I am looking for references defining contractible or null-homotopic in terms of endofunctors $\Delta\to\Delta$.
Let $[+1]:\Delta\to\Delta, n\mapsto n+1$ be the endofunctor of $\Delta$ adding a new least element. In $X_\bullet\in Ob\, sC$ of a simplicial category $sC$, the object $X_\bullet\circ [+1]$ is referred to the simplicial path space of $X_\bullet$, and comes equipped with two maps $Y\circ[+1]\xrightarrow{pr_0} disc(Y_0)$ (induced by $[0]\to[0<1<..<n]$) and $Y\circ[+1]\xrightarrow{pr_{1,2,...}} Y$ (induced by $[1<..<n]\to[0<1<..<n]$).
Is it true that in sSets,
$$Y_\bullet\circ [+1]=disc(Y_0)\times_{Y_\bullet} Y_\bullet$$ is a subobject of the path space
$Y^{\Delta[1]}=\operatorname{InternalHom}_{sSets}(\Delta[1],Y_\bullet)$
induced by maps $\Delta[n]\times \Delta[1]\to \Delta[n+1]$ contracting $\Delta[n]\times \Delta[0]$ for some fixed embedding $\Delta[0]\to \Delta[1]$ ? In what generality something like this holds ?
Call a map $X_\bullet \to Y_\bullet$ contractible iff it factors as $X_\bullet\to Y_\bullet\circ[+1]\xrightarrow{pr_{1,2,...}} Y_\bullet$ (if the fact above is true,
in sSets contractible means being homotopic to a map $X_\bullet\to disc(Y_0)$ where $disc(Y_0)$ denotes the constant (discrete) object $(Y_0,Y_0,...)$).
This definition makes sense in any category of simplicial objects.
In what generality does this notion has good properties ?
I am looking for a reference discussing this definition.
This probably easily follows from the reference (Lemma 1.5.1, K.Waldhausen, Waldhausen, Friedhelm Algebraic K-theory of spaces. Algebraic and geometric topology (New Brunswick, N.J., 1983), 318–419, Lecture Notes in Math., 1126, Springer, Berlin, 1985.) given by John Rognes in Defining homotopy via endofunctors of a simplicial category but I am looking now for something mentioning this more explicitly.
