Abstract

this article we show that these structures are still susceptible to cohomological investigation, by developing the theory in the absence of the symmetry condition. Later we shall assume that the monoidal structure is left distributive over coproducts and the category is an abelian category; this is the case for operads, our original motivating example. 1. Monoids and Modules We define monoids in monoidal categories and introduce the "module" objects which will be used later as coefficients in the cohomology of such monoids. We also give some of our motivating examples of monoidal categories and the monoids therein. Let us start by recalling that a monoidal category is a tuple V = (V; ffi; I; a; l; r) where V is a category, ffi : V \Theta V ! V is a functor, I is an object of V, and a = (a X;Y;Z : (X ffi Y ) ffi Z ! X ffi (Y ffi Z)) X;Y;Z2V ; l = (l X : I ffi X ! X)X2V ; r = (r X : X ffi I ! X)X2V are natural isomorphisms, required to satisfy certain conditions which we omit here (see e.g. [19]). In many examples our monoidal categories will be strictly associative and have strict units, in the sense that all aX;Y;Z and l X , r X are identity morphisms. The monoidal category V is abelian if the underlying category V is an abelian category. Suppose V has binary coproducts, denoted X t Y ; then the monoidal structure is left distributive if the canonical natural transformation (X 1 ffi Y ) t (X 2 ffi Y ) ! (X 1 t X 2 ) ffi Y is an isomorphism. Right distributivity is defined similarly. A strict monoidal functor between monoidal categories is a functor between the underlying categories preserving all the existing structure in the obvious way. Given such a V, a monoid in V, or a V-monoid, is a triple G = (G; ; j) where G 2 V, : G ffi G ! G, j : I ! G must satisfy the...

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