Or.. don’t, if these notes are insufficient to your lofty standards…

]]>How about these tables:

]]>Actually the fact is that if it did NOT, mathematicians would not be interested in THIS theory–so not “lucky” but necessary

]]>In a way, you can think of them in programming terms as casting up an inheritance hierarchy: yes, is a monoid, but if all we care about is its set of elements we can just consider .

The really interesting thing is that forgetful functors are usually one side of an adjunction, and the other side is a “free” functor, that gives the “most generic” way of building a new layer of structure on top of a given one. If is a set, then the free monoid on is the most generic monoid that includes the elements of .

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