This chapter will explore the general notion of a monoid in a category. As we have already seen in the introduction, an ordinary monoid in Set is defined by the usual diagrams relative to the cartesian product × in Set, while a ring is a monoid in Ab, relative to the tensor product ⊗ there. Thus we shall begin with categories B equipped with a suitable bifunctor such as × or ⊗, more generally denoted by □. These categories will themselves be called “monoidal” categories because the bifunctor □ : B × B→B is required to be associative. Usually it is associative only “up to” an isomorphism; for example, for the tensor product of vector spaces there is an isomorphism U⊗(V ⊗ W) ≅ (U⊗V)⊗ W. Ordinarily we simply “identify” these two iterated product spaces by this isomorphism. Closer analysis shows that more care is requisite in this identification — one must use the right isomorphism, and one must verify that the resulting identification of multiple products can be made in a “coherent” way.
KeywordsNatural Transformation Simplicial Object Monoidal Category Left Adjoint Forgetful Functor
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