I am looking for a reference describing the notions of homotopy and contactability 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 decalage 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 naive 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;
$$0\le 1\le ... \le n \longmapsto 0\le 1 \le ... \le n \le 0' \le 1'\le ...\le n'$$
$$n\mapsto 2n+1, \ \ f:n\to m \,\longmapsto\, f':2n+1\to 2m+1, f'(i+n+1)=f'(i)=f(i)\forall 0\leq i\leq n$$



> **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    $$sing
 F_\bullet(n):=Hom_\text{Top}( \Delta^n, F)$$ $$sing
 X_\bullet(n):=Hom_\text{Top}( \Delta^n, X)$$ $$sing
 X_\bullet\circ[+1](n)=Hom_\text{Top}( \Delta^n\times
 [0,1]/{\Delta^n\times\{1\}}, X)$$ where $n\geqslant 0$, $n\in \Delta$
> 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).