Clippings

…both posets and monoids are themselves special kinds of categories, which in a certain sense represent the two “dimensions” (objects and arrows) that a general category has. Many phenomena occurring in categories can best be understood as generalizations from posets or monoids.

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Any preorder P can be regarded as a category by taking the objects to be the elements of P and taking a unique arrow a → b if and only if a ≤ b. … A poset is evidently a preorder satisfying the additional condition of antisymmetry… It’s often useful to think of a category as a kind of generalized poset, one with with “more structure” than just p ≤ q. One can thus also think of a functor as a generalized monotone map.

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…a monoid is a category with just one object. The arrows of the category are the elements of the monoid. In particular, the identity arrow is the unit element u. Composition of arrows is the binary operation m⋅n of the monoid. …a monoid homomorphism from M to N is the same thing as a functor from M regarded as a category to N regarded as a category. In this sense, categories are also generalized monoids, and functors are generalized homomorphisms.