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OPERADS AND PROPS
, 2006
"... We review definitions and basic properties of operads, PROPs and algebras over these structures. ..."
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We review definitions and basic properties of operads, PROPs and algebras over these structures.
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.
On the axiomatisation of boolean categories with and without medial
 THEORY APPL. CATEG
, 2007
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Pasting Diagrams in nCategories with Applications to Coherence Theorems and Categories of Paths
, 1987
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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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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
The structure group for the associative identity
 J. Pure Appl. Algebra
, 1996
"... ABSTRACT. A group of elementary associativity operators is introduced so that the bracketing graphs which are the skeletons of Stasheff’s associahedra become orbits and can be constructed as subgraphs of the Cayley graph of this group. A very simple proof of Mac Lane’s coherence theorem is given, as ..."
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ABSTRACT. A group of elementary associativity operators is introduced so that the bracketing graphs which are the skeletons of Stasheff’s associahedra become orbits and can be constructed as subgraphs of the Cayley graph of this group. A very simple proof of Mac Lane’s coherence theorem is given, as well as an oriented version of this result. We also sketch a more general theory and compare the cases of associativity and left selfdistributivity. AMS Classification: 08A05, 20L10, 20M50. The general purpose of this paper can be summarized as the introduction of some algebraic structure on the faces of Stasheff’s associahedra which are CWcomplexes whose faces correspond to the complete bracketings of a given string (see [12]). We introduce a ‘structure group of associativity ’ G̃A so that the (skeletons of the) associahedra become orbits for some natural action of G̃A – exactly like the usual regular
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 12 (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
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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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.
SKEW MONOIDALES, SKEW WARPINGS AND QUANTUM CATEGORIES
"... Abstract. Kornel Szlachányi [28] recently used the term skewmonoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skewmonoidal structures on the category of onesided Rmodules for which the lax un ..."
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Abstract. Kornel Szlachányi [28] recently used the term skewmonoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skewmonoidal structures on the category of onesided Rmodules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudomonoids) in any monoidal bicategory M. These are skewmonoidal categories when M is Cat. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories [10] with base comonoid C in a suitably complete braided monoidal category V are precisely skew monoidales in Comod(V) with unit coming from the counit of C. Quantum groupoids (in the sense of [6] rather than [10]) are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping de ned in [3] to modify monoidal structures. 1.