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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.
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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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
PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES
 JOURNAL OF HOMOTOPY AND RELATED STRUCTURES, VOL. 2(2), 2007, PP.119–170
, 2007
"... 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 ..."
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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.
A THOMASON MODEL STRUCTURE ON THE CATEGORY
, 808
"... Abstract. We construct a cofibrantly generated Thomason model structure on the category of small nfold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An nfold functor is a weak equivalence if and only if the diagonal of its n ..."
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Abstract. We construct a cofibrantly generated Thomason model structure on the category of small nfold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An nfold functor is a weak equivalence if and only if the diagonal of its nfold nerve is a weak equivalence of simplicial sets. We introduce an nfold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the nfold nerve. As a consequence, the unit and counit of the adjunction between simplicial sets and nfold categories are natural weak equivalences.
Abstract. KAN EXTENSIONS IN DOUBLE CATEGORIES (ON WEAK DOUBLE CATEGORIES, PART III)
"... are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category. ..."
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are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category.
MAPPING SPACES OF GrayCATEGORIES
"... Abstract. We define a mapping space for Grayenriched categories adapted to higher gauge theory. Our construction differs significantly from the canonical mapping space of enriched categories in that it is much less rigid. The two essential ingredients are a path space construction for Graycategori ..."
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Abstract. We define a mapping space for Grayenriched categories adapted to higher gauge theory. Our construction differs significantly from the canonical mapping space of enriched categories in that it is much less rigid. The two essential ingredients are a path space construction for Graycategories and a kind of comonadic resolution of the 1dimensional structure of a given Graycategory obtained by lifting the resolution of ordinary categories along the canonical fibration of GrayCat over Cat.
LAX KAN EXTENSIONS FOR DOUBLE CATEGORIES (On weak double categories, Part IV)
"... Résumé. Les extensions de Kan à droite pour les catégories doubles (faibles) généralisent les limites doubles et d'autres constructions, appelées 'vertical companion ' et 'vertical adjoint', que nous avons étudiées dans des articles précédents. Nous prouvons ici que ces cas ..."
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Résumé. Les extensions de Kan à droite pour les catégories doubles (faibles) généralisent les limites doubles et d'autres constructions, appelées 'vertical companion ' et 'vertical adjoint', que nous avons étudiées dans des articles précédents. Nous prouvons ici que ces cas particuliers sont suffisants pour construire toutes les extensions de Kan à droite ponctuelles, le long de foncteurs lax doubles satisfaisant une condition 'de Conduché'. Les catégories doubles 'basées sur les profoncteurs ' sont complètes, dans le sens qu'elles admettent toutes ces constructions, tandis que la catégorie double des carrés commutatifs d'une catégorie complète ne l'est pas, en général.
UNIVERSITÉ DE NICESOPHIA ANTIPOLIS — UFR Sciences École Doctorale de Sciences Fondamentales et Appliquées THÈSE pour obtenir le titre de Docteur en Sciences
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Acknowledgments 6
"... Abstract. We develop a theory of enriched categories over a (higher) category M equipped with a class W of morphisms called homotopy equivalences. We call them Segal MWcategories. Our motivation was to generalize the notion of “uptohomotopy monoids ” in a monoidal category M, introduced by Leinst ..."
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Abstract. We develop a theory of enriched categories over a (higher) category M equipped with a class W of morphisms called homotopy equivalences. We call them Segal MWcategories. Our motivation was to generalize the notion of “uptohomotopy monoids ” in a monoidal category M, introduced by Leinster. The formalism adopted generalizes the classical Segal categories and extends the theory of enriched category