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May 6, 2008

Ambimorphic?

Posted by Urs Schreiber

A question on the interpretation of the fundamental path nn-groupoid as an ambimorphic object: an nn-groupoid valued co-presheaf:

At the moment I am trying to rework my various notes on action L L_\infty-algebroids and their relation to action nn-groupoids, with the prime example of interest the BRST complex (the action Lie nn-algebroid of the gauge nn-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 nn-Lie theory (integration of L L_\infty-algebras to smooth Lie nn-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 Ω \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 nn-groupoids, induced by homming out of Π n \Pi_n the nn-groupoid valued co-presheaf which sends each smooth space XX to its fundamental path nn-groupoid Π n(x)\Pi_n(x).

With my fingers on autopilot I kept typing about how this is an adjunction due to ambimorphicity of Π n\Pi_n. But then realized that it’s at least a bit more subtle, with Π n\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 Π n\Pi_n induces an adjunction between smooth spaces and smooth nn-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.

Posted at May 6, 2008 6:21 PM UTC

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