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How much do you want to stick to Chen's original definition of a "differentiable space"? I recommend to look for the very similar notion of a "diffeological space", for which more literature is availa …

For example, this is proved as: Lemma 3.34 in Lerman, E.: Orbifolds as stacks? Proposition 60 in Metzler, D.S.: Topological and Smooth Stacks

Thomas Nikolaus and Christoph Schweigert discuss the +-construction for $n=2$ in their paper Equivariance in Higher Geometry. They split it up into two steps (I think): first producing a pre-2-stack o …

The paper Extensions of symmetric cat-groups by D. Bourn and E.M. Vitale defines and studies a bicategory of extensions of 2-groups (called cat-groups in the paper). In section 13, it introduces a mon …

I know this description from following paper of Segal. He doesn't mention Milgram there, but relates it to Milnor's join construction. Also, $G$ is allowed to be any topological group, no need for dis …

Section 4 of Gawȩdzki, Krzysztof; Reis, Nuno, Basic gerbe over non-simply connected compact groups, J. Geom. Phys. 50, No. 1-4, 28-55 (2004). ZBL1067.22009. lists, in an absolutely concrete way, the s …

I think that the book David Bleecker: Gauge Theory and Variational Principles, Addison-Wesley, 1981 contains exactly what you are looking for.

Sorry, I just saw this question. One can consider Deligne cohomology for simplicial manifolds in the rather obvious way, namely by adding an additional "simplicial manifold" direction to the ususal do …

I am asking for references and discussions of statements of the form Every bounded hypercover can be refined by an ordinary cover By "bounded" I mean "finite height". E.g., are there conditions for …

Gawedzki and I have investigated this question for all compact simple Lie groups using the descent of multiplicative bundle gerbes from simply connected covers to quotients by subgroups of the center: …