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My former Master student Herman Stel (see master thesis Stel (schreiber)) finally came out with a short note that has grown out of his thesis:
This demonstrates how the normalization of the cosimplicial algebra of functions on the infinitesimal singular simplicial complex is the de Rham complex.
(In section 4.5.1.3 of dcct I discuss how this can be used to embed Lie algebroids into synthetic differential higher stacks.)
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