2 Translations of the Yoneda Lemma
Theorem stated by Nabuo Yoneda, and translated by Kavi Duvvoori
Note: all metaphors are to be used at the reader’s own risk. The translator takes no responsibility for any harm – social, academic, emotional, ethical, financial, physical, or otherwise – caused by improper use of the over-large generalizations and unjustified associations made here.
Let C be any locally small category, i.e. a class of objects Obj and a morphism Hom function from the class of objects squared to the class of all sets of sets, a composition operation on adjacent morphisms, and an identity function Id from the class of objects to the class of sets.
Let F be any functor from C to Set, the category of sets as objects with maps as arrows.
Let A be an object in C. Then, the covariant hom-functor at A is a functor from C to Set defined as follows: Hom at A of a morphism f from an object X to an object Y is taken to the map between the hom-set of A and X and the hom-set of A and Y that acts by precomposition of f.
The Yoneda Lemma states that the set of natural transformations between the covariant hom-functor at A and F is isomorphic with F of A, and this isomorphism is natural when both sides are considered as functors from the category of functors from C to Set cross C to Set!
First, learn cartography. No one changes road-signs. Trains leave on time. Flights follow their scheduled routes and blink their positions along monitors over your head and you try to sleep. Ships are registered with the proper authorities. Camels walk where they’re told.
There is language, call it Set. People turn words into words. Authors write encyclopedias, news reports, self-help books, and nursery rhymes which can be found in basements or the Library of Congress.
You start somewhere, C. Look around.
The things around you arrange themselves in terms of old words, remembered hallways that you can walk around the corners of when you close your eyes in the right weather. That house is green and boarded and wooden and high like your neighbor’s, which you walked into once and wished you hadn’t. The city is made out of streets arranged in a grid like you’ve seen in foldout and googled maps, made out of rectangles that you learned the name and shape of with wooden blocks that fit into each other in your Montessori pre-school. You see the way branches rise as the explanation your mother gave about photosynthesis when you, loud and fragile, asked how plants grow (or maybe what she told you about tree spirits waving to meet the running sun.)
You have a children’s dictionary, or an atlas, or a bird watching book, or a Guidebook for the Budget Traveler – F. It says some things.
Go somewhere. Pick something up, A. See how the world holds it. Consider what this billboard or bird or 11th story window, or cloud, or left sock sees. Call what it says A’s Hom, type it up, and put it on a shelf somewhere.
Now hold the stories side by side. Look at one, and then the other. Your left sock tells you about an earthworm experimenting with air, deciding on things, and now you step a little to the left. The book prefers the history of the sidewalk, the experimentations of the early family-run concrete companies, the century old firmness of the stone marked and dated by Herbert & Co, which a foot rests on.
Write an essay comparing and contrasting (but mostly comparing) A’s Hom and F. Make some bubble charts. Notice characters overlapping motivations, and influences in structure and worldview. Try out thesis statements, and support with well-cited examples your properly ordered topic sentences.
Now you’re drowning in a forest of metaphors but look around (says Yoneda). Make some maps – your wrists and fingers feel coastlines turning smoothly forward, then a sudden cutting back. Make islands play off of islands, trace names hovering around the bends in rivers which come out of your mouth like the way the water pumps forward, and which keep sounding right when you step in them. Consider the coastline of a chin or palm or democratic constitution. See how to get from here to there, what this says about that. The maps make matching shapes; they’re the same place. Maybe as you look you know where you are, how you got here.