Definition: A category with exactly one object is called a monoid.
In order to specify a category, we need to specify the objects, maps, which object is the domain of
each map, which object is the codomain of each map, which map is the identity map of each object and
which map is the composite of any two composable maps. And the following three laws should be satisfied:
bookkeeping laws, associative laws and identity laws.
Let’s define M for multiplication as * ->(n)-> *, where * is the only object in the category. And all the maps in the category are endomaps. Suppose taking all natural numbers as the maps, and taking the composition of two maps in the category is the product of the two numbers, then the identity is the number one 1, and the bookkeeping laws and the associative laws are satisfied in this category.
The object in M seems features. There are ways interpreting such category in sets, so that the object takes on a certain life. An interpretation will be denoted as: M -> S. One interpretation interprets the only object * of M as the set N of natural numbers, and each map in M is interpreted as a map from the set of natural numbers to itself: N ->(fn)-> N, defined by fn(x) = n * x. Then f1 is the identity, and fn◦fm = fnm is the composition of two maps. This shows the interpretation preserves the structure of the category.
Such a ‘structure-preserving’ interpretation of one category into another is called a functor (from the
first category to the second). A functor is also required to preserve the notions of domain and codomain.
Let’s define another monoid N, which has only one object , the maps are numbers again, while the composition is addition instead of multiplication. Now 1 must be 0. To give a functor N -> S means that we interpret
- as some set S and each map * ->(n)-> * in N as an endomap S ->(gn)-> S of the set S, in such a way that g0 = 1S and gn◦gm = g(n+m). We take S to be a set of numbers and define gn(x) = n + x.
All the above suggests the ‘standard example’ of interpretation of a monoid in sets, in which the object of the monoid is interpreted as the set of maps of the monoid itself. In this way, we get a standard functor from any monoid to the category of sets.
There are many functors from N to sets other than the standard one. Suppose we take a set X together with an endomap alpha, and interpret * as X and send each map n of N (a natural number) to the composite of alpha with itself n times, i.e. alpha^n. In order to preserve the identities, we send the number 0 to the identity map on X. In this way, we get a functor from n to sets, h: N -> S which can be summarized this way:
- h(*) = X
- h(n) = alpha^n
- h(0) = 1X
Then it is clear that h(n+m) = h(n)◦h(m).
In this way, whenever we specify a set-with-endomap X[endomap] we obtain a functorial interpretation of N in sets. This suggests another reasonable name for S[endomap] would be S[N] to suggest that an object is a functor from N to S.
