See nerve and nerve and realization for a clearer description of what the premise of the claim is probably gesturing at. (This seems comically useless for someone who can’t read the language, though I’ll try to briefly explain.)
That is, if there is a category C where an interesting object c (such as “the territory”) lives, we can observe that object by first setting up abstract measuring apparatus that takes the form of a category S, and then setting up a “realization” process F that “concretely instantiates” objects and morphisms of S in C, that is it’s a functor from S to C. Then, for any object s of S (a measuring device), we can observe the ways in which F(s) (realization of s in C) maps to the object of study c, and capture the whole collection [F(s), c] of ways in which it does so in C. In the straightforward case this is some set of maps, but more generally it could include more data than only a set (maybe C is an enriched category). Then we can take all our observations [F(s), c] taken in C using each abstract measuring device s, together with ways in which they are related in C, and assemble them into an object of an “abstract map” (“nerve”) of c that’s now written in the language of S and not in the language of C.
The datastructure that encodes this kind of “abstract map” is called a presheaf (on S). Presheaves can be understood as being built out of many abstract measuring devices (objects of S) by “glueing” them together into larger objects (taking coproducts). The Yoneda embedding indicates where the original primitive measuring devices from S exist as presheaves in the category of presheaves PSh(S), with no glueing applied. If understood as a “realization” process, Yoneda embedding can be used in an example of the same methodology for building “abstract maps”. That is, each object (measuring device) from S can be “instantiated” in PSh(S), and then these instances can be collectively used to observe arbitrary objects of PSh(S), creating their “abstract maps”. The Yoneda lemma then says that an “abstract map” of an “abstract map” (some object of PSh(S)) obtained through this method is that same original “abstract map” itself.
See nerve and nerve and realization for a clearer description of what the premise of the claim is probably gesturing at. (This seems comically useless for someone who can’t read the language, though I’ll try to briefly explain.)
That is, if there is a category C where an interesting object c (such as “the territory”) lives, we can observe that object by first setting up abstract measuring apparatus that takes the form of a category S, and then setting up a “realization” process F that “concretely instantiates” objects and morphisms of S in C, that is it’s a functor from S to C. Then, for any object s of S (a measuring device), we can observe the ways in which F(s) (realization of s in C) maps to the object of study c, and capture the whole collection [F(s), c] of ways in which it does so in C. In the straightforward case this is some set of maps, but more generally it could include more data than only a set (maybe C is an enriched category). Then we can take all our observations [F(s), c] taken in C using each abstract measuring device s, together with ways in which they are related in C, and assemble them into an object of an “abstract map” (“nerve”) of c that’s now written in the language of S and not in the language of C.
The datastructure that encodes this kind of “abstract map” is called a presheaf (on S). Presheaves can be understood as being built out of many abstract measuring devices (objects of S) by “glueing” them together into larger objects (taking coproducts). The Yoneda embedding indicates where the original primitive measuring devices from S exist as presheaves in the category of presheaves PSh(S), with no glueing applied. If understood as a “realization” process, Yoneda embedding can be used in an example of the same methodology for building “abstract maps”. That is, each object (measuring device) from S can be “instantiated” in PSh(S), and then these instances can be collectively used to observe arbitrary objects of PSh(S), creating their “abstract maps”. The Yoneda lemma then says that an “abstract map” of an “abstract map” (some object of PSh(S)) obtained through this method is that same original “abstract map” itself.