ABSTRACT

This chapter introduces factorization homology—or factorization algebras—in the topological setting. Factorization homology has three essential features making it technically advantageous: Local-to-global principle-excision, generalizing the Eilenberg—Steenrod axioms, Filtration-a generalization of the Goodwillie—Weiss embedding calculus and Duality-Poincare/Koszul duality. The chapter discusses a classification of sheaves on an ∞-category Mfldn of n-manifolds and embeddings among them: sheaves on Mfldn are n-dimensional tangential structures. It identifies values of factorization homology of Diskn-algebras in spaces, with its Cartesian monoidal structure, as twisted compactly supported mapping spaces. The chapter describes filtrations and cofiltrations of factorization homology, whose layers are explicit in terms of configuration spaces. These (co)filtrations offer access to identifying and controlling factorization homology.