Yoneda 's lemma

Formal statement

Pending task...

Yoneda’s perspective

The Yoneda perspective is the idea that objects in a category are understood through the morphisms into them.

• In Set, elements of a set $A$ are maps $1\to A$, from the singleton set.
  → A set is determined by its elements.
• In a general category, morphisms

are called generalized elements of $A$ of shape $X$.
→ Objects are determined by their generalized elements, i.e., the structured ways other objects map into them.

This expresses a kind of relational ontology: objects are not understood in isolation, but only through their relationships (morphisms) with the rest of the category.

Examples

• Groups:

  • Maps $\mathbb{Z}\to G$ correspond to elements of a group $G$.
  • The trivial map $1\to G$ only recovers the identity.
  • To “see” the group, one must consider all homomorphisms into it.

• Measurable spaces / Random variables:

  • A random variable $X:\mathrm{\Omega }\to \mathbb{R}$ is a generalized element of $\mathbb{R}$ of shape $\mathrm{\Omega }$.
  • It is a structured element, where the structure is given by the probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$.

Slogan

To know an object is to know all the ways other objects can map into it.

This is the true content of Yoneda’s lemma:

• The data of all morphisms into an object determines the object completely.
• In practice, this lets us “probe” objects by looking at their generalized elements.