The microlocal homology is a family of chain theories that interpolates between the Borel–Moore homology and singular cohomology of a complex variety in the case when that variety is singular and Poincaré duality fails. Such a device allows one to speak of the singular support of classes in Borel–Moore homology, which we show decategorifies the Arinkin–Gaitsgory singular support of coherent sheaves in a precise sense. 

 

The connection between these two singular support theories leverages a description of the microlocal homology in terms of the canonical perverse sheaf of vanishing cycles (the DT sheaf) on shifted cotangent bundles, as well as the known relation between vanishing cycles and categories of matrix factorizations.