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On clubs and data-type constructors. (English) Zbl 0789.18007

Fourman, M. P. (ed.) et al., Applications of categories in computer science. Proceedings of the LMS symposium, held at the University of Durham, England, UK, from 20 to 30 July, 1991. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 177, 163-190 (1992).
Let \({\mathcal A}\) be a category with finite limits, \([{\mathcal A}, {\mathcal A}]\) the category of its endofunctors, and \(\Delta:{\mathcal A} \to [{\mathcal A}, {\mathcal A}]\) the functor sending \(A\) to the constant functor \(\Delta A\) at \(A\). Then \(\Delta\) has a left adjoint \(E\) given by evaluation at the terminal object 1 of \({\mathcal A}\), so that \({\mathcal A}\) is a localization of \([{\mathcal A}, {\mathcal A}]\). As is well known, there is a factorization system \(({\mathcal E}, {\mathcal M})\) on \([{\mathcal A}, {\mathcal A}]\), where \({\mathcal E}\) is the class of maps inverted by the localization and \({\mathcal M}\) is easily described. Composition of functors makes \([{\mathcal A}, {\mathcal A}]\) a monoidal category, and thus we get a monoidal structure on \([{\mathcal A}, {\mathcal A}]/S\) for any monad \(S\) on \({\mathcal A}\). The monad \(S\) is said to be a club if this monoidal structure restricts to \({\mathcal M}/S\); in which case a monoid \(\alpha:T \to S\) for this structure is a monad \(T\) with a monad-map \(\alpha\) in \({\mathcal M}\). Such a \(T\) is itself a club; and the category of monoids in \({\mathcal M}/S\) is the category of clubs over \(S\). There is an equivalence of categories \({\mathcal M}/S \simeq {\mathcal A} /S1\); one easily describes the monoidal structure – now transferred to the latter – and thus gives an elementary description of clubs over \(S\). Since \(\alpha:T \to S\) in \({\mathcal M} /S\) is sent to \(\alpha 1:T1 \to S1\) in \({\mathcal A} /S1\), we see that the monad \(T\) (and so the free \(T\)- algebras) are fully determined by a knowledge of the free \(T\)-algebra \(T1\) on 1, together with its augmentation \(\alpha 1\) – which in practice is easily read off from any basic description of the \(T\)-algebras.
There are many examples in computer science, and we give several in the case where \(S\) is the “list” monad. The whole thing works for enriched categories, and in particular for 2-categories; now the clubs correspond to 2-monads whose algebras are categories with certain kinds of structure, such as symmetric monoidal closed ones. These latter cases were in fact those first considered by the author in 1972; they are recalled here as motivation, before the newer abstract theory is presented.
For the entire collection see [Zbl 0771.00028].
Reviewer: G.M.Kelly

MSC:

18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
68Q55 Semantics in the theory of computing
68Q65 Abstract data types; algebraic specification
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
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