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Monoids versus semigroups

Sunday May 28 2023.

Read the following passage from the nLab page for semigroups under the heading “Attitudes towards semigroups”

Some mathematicians consider semigroups to be a case of centipede mathematics. Category theorists sometimes look with scorn on semigroups, because unlike a monoid, a semigroup is not an example of a category.

However, a semigroup can be promoted to a monoid by adjoining a new element and decreeing it to be the identity. This gives a fully faithful functor from the category of semigroups to the category of monoids. So, a semigroup can actually be seen as a monoid with extra property.

How weird! Who wrote this anyway? By checking the revision history we can see the phrase “Category theorists sometimes look with scorn on semigroups” was added in revision #2 and the comment about “centipede mathematics” in revision #3, both in 2009.

So, what other sorts of mathematical objects do category theorists scorn? Apparently the notion of centipede mathematics is well known enough to get its own wiki page, which cites the semigroup as centipede mathematics with a link to nLab right away.

The following quote summarises the value and usefulness of the concept: “The term ‘centipede mathematics’ is new to me, but its practice is surely of great antiquity. The binomial theorem (tear off the leg that says that the exponent has to be a natural number) is a good example. A related notion is the importance of good notation and the importance of overloading, aka abuse of language, to establish useful analogies.” — Gavin Wraith

There we find a 2000 blog post from John Baez accusing all sorts of objects of being centipede mathematics, including ternary rings, nearfields, quasifields, Moufang loops, Cartesian groups, and so on!