Higher Hochschild Homology, Topological Chiral Homology and Factorization Algebras

Abstract

Abstract. We study the higher Hochschild chain functor, factorization algebras and their relationship with topological chiral homology. To this end, we emphasize that the higher Hochschild complex is a (∞, 1)-functor sSet∞ × CDGA∞ where sSet∞ and CDGA∞ are the (∞, 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. A byproduct of our proof yields a similar statement about the relationship between the blob complex and topological chiral homology of unoriented manifolds. 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) En-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.

Homological stability for topological chiral homology of completions

“…Topological chiral homology has applications to higher dimensional string topology [GTZ12], quantum field theory [CG12] and the geometric Langlands program [Gai13]. References for topological chiral homology include [And10,Fra11,AF15,GTZ14,Lur09,Sal01].…”

Factorization homology of topological manifolds

“…Factorization homology and related ideas have recently become the subject of closer study; in addition to Lurie's originating work, see [30,31], and Costello and Gwiliam [12]; see also [1,20,22,36]. We expect the theory of factorization homology to be a source of interesting future mathematics and to carry many important manifold invariants.…”

Higher Structures in Algebraic Quantum Field Theory

“…Then the derived extension may be computed object‐wise for each by where is the simplicial set of singular simplices in M and is the derived ‐tensoring for ‐algebras, cf. [].…”

“…Remark In [] the ‐algebra is also referred to as the derived higher Hochschild chains on Sing( M ) with coefficients in A .…”

“…where Sing(M) ∈ sSet is the simplicial set of singular simplices in M and L ⊗ is the derived sSet-tensoring for E ∞ -algebras, cf. [63,64].…”