{gather*} where $n\ge 0$, $n\in \Delta$ is the linear order with $n+1$ elements, and $ \Delta^n\times [0,1]/{\Delta^n\times\{1\}}$ is the cone of $n$-simplex $\Delta^n$.

To define a lifting $h_\bullet$, take each map $\delta: \Delta^n \to F$ in $F_\bullet(n)$ to a map $$ h_*(\delta):\Delta^n\times [0,1]/\Delta^n\times\{1\}\to X $$

in $X_\bullet(n+1)$ defined by $$h_*(\delta)( x,t ):= h(\delta(x),t).$$

To see the other direction, note that $h_\bullet:F_\bullet\to X_\bullet[+1]$ takes a singular simplex $\delta:\Delta^n\to F$ into $h_\bullet(\delta):\Delta^{n+1}=\Delta^n\times [0,1]/\Delta^n\times\{1\}\to X$ such that $\delta\circ h_0=h_\bullet(\delta)_{|\Delta^n\times \{0\}}$, i.e. each $\delta:\Delta^n\to F\to X$ factors through the cone of $\Delta^n$.
A verification using functoriality shows that the same factorisation holds for $\mathbb S^n = \partial \Delta^{n+1}$, which means exactly that $h_0$ is weakly contractible, and for “nice” topological spaces contractible and weakly contractible are equivalent.

I am also looking for a reference to this particular claim (if it is true).