Page 17 - Textos de Matemática Vol. 46

COLIMITS OF MONADS
J I Rˇ ´I A D A´ M E K
Dedicated to the seventieth birthday of Manuela Sobral
Abstract. The category of all monads over many-sorted sets (and over other “set-like” categories) is proved to have coequalizers. And a gen- eral diagram has a colimit whenever all the monads involved preserve monomorphisms and have arbitrarily large joint pre-fixpoints. In con- trast, coequalizers fail to exist e.g. for monads over the (presheaf) cate- gory of graphs.
For more general categories we extend the results on coproducts of monads from [3]. We call a monad separated if, when restricted to monomorphisms, its unit has a complement. We prove that every col- lection of separated monads with arbitrarily large joint pre-fixpoints has a coproduct. And a concrete formula for these coproducts is presented.
1. Introduction
Whereas limits in the category Monad(A) of monads over a complete cate- gory A are easy, since the forgetful functor into the category of all endofunctors creates limits, colimits are more interesting. For example, a coproduct of two monads need not exist in Monad (A) – in fact, there are only four (trivial) types of monads over Set having a coproduct with every monad, as proved in [3], see also Theorem 4.4 below. In that paper a formula for coproducts of monads over Set was presented, and we extend it to coproducts of separated monads over general categories A. Separatedness means that a complement of the unit of the monad exists if we restrict ourselves to the category Am of objects and monomorphisms of A. All consistent monads over Set are separated, see [3], where a monad is called consistent if its unit is monic. (The only inconsis- tent monads over Set are the terminal monad, constantly 1, and its submonad given by ; 7! ;.) For other base categories many interesting monads fail to be separated.
Received: 6 September 2014 / Accepted: 11 December 2014.
2010 Mathematics Subject Classification. 18C15, 18C50, 18A30, 08C05. Key words and phrases. Monad, colimit, coproduct.
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