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Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
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We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.
Introduction to Ainfinity algebras and modules
 Homology, Homotopy and Applications
"... Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞algebras, ..."
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Cited by 68 (6 self)
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Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞algebras,
On Operad Structures of Moduli Spaces and String Theory
, 1994
"... We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and BatalinVilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a ..."
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Cited by 49 (13 self)
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We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and BatalinVilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory.
RozanskyWitten invariants via Atiyah classes
 Compositio Math
, 1999
"... Recently, L.Rozansky and E.Witten [RW] associated to any hyperKähler manifold X an invariant of topological 3manifolds. In fact, their construction gives a system of weights cΓ(X) associated to 3valent graphs Γ and the corresponding invariant of a 3manifold Y is obtained as the sum ∑ cΓ(X)IΓ(Y) ..."
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Cited by 46 (1 self)
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Recently, L.Rozansky and E.Witten [RW] associated to any hyperKähler manifold X an invariant of topological 3manifolds. In fact, their construction gives a system of weights cΓ(X) associated to 3valent graphs Γ and the corresponding invariant of a 3manifold Y is obtained as the sum ∑ cΓ(X)IΓ(Y) where IΓ(Y) is the standard integral of the product of linking forms. So the new ingredient is the system of invariants cΓ(X) of hyperKähler manifolds X, one for each trivalent graph Γ. They are obtained from the Riemannian curvature of the hyperKähler metric. In this paper we give a reformulation of the cΓ(X) in simple cohomological terms which involve only the underlying holomorphic symplectic manifold. The idea is that we can replace the curvature by the Atiyah class [At] which is the cohomological obstruction to the existence of a global holomorphic connection. The role of what in [RW] is called “Bianchi identities in hyperKähler geometry ” is played here by an identity for the square of the Atyiah class expressing the existence of the fiber bundle of second order jets. The analogy between the curvature and the structure constants of a Lie algebra observed in [RW] in fact holds even without any symplectic structure, and we study the
Vishik: Determinants of elliptic pseudo–differential operators
, 1994
"... Abstract. Determinants of invertible pseudodifferential operators (PDOs) close to positive selfadjoint ones are defined through the zetafunction regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained a ..."
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Cited by 34 (1 self)
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Abstract. Determinants of invertible pseudodifferential operators (PDOs) close to positive selfadjoint ones are defined through the zetafunction regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural class of PDOs on odddimensional manifolds generalizing the class of elliptic differential operators, the multiplicative anomaly is identically 1. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a quadratic nonlinearity hidden in the definition of determinants of PDOs through zetafunctions. The natural explanation of this nonlinearity follows from complexanalytic properties of a new trace functional TR on PDOs of noninteger orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain several new results. In particular, we describe a structure of derivatives of zetafunctions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zetaregularized determinants to general elliptic PDOs. For elliptic PDOs of nonzero complex orders we introduce a canonical determinant in its natural domain of definition. Contents
A Nonconnecting Delooping of Algebraic KTheory
"... Given a ring R, it is known that the topological space BGl(R) + is an infinite loop space. One way to construct an infinite loop structure is to consider the category F of free Rmodules, or rather its classifying space BF, as food for suitable infinite loop space machines. These machines produce co ..."
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Cited by 30 (3 self)
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Given a ring R, it is known that the topological space BGl(R) + is an infinite loop space. One way to construct an infinite loop structure is to consider the category F of free Rmodules, or rather its classifying space BF, as food for suitable infinite loop space machines. These machines produce connective spectra whose zeroth space is (BF Z × BGl(R) +. In this paper we consider categories C (F) = F, C (F),... of parameterized =1 = = =1 = free modules and bounded homomorphisms and show that the spaces (BC) =0 + = (BF)
Koszul duality for dioperads
"... Abstract. We introduce the notion of a dioperad to describe certain operations with multiple inputs and multiple outputs. The framework of Koszul duality for operads is generalized to dioperads. We show that the Lie bialgebra dioperad is Koszul. The current interests in the understanding of various ..."
