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28
A 2Categorical Approach To Change Of Base And Geometric Morphisms II
, 1998
"... We introduce a notion of equipment which generalizes the earlier notion of proarrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the Vpro arising from a suitable monoidal category V. We further exhibi ..."
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Cited by 45 (7 self)
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We introduce a notion of equipment which generalizes the earlier notion of proarrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the Vpro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2category, in such a way that arbitrary functors F: L ✲ K induce equipment arrows relF: relL ✲ relK, spnF: spnL ✲ spnK, and so on, and similarly for arbitrary monoidal functors V ✲ W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locallyfullyfaithful 2functor to the 2category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.
Representable Multicategories
 Advances in Mathematics
, 2000
"... We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe ..."
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Cited by 33 (6 self)
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We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...
Pasting Diagrams in nCategories with Applications to Coherence Theorems and Categories of Paths
, 1987
"... This document was typeset using L ..."
From Coherent Structures to Universal Properties
 J. Pure Appl. Algebra
, 1999
"... Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coh ..."
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Cited by 13 (2 self)
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Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coherent structures (pseudoTalgebras) are transformed into universally characterised ones (adjointpseudoSalgebras). The 2category L consists of lax algebras for the pseudomonad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudoSalgebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudoalgebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudofunctors into Cat.
Notes on enriched categories with colimits of some class
 Theory Appl. Categ
"... The paper is in essence a survey of categories having φweighted colimits for all the weights φ in some class Φ. We introduce the class Φ + of Φflat weights which are those ψ for which ψcolimits commute in the base V with limits having weights in Φ; and the class Φ − of Φatomic weights, which are ..."
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Cited by 9 (0 self)
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The paper is in essence a survey of categories having φweighted colimits for all the weights φ in some class Φ. We introduce the class Φ + of Φflat weights which are those ψ for which ψcolimits commute in the base V with limits having weights in Φ; and the class Φ − of Φatomic weights, which are those ψ for which ψlimits commute in the base V with colimits having weights in Φ. We show that both these classes are saturated (that is, what was called closed in the terminology of [AK88]). We prove that for the class P of all weights, the classes P + and P − both coincide with the class Q of absolute weights. For any class Φ and any category A, we have the free Φcocompletion Φ(A) of A; and we recognize Q(A) as the Cauchycompletion of A. We study the equivalence between (Q(A op)) op and Q(A), which we exhibit as the restriction of the Isbell adjunction between [A, V] op and [A op, V] when A is small; and we give a new Morita theorem for any class Φ containing Q. We end with the study of Φcontinuous weights and their relation to the Φflat weights. 1
Monads And Interpolads In Bicategories
, 1997
"... . Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The local ..."
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Cited by 8 (4 self)
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. Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchycomplete, but have a wellknown Cauchycompletion in common. This prompts us to formulate a concept of Cauchycompleteness for bicategories that are not locally ordered and suggests a weakening of the notion of monad. For this purpose, we develop a calculus of general modules between unstructured endo1cells. These behave well with respect to composition, but in general fail to have identities. To overcome this problem, we do not need to impose the full structure of a monad on endo1cells. We show that associative coequalizing multiplications suffice and call the resulting structures interpolads. Together with str...
Constructive complete distributivity IV
 Appl. Cat. Struct
, 1994
"... A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from downclosed subobjects of L to L has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and suppreserving arrows. There is a restrict ..."
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Cited by 7 (5 self)
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A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from downclosed subobjects of L to L has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and suppreserving arrows. There is a restriction to order ideals and "totally algebraic" lattices. Both biequivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of (CCD) lattices in a topos. Research partially supported by grants from NSERC Canada. Diagrams typeset using Michael Barr's diagram package. AMS Subject Classification Primary: 06D10 Secondary 18B35, 03G10. Keywords: completely distributive, adjunction, projective, nuclear Introduction Idempotents do not split in the category of rel...