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Framed Bicategories and Monoidal Fibrations
, 2007
"... Abstract. In some bicategories, the 1cells are ‘morphisms ’ between the 0cells, such as functors between categories, but in others they are ‘objects ’ over the 0cells, such ..."
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Abstract. In some bicategories, the 1cells are ‘morphisms ’ between the 0cells, such as functors between categories, but in others they are ‘objects ’ over the 0cells, such
ISOTROPY AND CROSSED TOPOSES
"... Abstract. Motivated by constructions in the theory of inverse semigroups and étale groupoids, we define and investigate the concept of isotropy from a topostheoretic perspective. Our main conceptual tool is a monad on the category of grouped toposes. Its algebras correspond to a generalized notion ..."
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Abstract. Motivated by constructions in the theory of inverse semigroups and étale groupoids, we define and investigate the concept of isotropy from a topostheoretic perspective. Our main conceptual tool is a monad on the category of grouped toposes. Its algebras correspond to a generalized notion of crossed module, which we call a crossed topos. As an application, we present a topostheoretic characterization and generalization of the ‘Clifford, fundamental ’ sequence associated with an inverse semigroup.
COMPLETENESS RESULTS FOR QUASICATEGORIES OF ALGEBRAS, HOMOTOPY LIMITS, AND RELATED GENERAL CONSTRUCTIONS
"... Abstract. Consider a diagram of quasicategories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasicategory admitting these (co)limits and that they are preserved by th ..."
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Abstract. Consider a diagram of quasicategories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasicategory admitting these (co)limits and that they are preserved by the functors in the limit cone. In particular, the BousfieldKan homotopy limit of a diagram of quasicategories admit any limits or colimits existing in and preserved by the functors in that diagram. In previous work, we demonstrated that the quasicategory of algebras for a homotopy coherent monad could be described as a weighted limit with projective cofibrant weight, so these results immediately provide us with important (co)completeness results for quasicategories of algebras. These generalise most of the classical categorical results, except for a well known theorem which shows that limits lift to the category of algebras for any monad, regardless of whether its functor part preserves those limits. The second half of this paper establishes this more general result in the quasicategorical setting: showing that the monadic forgetful functor of the quasicategory of algebras for a homotopy coherent monad creates all limits that exist in the base quasicategory, without further assumption on the monad. This proof relies upon a more delicate and explicit analysis of the particular weight used to define quasicategories of algebras. Contents