Structured Geometric Sheaves in Higher Category Theory

Much like ordinary topos theory is the theory of sheaves on a category equipped with a topology, higher topos theory can be understood as the theory of homotopy-coherent sheaves on a higher category equipped with a "structured" topology. In essence, the latter notion replaces coherent families of covering sieves with coherent families of fibered covering structures.

In this talk we make use of that additional freedom in the definition of higher sites to move away from the classical sheaf condition over topological spaces -- and over geometric categories more generally -- and introduce a stronger limit-preserving property. We therefore define and study a new class of higher toposes: the structured geometric sheaf theories on suitably equipped higher categories. We will see that the two notions of structured and unstructured (i.e. ordinary) geometric sheaves differ only by a subtle cotopological fragment. Yet it turns out that this fragment is crucial in various aspects. As a case in point, we will show that every higher topos is the theory of structured geometric sheaves over itself, while the same is generally not true for the ordinary geometric sheaves over itself.