Results 11  20
of
70
Elementary remarks on units in monoidal categories
, 2006
"... We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1categorical sense). This notion is more economical than the usual notion in terms of leftright constraints, and is motivated by higher category theory ..."
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Cited by 5 (3 self)
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We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1categorical sense). This notion is more economical than the usual notion in terms of leftright constraints, and is motivated by higher category theory. To start, we describe the semimonoidal category of all possible unit structures on a given semimonoidal category and observe that it is contractible (if nonempty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent nonalgebraic notion of fair monoidal category, 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. The paper is selfcontained. All arguments are elementary, some of them of a certain beauty.
Computing Critical Pairs in 2Dimensional Rewriting Systems
, 2010
"... Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for ..."
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Cited by 5 (2 self)
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Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higherdimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of ncategories. Here, we are interested in proving confluence for polygraphs presenting 2categories, which can be seen as a generalization of termrewriting systems. For this purpose, we propose an adaptation of the usual algorithm for computing critical pairs. Interestingly, this framework is much richer than term rewriting systems and requires the elaboration of a new theoretical framework for representing critical pairs, based on contexts in compact 2categories. Term rewriting systems have proven very useful to reason about terms modulo equations. In some cases, the equations can be oriented and completed in a way giving rise to a normalizing (i.e. confluent and terminating) rewriting system, thus providing a notion of canonical representative of equivalence classes of terms. Usually, terms are freely generated by a signature (Σn)n∈N, which consists of a family of sets Σn of generators of arity n, and one considers equational theories on such a signature, which are formalized by sets of pairs of terms called equations. For example, the equational theory of monoids contains two generators m and e, whose arities are respectively 2 and 0, and three equations
Model synchronization: mappings, tile algebra, and categories
 In: Postproc. GTTSE
, 2009
"... Abstract. The paper presents a novel algebraic framework for specification and design of model synchronization tools. The basic premise is that synchronization procedures, and hence algebraic operations modeling them, are diagrammatic: they take a configuration (diagram) of models and mappings as th ..."
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Cited by 4 (4 self)
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Abstract. The paper presents a novel algebraic framework for specification and design of model synchronization tools. The basic premise is that synchronization procedures, and hence algebraic operations modeling them, are diagrammatic: they take a configuration (diagram) of models and mappings as their input and produce a diagram as the output. Many important synchronization scenarios are based on diagram operations of square shape. Composition of such operations amounts to their tiling, and complex synchronizers can thus be assembled by tiling together simple synchronization blocks. This gives rise to a visually suggestive yet precise notation for specifying synchronization procedures and reasoning about them. 1
Operads, clones, and distributive laws
, 2008
"... Abstract We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory ..."
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Abstract We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory of generalized species of structures,but, for the multicategory case, the general idea goes back to Burroni's Tcategories (1971). We show how other previous categorical analysesof operad (via Day's tensor products, or via analytical functor) fit with the profunctor approach.
Axiomatic Rewriting Theory I  A Diagrammatic Standardization Theorem
, 2001
"... Machine translation ## calculus interpretation ## calculus Formally, the calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation. ..."
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Machine translation ## calculus interpretation ## calculus Formally, the calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation.
Associative algebras related to conformal algebras
 Appl. Categ. Structures
"... Abstract. In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TCalgebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we generally mean what is known as Hpseudoalgebra over t ..."
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Abstract. In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TCalgebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we generally mean what is known as Hpseudoalgebra over the polynomial Hopf algebra H = k[T1,..., Tn]. Some recent results in structure theory of conformal algebras are applied to get a description of TCalgebras. 1.
Twisted differential nonabelian cohomology Twisted (n−1)brane nbundles and their ChernSimons (n+1)bundles with characteristic (n + 2)classes
, 2008
"... We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shif ..."
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We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian ngroup B n−1 U(1). Notable examples are String 2bundles [9] and Fivebrane 6bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spinstructures to Stringstructures [13] and further to Fivebranestructures [133, 52], are abelian ChernSimons 3 and 7bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞Lieintegrating the L∞algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2 and twisted Fivebrane 6bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted Ktheory. We explain the GreenSchwarz mechanism in heterotic string theory in terms of twisted String 2bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6bundles. We close by transgressing differential cocycles to mapping
Coherence for categorified operadic theories
"... It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak Pcategory for any strongly regular (operadic) ..."
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It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak Pcategory for any strongly regular (operadic) theory P, and show that every weak Pcategory is equivalent via Pfunctors and Ptransformations to a strict Pcategory. This strictification functor is then shown to have an interesting universal property. 1
The categorification of a symmetric operad is independent of signature
"... Given a symmetric operad P, and a signature (or generating sequence) Φ for P, we define a notion of the categorification (or weakening) of P with respect to Φ. When P is the symmetric operad whose algebras are commutative monoids, with the standard signature, we recover the notion of symmetric monoi ..."
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Given a symmetric operad P, and a signature (or generating sequence) Φ for P, we define a notion of the categorification (or weakening) of P with respect to Φ. When P is the symmetric operad whose algebras are commutative monoids, with the standard signature, we recover the notion of symmetric monoidal categories. We then show that this categorification is independent (up to equivalence) of the choice of signature. 1
Towards Unifying Structures in Higher Spin Gauge Symmetry
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2008
"... This article is expository in nature, outlining some of the many still incompletely understood features of higher spin field theory. We are mainly considering higher spin gauge fields in their own right as freestanding theoretical constructs and not circumstances where they occur as part of another ..."
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This article is expository in nature, outlining some of the many still incompletely understood features of higher spin field theory. We are mainly considering higher spin gauge fields in their own right as freestanding theoretical constructs and not circumstances where they occur as part of another system. Considering the problem of introducing interactions among higher spin gauge fields, there has historically been two broad avenues of approach. One approach entails gauging a nonAbelian global symmetry algebra, in the process making it local. The other approach entails deforming an already local but Abelian gauge algebra, in the process making it nonAbelian. In cases where both avenues have been explored, such as for spin 1 and 2 gauge fields, the results agree (barring conceptual and technical issues) with Yang–Mills theory and Einstein gravity. In the case of an infinite tower of higher spin gauge fields, the first approach has been thoroughly developed and explored by M. Vasiliev, whereas the second approach, after having lain dormant for a long time, has received new attention by several authors lately. In the present paper we briefly review some aspects of the history of higher spin gauge fields as a backdrop to an attempt at comparing the gauging vs. deforming approaches. A common unifying structure of strongly homotopy Lie algebras