The n-Category Café

At the moment I am trying to rework my various notes on action $L_\infty$-algebroids and their relation to action $n$-groupoids, with the prime example of interest the BRST complex (the action Lie $n$-algebroid of the gauge $n$-group acting on the space of fields):

On action Lie $\infty$-groupoids and action Lie $\infty$-algebroids (pdf in preparation)

In an attempt to further conceptualize the point of $n$-Lie theory (integration of $L_\infty$-algebras to smooth Lie $n$-groupoids and differentiation of the former to the latter) I drew figure 1 on p. 5.

On the right in this figure we see the adjunction between smooth spaces and differential graded commutative algebras induced by homming into the ambimorphic object $$\Omega^\bullet$$ (a DGCA-valued smooth space) that, thankfully, Todd Trimble has helped discussing here a lot, a while ago.

On the left of this figure I have inserted a similar pair of morphisms, now relating smooth spaces with smooth $n$-groupoids, induced by homming out of $$\Pi_n$$ the $n$-groupoid valued co-presheaf which sends each smooth space $X$ to its fundamental path $n$-groupoid $\Pi_n(x)$.

With my fingers on autopilot I kept typing about how this is an adjunction due to ambimorphicity of $\Pi_n$. But then realized that it’s at least a bit more subtle, with $\Pi_n$ not being an ambimorphic space, but an ambimorphic quantity (as in space and quantity).

I bet it is still right that homming out of $\Pi_n$ induces an adjunction between smooth spaces and smooth $n$-groupoids .But I got to call it quits for today. I thought maybe sombody could help me with a hint while I catch my train back home.