the $d_0$ of $G(K)$? Why do the source and target maps at each level in $G(K)$ end up at the zeroth and first vertex rather than the seemingly more natural zeroth and $n^{th}$ that occur in $S(\mathbb{A})$? In fact is there a variant of the $G(K)$ construction that is nearer to the $S(\mathbb{A})$ construction without being merely artificially so?

Structure
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The ‘homotopy’ category
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If $\mathcal{C}$ is an $\mathcal{S}$-category, we can form a category $\pi_0\mathcal{C}$ with the same objects and having $$(\pi_0\mathcal{C})(X,Y) =
\pi_0({\mathcal{C}}(X,Y)).$$ For instance, if $\mathcal{C} =
\mathcal{CW}$, the category of CW-complexes, then $\pi_0\mathcal{CW} = {Ho\mathcal(CW)}$, the corresponding homotopy category. Similarly we could obtain a groupoid enriched category using the fundamental groupoid (cf. Gabriel and Zisman, [@gabrielzisman]).

One can ‘do’ some elementary homotopy theory in any $\mathcal{S}$-category, $\mathcal{C}$, by saying that two maps $f_0, f_1 : X \to Y$ in $\mathcal{C}$ are homotopic if there is an $H \in {\mathcal{C}}(X,Y)_1$ with $d_0H =
f_1$, $d_1H = f_0$.

This theory will not be very rich unless at least some low dimensional Kan conditions are satisfied. The $\mathcal{S}$-category, $\mathcal{C}$, is called *locally Kan* if each $\mathcal{C}(X,Y)$ is a Kan complex, *locally weakly Kan* if …, etc. (If you have 