Abstract
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.
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© 1971 Springer-Verlag New York Inc.
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Lane, S.M. (1971). Monoids. In: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol 5. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-9839-7_8
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DOI: https://doi.org/10.1007/978-1-4612-9839-7_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90036-0
Online ISBN: 978-1-4612-9839-7
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