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<span style="font-family: JuliaMono">These six factored functors P⃗ , P⃡ , P : D(∞-Grpd₀), p⃗ (𝟙 C), p⃡ (𝟙 X), p are each fully faithful and produce categorical equivalences; we later construct functors B⃗, B⃡, B, b⃗, b⃡, b defined on the essential image of these six, which are inverse to them up to natural isomorphism.\\

The directed path space, the path space, and loop space form components of the functors P⃗, P⃡, and P, which are valued in internal categories, internal groupoids, and internal groups respectively.
\begin{enumerate}
\item P⃗ : D(∞-Cat) ⭢ Cat D(∞-Cat)
\item P⃡ : D(∞-Grpd) ⭢ Grpd D(∞-Grpd)
\end{enumerate}</span>

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