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Monads and interpolads in bicategories. (English) Zbl 0889.18003
Work in a bicategory \({\mathcal W}\) with local coequalizers preserved by composition. An interpolad \((A,s,\mu)\) is defined in this paper to consist of an endo-arrow \(s:A\to A\) together with a 2-cell \(\mu: ss\Rightarrow s\) which is the coequalizer of \(s\mu\), \(\mu s: sss\Rightarrow ss\). An interpolad morphism \(f= (f,\lambda,\rho): (A,s,\mu)\to (B,t,\nu)\) consists of an arrow \(f:A\to B\) and 2-cells \(\rho: f s\Rightarrow f\), \(\lambda: t f\Rightarrow f\) such that \(\rho\) is the coequalizer of \(f\mu\), \(\rho s\), and \(\lambda\) is the coequalizer of \(\nu f\), \(t\lambda\). There is a bicategory \({\mathcal W}\)-\({\mathcal I}nt\) of interpolads; it contains \({\mathcal W}\) as a full subbicategory. An idempolad is an interpolad \((A,s,\mu)\) with \(\mu\) invertible. There is a natural notion of splitting for idempolads in \({\mathcal W}\); indeed, all idempolads in \({\mathcal W}\)-\({\mathcal I}nt\) do split. A further result is that \({\mathcal W}\)-\({\mathcal I}nt\) has all right extensions if \({\mathcal W}\) does, provided \({\mathcal W}\) also has local equalizers (much the same as for \({\mathcal W}\)-\({\mathcal M}od\) as discussed by R. Street [“Enriched categories and cohomology”, Quaest. Math. 6, 265-283 (1983; Zbl 0523.18007)]).

MSC:
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
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