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19
A Darboux theorem for Hamiltonian operators in the formal calculus of variations
 Duke Math. J
"... We prove a formal Darbouxtype theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain a ..."
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Cited by 36 (1 self)
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We prove a formal Darbouxtype theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras g concentrated in degrees [−1, ∞); the formal deformations of g are parametrized by a 2groupoid that we call the Deligne 2groupoid of g, and quasiisomorphic differential graded Lie algebras have equivalent Deligne 2groupoids. The Darboux theorem states that all symplectic structures on an affine space are locally isomorphic. Hamiltonian operators are a generalization of symplectic forms introduced by I. Gelfand and I. Dorfman [3] and are important in the study of integrable hierarchies such as the Korteweg–de Vries (KdV) and KodomtsevPetviashvili (KP) equations. It is natural to ask whether an analogue of the Darboux theorem holds for Hamiltonian operators. The problem is considerably simplified by restricting attention to formal deformations of a given Hamiltonian operator H. The study of the moduli space of deformations is then controlled by a differential graded (dg) Lie algebra, the Schouten Lie algebra, with differential ad(H). The study of formal deformations is closely related to the problem of calculating the cohomology of this dg Lie algebra, which was posed by P. Olver [9].
Simplicial Matrices And The Nerves Of Weak nCategories I: Nerves Of Bicategories
, 2002
"... To a bicategory B (in the sense of Benabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2Coskeleton and itself isomorphic to its 3Coskeleton, what we call a 2dimensio ..."
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Cited by 26 (1 self)
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To a bicategory B (in the sense of Benabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2Coskeleton and itself isomorphic to its 3Coskeleton, what we call a 2dimensional Postnikov complex. We then give, somewhat more delicately, a complete characterization of those simplicial sets which are the nerves of bicategories as certain 2dimensional Postnikov complexes which satisfy certain restricted "exact hornlifting" conditions whose satisfaction is controlled by (and here defines) subsets of (abstractly) invertible 2 and 1simplices. Those complexes which have, at minimum, their degenerate 2simplices always invertible and have an invertible 2simplex # 1 2 (x 12 , x 01 ) present for each "composable pair" (x 12 , , x 01 ) # # 1 2 are exactly the nerves of bicategories. At the other extreme, where all 2 and 1simplices are invertible, are those Kan complexes in which the Kan conditions are satisfied exactly in all dimensions > 2. These are exactly the nerves of bigroupoids  all 2cells are isomorphisms and all 1cells are equivalences. Contents
Pasting Schemes for the Monoidal Biclosed Structure on
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a gen ..."
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a general and uniform definition of higher dimensional lax natural transformations, and a nice and transparent description of the corresponding internal homs. Further consequences will be in the development of a theory for weak ncategories, since both tensor products and lax structures are crucial in this. Contents 1 Introduction 3 2 Cubes and cubical sets 5 2.1 Cubes combinatorially : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 A model category for cubes : : : : : : : : : : : : : : : : : : : : : 6 2.3 Generating the model category for cubes : : : : : : : : : : : : : : 7 2.4 Cubical sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.5 Duality : : : : : : : : : : : : : ...
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using ..."
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Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.
and B.Noohi, Uniformization of DeligneMumford curves
"... Abstract. We compute the fundamental groups of nonsingular analytic DeligneMumford curves, classify the simply connected ones, and classify analytic DeligneMumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary DeligneMumford curve as a quotien ..."
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Abstract. We compute the fundamental groups of nonsingular analytic DeligneMumford curves, classify the simply connected ones, and classify analytic DeligneMumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary DeligneMumford curve as a quotient stack. Along the way, we compute the automorphism 2groups of weighted projective stacks P(n1, n2, · · · , nr). We also discuss connections with the theory of Fgroups, 2groups, and BassSerre theory of graphs of groups. 1.
Semistrict models of connected 3types and Tamsamani’s weak 3groupoids
, 2006
"... Homotopy 3types can be modelled algebraically by Tamsamani’s weak 3groupoids as well as, in the path connected case, by cat 2groups. This paper gives a comparison between the two models in the pathconnected case. This leads to two different semistrict algebraic models of connected 3types usin ..."
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Cited by 7 (3 self)
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Homotopy 3types can be modelled algebraically by Tamsamani’s weak 3groupoids as well as, in the path connected case, by cat 2groups. This paper gives a comparison between the two models in the pathconnected case. This leads to two different semistrict algebraic models of connected 3types using Tamsamani’s model. Both are then related to Gray groupoids.
On the geometry of 2categories and their classifying spaces, KTheory 29
, 2003
"... Abstract. In this paper we prove that realizations of geometric nerves are classifying spaces for 2categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen’s Theorem A. ..."
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Abstract. In this paper we prove that realizations of geometric nerves are classifying spaces for 2categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen’s Theorem A.
NOTES ON 2GROUPOIDS, 2GROUPS AND CrossedModules
, 2005
"... This paper contains some basic results on 2groupoids, with special emphasis on computing derived mapping 2groupoids between 2groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author’s k ..."
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Cited by 5 (1 self)
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This paper contains some basic results on 2groupoids, with special emphasis on computing derived mapping 2groupoids between 2groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author’s knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2groupoids introduced by Moerdijk and Svensson.
ON WEAK MAPS BETWEEN 2GROUPS
, 2008
"... We give an explicit handy cocyclefree description of the groupoid of weak maps between two crossedmodules using what we call a butterfly (Theorem 8.4). We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2category of pointed homotopy 2types. ..."
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Cited by 5 (2 self)
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We give an explicit handy cocyclefree description of the groupoid of weak maps between two crossedmodules using what we call a butterfly (Theorem 8.4). We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2category of pointed homotopy 2types. This has applications in the study of 2group actions (say, on stacks), and in the theory of gerbes bound by crossedmodules and principal2bundles).