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Pasting Diagrams in nCategories with Applications to Coherence Theorems and Categories of Paths
, 1987
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WHEELED PROPS, GRAPH COMPLEXES AND THE MASTER EQUATION
, 2007
"... We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in onetoone correspondence with formal germs of ..."
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We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in onetoone correspondence with formal germs of SPmanifolds, key geometric objects in the theory of BatalinVilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and As s as rather nonobvious extensions of Com ∞ and As s∞, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich’s complex of ribbon graphs.
Inferring Type Isomorphisms Generically
 Proceedings of the 7th International Conference on Mathematics of Program Construction, MPC 2004, volume 3125 of LNCS
"... Datatypes which di#er inessentially in their names and structure are said to be isomorphic; for example, a ternary product is isomorphic to a nested pair of binary products. In some canonical cases, the conversion function is uniquely determined solely by the two types involved. ..."
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Cited by 11 (7 self)
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Datatypes which di#er inessentially in their names and structure are said to be isomorphic; for example, a ternary product is isomorphic to a nested pair of binary products. In some canonical cases, the conversion function is uniquely determined solely by the two types involved.
From Operads to `Physically' Inspired Theories
"... Introduction As evidenced by these conferences (Hartford and Luminy), operads have had a renaissance in recent years for a variety of reasons. Originally studied entirely as a tool in homotopy theory, operads have recently received new inspirations from homological algebra, category theory, algebra ..."
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Cited by 11 (1 self)
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Introduction As evidenced by these conferences (Hartford and Luminy), operads have had a renaissance in recent years for a variety of reasons. Originally studied entirely as a tool in homotopy theory, operads have recently received new inspirations from homological algebra, category theory, algebraic geometry and mathematical physics. I'll try to provide a transition from the foundations to the frontier with mathematical physics. For me, the transition occurred in two stages. First, there is the generalization of Lie algebra cohomology known as BRST (BecchiRouetStora Tyutin) cohomology, which turned out to be very closely related to strong homotopy Lie (L1 ) algebras, which I will describe later in homological algebraic terms  along the lines of Balavoine's talk at this conference [5]. That description makes no use of operads, but the relevance of operads appeared later in the work of Hinich and Schechtman [25]. Operads rev
OPERADS AND PROPS
, 2006
"... We review definitions and basic properties of operads, PROPs and algebras over these structures. ..."
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Cited by 8 (0 self)
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We review definitions and basic properties of operads, PROPs and algebras over these structures.
Two Signed Associahedra
 New York J. Math. (electronic
, 1998
"... The associahedron is a convex polytope whose vertices correspond to triangulations of a convex polygon. We define two signed or hyperoctahedral analogues of the associahedron, one of which is shown to be a simple convex polytope, and the other a regular CWsphere. ..."
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Cited by 7 (1 self)
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The associahedron is a convex polytope whose vertices correspond to triangulations of a convex polygon. We define two signed or hyperoctahedral analogues of the associahedron, one of which is shown to be a simple convex polytope, and the other a regular CWsphere.
Perverse sheaves on real loop Grassmannians
 Invent. Math
"... Abstract. The aim of this paper is to identify a certain tensor category of perverse sheaves on the real loop Grassmannian GrR of a real form GR of a connected reductive complex algebraic group G with the category of finitedimensional representations of a reductive complex algebraic subgroup H of t ..."
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Cited by 6 (3 self)
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Abstract. The aim of this paper is to identify a certain tensor category of perverse sheaves on the real loop Grassmannian GrR of a real form GR of a connected reductive complex algebraic group G with the category of finitedimensional representations of a reductive complex algebraic subgroup H of the dual group ˇ G. The root system of H is closely related to the restricted root system of GR. The fact that H is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition Theorem of BeilinsonBernsteinDeligne. 1.
Elementary remarks on units in monoidal categories
"... We gather some notverywellknown remarks on units in monoidal categories, motivated by (but independent of) higherdimensional viewpoints. All arguments are elementary, some of them of a certain beauty. The first theme is uniqueness of units: we describe the semimonoidal category of all possible ..."
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Cited by 6 (3 self)
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We gather some notverywellknown remarks on units in monoidal categories, motivated by (but independent of) higherdimensional viewpoints. All arguments are elementary, some of them of a certain beauty. The first theme is uniqueness of units: we describe the semimonoidal category of all possible unit structures on a given semimonoidal category and show that it is contractible (if nonempty). The second theme is a redundancy in the classical definition of units, which is exhibited with clarity by comparison with an alternative definition of unit originally due to Saavedra: a Saavedra unit is a cancellable idempotent, in a certain sense. It is shown that the two notions are isomorphic in a strong functorial sense. One corollary of this comparison is that a (strong) semimonoidal functor is compatible with the left constraint if and only if it is compatible with the right constraint, and in fact this compatibility can be measured on I alone. The unit compatibility condition for a (strong) monoidal functor is shown to be precisely the condition for the functor to lift to the categories of units. The notion of Saavedra unit leads naturally to the equivalent nonalgebraic notion of fair monoidal category (treated elsewhere), where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered.
Physics, Topology, Logic and Computation: A Rosetta Stone
, 2009
"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."
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Cited by 5 (1 self)
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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps
R.: Obstruction theory for extensions of categorical groups
 Appl. Categ. Structures
"... Abstract. For any categorical group H, we introduce the categorical group Out(H) andthenthe wellknown group exact sequence 1 → Z(H) → H → Aut(H) → Out(H) → 1israisedtoa categorical group level by using a suitable notion of exactness. Breen’s Schreier theory for extensions of categorical groups i ..."
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Cited by 5 (4 self)
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Abstract. For any categorical group H, we introduce the categorical group Out(H) andthenthe wellknown group exact sequence 1 → Z(H) → H → Aut(H) → Out(H) → 1israisedtoa categorical group level by using a suitable notion of exactness. Breen’s Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg–Mac Lane obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology.