(currently the file that used to be here is a subsection of the one linked to at Nonabelian cocycles and their sigma model QFTs)
We formulate nonabelian cohomology and its classification of fiber bundles in a general context of enriched homotopy theory, i.e. in homotopy coherent category theory, and discuss general issues such as lifting and extension problems.
We list examples and applications with enrichment over higher categories which describe higher principal bundles and higher vector bundles, possibly equivariant, possibly with connection.
This further develops and expands on the parts related to higher bundles appearing in Differential Nonabelian Cohomology and Nonabelian cocycles and their sigma model QFTs. Here we start changing strategy in that everything is first formulated in a general enriched homotopical category and only when looking at examples and applications do we choose concrete models and obtain -bundles proper.