# Yoneda and the Apple-Shaped Figure

In Appendix A1 of the 2017 paper "Buddhist Thought on Emptiness and Category Theory" by Venkata Rayudu Posina and Sisir Roy, there's an explanation of the Yoneda Lemma. It uses the metaphors of "A-shaped figure in B" and "B-valued property in A":

... Broadly speaking, the Yoneda lemma is about [properties of] objects [of categories] and their mutual determination.

First, let us consider a function

f: A → B

We can think of the function f as (i) a figure of shape A in B, i.e., an A-shaped figure in B. For example, in the category of graphs, a map

d: D → G

from a graph D (consisting of one dot) to any graph G is a D-shaped figure in G, i.e., a dot in the graph G. We can also think of the same function f as (ii) a property of A with values in B, i.e., a B-valued property of A (Lawvere and Schanuel, 2009, pp. 81-85). For example, with sets, say, Fruits = {apple, grape} and Color = {red, green}, a function

c: Fruits → Color

(with c(apple) = red and c(grape) = green) can be viewed as a Color-valued property of Fruits. ...

(cf Yoneda Perspective (2018-10-03), If You Need a Theorem (2018-11-08), Yoneda Friend (2019-11-22), Yoneda Explained (2020-01-05), ...) - ^z - 2020-10-06