Derived Higher Hochschild Homology, Topological Chiral Homology and Factorization algebras

In this paper, we study the higher Hochschild functor and its relationship with factorization algebras and topological chiral homology. To this end, we emphasize that the higher Hochschild complex is a $(infty,1)$-functor from the category $hsset times hcdga$ to the category $hcdga$ (where $hsset$ and $hcdga$ are the $(infty,1)$-categories of simplicial sets and commutative differential graded algebras) and give an axiomatic characterization of this functor. From the axioms we deduce several properties and computational tools for this functor. We study the relationship between the higher Hochschild functor and factorization algebras by showing that, in good cases, the Hochschild functor determines a constant commutative factorization algebra. Conversely, every constant commutative factorization algebra is naturally equivalent to a Hochschild chain factorization algebra. Similarly, we study the relationship between the above concepts and topological chiral homology. In particular, we show that on their common domains of definition, the higher Hochschild functor is naturally equivalent to topological chiral homology. Finally, we prove that topological chiral homology determines a locally constant factorization algebra and, further, that this functor induces an equivalence between locally constant factorization algebras on a manifold and (local system of) $E_n$-algebras. We also deduce that Hochschild chains and topological chiral homology satisfies an exponential law, i.e., a Fubini type Theorem to compute them on products of manifolds.