In more detail, what I am after is described around def. 4.7 of this note on parameterized WZW terms (pdf). My real goal is to get a useful condition that makes the necessary obstruction of theorem 4.5 also a sufficient obstruction, as in corollary 4.11. Possibly there is an altogether better way to do this than I have there right now.
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]]>So concretely, I am looking at the following situation: a -manifold (as here), its infinitesimal disk bundle (as here), a formally étale 1-epi, and the pullback in question being the one that exhibits a local trivialization of the formal disk bundle
So as well as are -equivalences ( the reduction modality). This means that applying to this diagram it does become a homotopy pushout. Since preserves homotopy pushouts, this means that the homotopy pushout differs from at most in some infinitesimal extension.
So for achieving the desired I am reduced to arguing (or else to try to arrange by further assumptions) that also this infinitesimal difference vanishes. This should follow since is something like a fiberwise equivalence. Hm…
]]>That’s true of course, stupid me. Thanks.
I’ll need to consider some further conditions.
]]>If I’m not mistaken, the -analogue is false. For instance,
is a (homotopy) pullback square of effective epimorphisms (of sets) in but not a homotopy pushout square. (The homotopy pushout is a circle.)
]]>have added to regular epimorphism the statement (here) that a pullback square of regular epis is also a pushout.
This must be true for effective epimorphisms in an -topos, too. What’s the proof?
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