The “alpha” version of factorization homology pairs framed n-manifolds with $E_n$-algebras. This construction generalizes the classical homology of a manifold, yields novel results concerning configuration spaces of points in a manifold, and supplies a sort of state-sum model for sigma-models (i.e., mapping spaces) to (n-1)-connected targets. This “alpha” version of factorization homology novelly extends Poincaré duality, shedding light on deformation theory and dualities among field theories. Being defined using homotopical mathematical foundations, “alpha” factorization homology is manifestly functorial and continuous in all arguments, notably in moduli of manifolds and embeddings between them, and it satisfies a local-to-global expression that is inherently homotopical in nature.

Now, $E_n$-algebras can be characterized as $(∞,n)$-categories equipped with an (n-1)-connected functor from a point. The (full) “beta” version of factorization homology pairs framed n-manifolds with pointed $(∞,n)$-categories with adjoints. Applying 0th homology, or $π_0$, recovers a version of the string net construction on surfaces, as well as skein modules of 3-manifolds. In some sense, the inherently homotopical nature of (full) “beta” factorization homology affords otherwise unforeseen continuity in all arguments, and local-to-global expressions.

In this talk, I will outline a definition of “beta” factorization homology, focusing on low-dimensions and on suitably reduced $(∞,n)$-categories (specifically, braided monoidal categories). I will outline some examples, and demonstrate some features of factorization homology. Some of this material is established in the literature, some a work in progress, and some conjectural — the status of each assertion will be made clear. I will be especially interested in targeting this talk to those present, and so will welcome comments and questions.