I said that adjoint functors are like dictionaries that translate back and forth between different categories. How far can we take that analogy?
In the common understanding of dictionaries, we assume that the two languages … are equally expressive, and that a good dictionary will be an even exchange of ideas. But in category theory we often have two categories that are not on the same conceptual level. This is most clear in the case of so-called free-forgetful adjunctions. [In a] sense … each adjunction provides a dictionary between two categories that are not necessarily on an equal footing, so to speak.
Consider the category of monoids and the category of sets. A monoid
(M,e,∗) is a set with an identity element and a multiplication formula that is associative. A set is just [stuff in a box]. A dictionary between
Set should not be required to set up an even exchange, but instead an exchange that is appropriate to the structures at hand. …
Let’s bring it down to earth with an analogy. A one-year-old can make repeatable noises and an adult can make repeatable noises. One might say “after all, talking is nothing but making repeatable noises.”
But the adult’s repeatable noises are called words, they form sentences, and these sentences can cause nuclear wars. There is something more in adult language than there is simply in repeatable sounds.
In the same vein, a tennis match can be viewed as physics, but you won’t see the match. So we have something analogous to two categories here:
((repeated noises)) and
We are looking for adjoint functors going back and forth, serving as the appropriate sort of dictionary. To translate baby talk into adult language we would make every repeated noise a kind of word, thereby granting it meaning.
We don’t know what a given repeated noise should mean, but we give it a slot in our conceptual space, always pondering “I wonder what she means by Konnen..”
On the other hand, to translate from meaningful words to repeatable noises is easy. We just hear the word as a repeated noise, which is how the baby probably hears it.
Adjoint functors often come in the form of “free” and “forgetful”. Here we freely add Konnen to our conceptual space without having any idea how it adheres to the rest of the child’s noises or feelings. But it doesn’t act like a sound to us, it acts like a word; we don’t know what it means but we figure it means something. Conversely, the translation going the other way is “forgetful”, forgetting the meaning of our words and just hearing them as sounds.
The baby hears our words and accepts them as mere sounds, not knowing that there is anything extra to get.
Back to sets and monoids, the sets are like the babies from our story: they are simple objects full of unconnected dots. The monoids are like adults, forming words and performing actions. In the monoid, each element means something and combines with other elements in some way. There are lots of different sets and lots of different monoids, just as there are many babies and many adults, but there are patterns to the behavior of each kind and we put them in different categories.
Applying a free functor … makes every element… a word…. Since a set … carries no information about the meaning or structure of its various elements, the free monoid … does not relate different words in any way.
To apply a forgetful functor … to a monoid, even a structured one, is to … [remove all the arrows, all the interrelationships which give the elements meaning relative to each other]. It sends a monoid … to a set. The analogy is complete.