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started a section Preservation of some homotopy fiber products spelling out how the shape modality does not preserve all homotopy fiber products but does preserve the two special cases of (a) looping of delooped objects and (b) of homotopy fiber products over cohesively discrete objects.
Next I’ll add a little new statement, resulting from the combination of these two and the “reverse pasting law”.
I have added (here) discussion of a key technical result from traditional equivariant principal bundle theory, recast as a condition of the shape modality commuting with looping when applied to mapping stacks between deloopings of certain topological groups.
(This discussion might also and maybe better go elsewhere, such as at nearby homomorphisms from compact Lie groups are conjugate. Maybe I’ll put the section into an !include
file in order to conveniently have it in both places. )
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