What topology is seen in all smooth resolutions of a singular algebraic variety? In this talk I'll propose one answer; canonical (complexes of) sheaves on singular varieties which we call geometric extensions. These sheaves are characterised by their occurrence as a summand in the cohomology of any resolution of singularities. Our proof of their existence is simple and formal, valid in any suitably finite six functor formalism with fundamental classes. In the setting of rational coefficients, they are intersection cohomology sheaves, in the modular setting they generalise parity sheaves, and using K theory coefficients they give a sheaf theoretic definition of intersection K theory. This joint work with Geordie Williamson.