Geometry and Topology
Speaker: Adrian Clough (NYU Abu Dhabi)
"The homotopy theory of differentiable sheaves"
Time: 15:30
Room: MC 107
Many of the topological spaces involved in geometric topology, such as spaces of embedded manifolds, have complicated topologies that are difficult to specify and to manipulate. As already observed in the work of Galatius-Madsen-Tillmann-Weiss as well as Kupers, these spaces are often much easier to construct in the topos Diff≤0 of set-valued sheaves on manifolds, and may moreover be endowed with naturally occurring smooth structures.
Viewing Diff≤0 as a subcategory of the infinity topos Diff of homotopy-type valued sheaves - the eponymous differentiable sheaves - we will give conceptual proofs of how Diff≤0 provides a model for the theory of homotopy types, and exhibit many good formal properties of Diff≤0, such as the fact that all filtered colimits are homotopy colimits in Diff≤0. By endowing Diff with certain homotopical calculi, we are moreover able to obtain a generalisation of Berwick-Evans, Boavida de Brito, and Pavlov's result that for any (paracompact Hausdorff) manifold A, and homotopy-type valued Sheaf X the mapping sheaf Diff(A,X) computes the mapping space of the underlying homotopy types of A and X.
Mathematics Calendar 
Week of September 17, 2023
