2010/11/30 by Ginot, Gregory, Thomas Tradler, Tradler, Thomas +2 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1011.6483
openalex publication_date 2010/11/30 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
In this paper, we study the higher Hochschild functor and its relationship\nwith factorization algebras and topological chiral homology. To this end, we\nemphasize that the higher Hochschild complex is a (\∞,1)-functor from the\ncategory hsset \× hcdga to the category hcdga (where hsset and\n hcdga are the (\∞,1)-categories of simplicial sets and commutative\ndifferential graded algebras) and give an axiomatic characterization of this\nfunctor. From the axioms we deduce several properties and computational tools\nfor this functor. We study the relationship between the higher Hochschild\nfunctor and factorization algebras by showing that, in good cases, the\nHochschild functor determines a constant commutative factorization algebra.\nConversely, every constant commutative factorization algebra is naturally\nequivalent to a Hochschild chain factorization algebra. Similarly, we study the\nrelationship between the above concepts and topological chiral homology. In\nparticular, we show that on their common domains of definition, the higher\nHochschild functor is naturally equivalent to topological chiral homology.\nFinally, we prove that topological chiral homology determines a locally\nconstant factorization algebra and, further, that this functor induces an\nequivalence between locally constant factorization algebras on a manifold and\n(local system of) En-algebras. We also deduce that Hochschild chains and\ntopological chiral homology satisfies an exponential law, i.e., a Fubini type\nTheorem to compute them on products of manifolds.\n