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24
Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
Coalgebraic Components in a ManySorted Microcosm
"... Abstract. The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a ..."
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Abstract. The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a manysorted setting. Then we can show that the coalgebraic component calculi of Barbosa are examples, with compositionality of behavior following from microcosm structure. The algebraic structure on these coalgebraic components corresponds to variants of Hughes’ notion of arrow, introduced to organize computations in functional programming. 1
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Cited by 5 (3 self)
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
I.Kriz: Laplaza sets, or how to select coherence diagrams for pseudo algebras
"... Abstract. We define a general concept of pseudo algebras over theories and 2theories. A more restrictive such notion was introduced in [5], but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, ..."
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Cited by 5 (2 self)
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Abstract. We define a general concept of pseudo algebras over theories and 2theories. A more restrictive such notion was introduced in [5], but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, allowing more flexibility in selecting coherence diagrams for pseudo algebras. 1.
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
Compactly generated stacks: a cartesianclosed theory of topological stacks, Adv
 Math
"... Abstract. A convenient 2category of topological stacks is constructed which is both complete and Cartesian closed. This 2category, called the 2category of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an e ..."
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Abstract. A convenient 2category of topological stacks is constructed which is both complete and Cartesian closed. This 2category, called the 2category of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. In fact, there is an equivalence of 2categories between compactly generated stacks and those classical topological stacks which admit locally compact atlases. Compactly generated stacks are also equivalent to a bicategory of topological groupoids and principal bundles, just as in the classical case. If a classical topological stack and a compactly generated stack have a presentation by the same topological groupoid, then they restrict to the same stack over locally compact Hausdorff spaces and are
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.
QUASISYMMETRIC SEWING IN RIGGED TEICHMÜLLER SPACE
, 2005
"... Abstract. One of the basic geometric objects in conformal field theory (CFT) is the the moduli space of Riemann surfaces whose n boundaries are “rigged ” with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alterna ..."
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Abstract. One of the basic geometric objects in conformal field theory (CFT) is the the moduli space of Riemann surfaces whose n boundaries are “rigged ” with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of npunctured Riemann surfaces together with local biholomorphic coordinates at the punctures. We refer to both of these moduli spaces as the “rigged Riemann moduli space”. By generalizing to quasisymmetric boundary parametrizations, and defining rigged Teichmüller spaces in both the border and puncture pictures, we prove the following results: (1) The Teichmüller space of a genusg surface bordered by n closed curves covers the rigged Riemann and rigged Teichmüller moduli spaces of surfaces of the same type, and induces complex manifold structures on them. (2) With this complex structure the sewing operation is holomorphic. (3) The border and puncture pictures of the rigged moduli and rigged Teichmüller spaces are biholomorphically equivalent. These results are necessary in rigorously defining CFT (in the sense of G. Segal), as well
WHAT IS THE JACOBIAN OF A RIEMANN SURFACE WITH BOUNDARY?
, 2008
"... Abstract. We define the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of “open abelian varieties ” which satisfies gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of “conformal field theory ..."
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Abstract. We define the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of “open abelian varieties ” which satisfies gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of “conformal field theory ” to be defined on this space. We further prove that chiral conformal field theories corresponding to even lattices factor through this moduli space of open abelian varieties. 1.