Derived Higher Hochschild Homology, Topological Chiral Homology and Factorization algebras

Details Derived Higher Hochschild Homology, Topological Chiral Homology and Factorization algebras Derived Higher Hochschild Homology, Topological Chiral Homology and Factorization algebras

2010-11-30

arxiv.org/abs/1011.6483v2

Mathematics

Quantum Algebra

59 pages

Scientific paper

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 then study the relationship of the higher Hochschild functor with factorization algebras by showing that in reasonable cases, the Hochschild functor determines a constant commutative factorization algebra. Conversely, we show that every constant commutative factorization algebra is naturally equivalent to the Hochschild chain factorization algebra. Similarly, we also study the relationship with topological chiral homology. In particular, we show that the higher Hochschild functor is naturally equivalent to topological chiral homology, whenever both are defined. As a corollary, we also get a similar statement of the relationship with blob homology. Finally, we prove that topological chiral homology determines a locally constant factorization algebra and that any locally constant factorization algebra on a manifold essentially arises in this way.

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