# zbMATH — the first resource for mathematics

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)
Full Text: