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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
FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS MICHAEL SHULMAN
, 706
"... 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 as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, becau ..."
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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 as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the ‘morphisms between 0cells’, such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1cells. We avoid complicated coherence problems by describing base change ‘nonalgebraically’, using categorical fibrations. The resulting ‘framed bicategories ’ assemble into 2categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a ‘monoidal fibration’, meaning a parametrized family of monoidal categories, and produces an analogue of the framed bicategory of spans. Combining the two, we obtain a construction which includes both enriched and internal categories as special cases.
FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS MICHAEL SHULMAN
, 706
"... 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 as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, becau ..."
Abstract
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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 as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the ‘morphisms between 0cells’, such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1cells. We avoid complicated coherence problems by describing base change ‘nonalgebraically’, using categorical fibrations. The resulting ‘framed bicategories ’ assemble into 2categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a ‘monoidal fibration’, meaning a parametrized family of monoidal categories, and produces an analogue of
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.