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Monads And Interpolads In Bicategories (1997)

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by Jürgen Koslowski , Transmitted R. J. Wood
Citations:8 - 4 self
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@MISC{Koslowski97monadsand,
    author = {Jürgen Koslowski and Transmitted R. J. Wood},
    title = {Monads And Interpolads In Bicategories},
    year = {1997}
}

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Abstract

. Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y -mnd by using lax functors from the generic 0-cell, 1-cell and 2-cell, respectively, into Y . Any lax functor into Y factors through Y -mnd and the 1-cells turn out to be the familiar bimodules. The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchy-complete, but have a well-known Cauchycompletion in common. This prompts us to formulate a concept of Cauchy-completeness 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 endo-1-cells. 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 endo-1-cells. We show that associative coequalizing multiplications suffice and call the resulting structures interpolads. Together with str...

Citations

124 The geometry of tensor calculus - Joyal, Street - 1991
94 Metric spaces, generalized logic, and closed categories - Lawvere - 1974
86 Handbook of Categorical Algebra 1: Basic Category Theory - Borceux - 1994
63 Introduction to bicategories, in Reports of the Midwest Category Seminar - Bénabou - 1967
30 Yoneda structures on 2-categories - Street, Walters - 1978
18 Enriched categories and cohomology - Street - 1981
13 An axiomatics for bicategories of modules - Carboni, Kasangian, et al. - 1987
11 Information systems for continuous posets - Vickers - 1993
10 Adjunction of semifunctors: categorical structures in nonextensional lambda calculus - Hayashi - 1985
7 A remark on the theory of semi-functors - Hoofman, Moerdijk - 1995
7 Constructive complete distributivity IV - Rosebrugh, Wood - 1994
6 Information Systems for Continuous Posets, Theoretical Computer Science 114 - Vickers - 1993
1 Coherence for Tricategories. No - Gordon, Power, et al. - 1995
1 Science 5(1995), 1{8. Theory and Applications of Categories - Comp
1 Science 5 - Comp - 1995
The National Science Foundation
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