DISTRIBUTIVE LAWS AND FACTORIZATION Dedicated to Max Kelly on his th birthday

This article shows that the distributive laws of Beck in the bicategory of sets and matrices wherein monads are categories determine strict factorization systems on their composite monads Conversely it is shown that strict factorization systems on categories give rise to distributive laws Moreover these processes are shown to be mutually inverse in a precise sense Strict factorization systems are shown to be the strict algebras for the monad on the category of categories Further an extension of the distributive law concept provides a correspondence with the classical factorization systems Introduction In this paper we understand a factorization system on a category K to mean a pair of subcategories E M each containing all the isomorphisms of K satisfying the diagonal ll in condition and further satisfying K EM Of course the equation is intended to be understood in the sense of what is called set multiplication in elementary modern algebra texts Part of the goal of this paper is to take that equation more seriously To put it another way factorization in the widest sense should be seen as a section for multiplication or composition This raises the question of how categories might be multiplied or composed Categories are monads in a certain bicategory and after Beck BEK we know that monads in the category of categories are composed with the help of distributive laws There is much that can be said about distributive laws in any bicategory and in particular Beck s correspondence between distributive laws and composite monad structures holds quite generally We refer the reader to STR L S MRW and R W for other general results about distributive laws In this article we show the equivalence of three concepts distributive laws in the bicategory of set valued matrices wherein monads correspond to categories strict factor ization systems on categories and strict algebras for the monad on CAT given by and the structure induced by the cocommutative comonoid The important paper K T showed that factorization systems on categories are equiv alent to normal pseudo algebras for the monad We extend the notion of distribu tive law in the bicategory of set valued matrices to give a third concept equivalent to that of factorization system The authors gratefully acknowledge nancial support from the Canadian NSERC Diagrams typeset using M Barr s diagram macros Mathematics Subject Classi cation A D