Decomposition spaces, right fibrations, and edgewise subdivision

Decomposition spaces are simplicial infinity-groupoids subject to an
exactness condition weaker than the Segal condition. Where the Segal
condition expresses composition, the weak condition expresses
decomposition. The motivation for studying decomposition spaces is that
they have incidence coalgebras and Möbius inversion. The most important
class of simplicial maps for decomposition spaces are the CULF maps
(standing for ‘conservative’ and ‘unique-lifting-of-factorisation’), first
studied by Lawvere; they induce coalgebra homomorphisms. The theorem I want
to arrive at in the talk says that the infinity-category of (Rezk-complete)
decomposition spaces and CULF maps is locally an infinity-topos. More
precisely for each (Rezk-complete) decomposition space D, the slice
infinity-category Decomp/D is equivalent to PrSh(Sd(D)), the infinity-topos
of presheaves on the edgewise subdivision of D. Most of the talk will be
spent on explaining preliminaries, though.

This is joint work with Philip Hackney.