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Cited by 29 (1 self)
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Abstract. We introduce the notion of a dioperad to describe certain operations with multiple inputs and multiple outputs. The framework of Koszul duality for operads is generalized to dioperads. We show that the Lie bialgebra dioperad is Koszul. The current interests in the understanding of various algebraic structures using operads is partly due to the theory of Koszul duality for operads; see eg. [K] or [L] for surveys. However, algebraic structures such as bialgebras and Lie bialgebras, which involve both multiplication and comultiplication, or bracket and cobracket, are defined using PROP’s (cf. [Ad]) rather than operads. Inspired by the theory of string topology of ChasSullivan ([ChS], [Ch], [Tr]), Victor Ginzburg suggested to the author that there should be a theory of Koszul duality for PROP’s. The present paper results from the observation that when the defining relations between the generators of a PROP are spanned over trees, then the ”treepart ” of the PROP has the structure of a dioperad. We show that one can set up a theory of Koszul duality for dioperads. In §1, we give the definition of a dioperad and other generalities. In §2, we define the notion of a quadratic dioperad, its quadratic dual, and introduce our main example of Lie bialgebra dioperad. In §3, we define the cobar dual of a dioperad. A quadratic dioperad is Koszul if its cobar dual is quasiisomorphic to its quadratic dual. The formalism in §2 and §3, in the case of operads, is due to GinzburgKapranov [GiK]. In §4, we prove a proposition to be used later in §5. This proposition is a generalization of a result of ShniderVan Osdol [SVO]. In §5, we prove that Koszulity of a quadratic dioperad is equivalent to exactness of certain Koszul complexes. In the case of operads, this is again due to GinzburgKapranov, with a different proof by ShniderVan Osdol. The Koszulity of the Lie bialgebra dioperad follows from this and an adaptation of results of Markl [M2].
Operads, homotopy algebra, and iterated integrals for double loop spaces
 15 T. KASHIWABARA – ON THE HOMOTOPY TYPE OF CONFIGURATION COMPLEXES, CONTEMP. MATH. 146
, 1995
"... Chen's theory of iterated integrals provides a remarkable model for the di erential forms on the based loop space M of a di erentiable manifold M (Chen [10]; see also HainTondeur [23] and GetzlerJonesPetrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of di ..."
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Cited by 26 (0 self)
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Chen's theory of iterated integrals provides a remarkable model for the di erential forms on the based loop space M of a di erentiable manifold M (Chen [10]; see also HainTondeur [23] and GetzlerJonesPetrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of di erentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic framework for understanding twodimensional topological eld theories. Our approach is to use the formalism of operads. Operads can be de ned in any symmetric monoidal category, although we will mainly be concerned with dgoperads (di erential graded operads), that is, operads in the category of chain complexes with monoidal structure de ned by the graded tensor product. An operad is a sequence of objects a(k), k 0, carrying an action of the symmetric group Sk, with products a(k) a(j1) : : : a(jk) �! a(j1 + + jk) which are equivariant and associative  we give a precise de nition in Section 1.2. An operad such that a(k) = 0 for k 6 = 1 is a monoid: in this sense, operads are a nonlinear generalization of monoids. If V is a chain complex, we may de ne an operad with EV (k) = Hom(V (k) ; V); where V (k) is the kth tensor power of V. The symmetric group Sk acts on EV (k) through its action on V (k) , and the structure maps of EV are the obvious ones. This operad plays the same role in the theory of operads that the algebra End(V) does in the theory of associative algebras. An algebra over an operad a (or aalgebra) is a chain complex A together with a morphism of operads: a �! EA. In other words, A is equipped with structure maps k: a(k)
Modules and Morita theorem for operads
 Am. J. of Math
"... (0.1) Morita theory. Let A, B be two commutative rings. If their respective categories of modules are equivalent, then A and B are isomorphic. This is not anymore true if A and/or B are not assumed to be commutative. Morita theory ..."
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(0.1) Morita theory. Let A, B be two commutative rings. If their respective categories of modules are equivalent, then A and B are isomorphic. This is not anymore true if A and/or B are not assumed to be commutative. Morita theory
An introduction to ncategories
 In 7th Conference on Category Theory and Computer Science
, 1997
"... ..."