Monads and Interpolads in Bicategories

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 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. Introduction Part of the original motivation for this work was to better understand, why the bicategory idl of pre-ordered sets, order-ideals, and inclusions inherits good properties from the bicategory rel of sets, relations and inclusions. Of particular interest was closedness with respect to 1-cell composition, also known as the existence of all right liftings and right extensions. The key observation is that pre-ordered sets can be viewed as monads in rel . Benabou [1] explicitly designed what is now known as lax functors to subsume the notion of monad, in this case as a lax functor from the terminal bicategory 1 to rel . To keep the paper reasonably self-contained, in Section 1 we recall the relevant definitions and establish our notation. Order-ideals, i.e., relations compatible with the orders on domain and codomain, do not correspond to the kind of morphisms that are usually considered between monads. Since ordinary morphisms in a category correspond to functors with domain 2 , the two-element chain or generic 1-cell, we introduce 1-cells between monads in terms of Received by the editors 1997 January 22 and, in revised form, 1997 September 24. Published on 1997 October 14 1991 Mathematics Subject Classification : 18D05.