Monads and interpolads in bicategories

Jurgen Koslowski

Given a bicategory, 2, 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 Cauchy-completion 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 structure-preserving i-modules these form a bicategory Y-int that is indeed Cauchy-complete, in our sense, and contains the bicategory of monads as a not necessarily full sub-bicategory. Interpolads over rel are idempotent relations, over the suspension of set they correspond to interpolative semi-groups, and over spn they lead to a notion of ``category without identities'' also known as ``taxonomy''. If Y locally has equalizers, then modules in general, and the bicategories Y-mnd and Y-int in particular, inherit the property of being closed with respect to 1-cell composition.

Keywords: bicategory, closed bicategory, Cauchy-complete bicategory, lax functor, monad, module, interpolation property, taxonomy.

1991 MSC: 18D05.

Theory and Applications of Categories, Vol. 3, 1997, No. 8, pp 182-212.

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