Book Notes / Category Theory in Context
Emily Riehl / Reading notes
The maps determine what counts as the same
- A category specifies objects, maps between them, identities, and associative composition. The choice of maps determines what structure a comparison must preserve. Sets admit functions; groups admit homomorphisms. The same number of elements does not settle whether two groups are isomorphic, because a bijection need not respect multiplication.
- An isomorphism has an inverse among the permitted maps. This is stronger than an invertible-looking correspondence between underlying elements. The model's notion of sameness follows from the structure it requires maps to preserve.
A product is a requirement on every incoming pair
- A product of A and B is an object P with projections to A and B. For any object X and pair of maps f: X โ A and g: X โ B, there must be exactly one map from X to P whose two projections recover f and g. The quantifier matters: the requirement applies to every eligible pair, not just the examples that suggested the construction.
- In sets, A ร B satisfies the requirement because each x must go to (f(x), g(x)). That assignment exists, and the projections leave no alternative. This proves the familiar construction works. Merely giving an object two projection maps would not prove either existence or uniqueness of the required assignment.
- Adding an unused bit exposes the difference. For nonempty sets A and B, let P = A ร B ร {0, 1}, with projections that ignore the bit. Every pair f and g can still be recovered, but a map into P can choose either bit. For nonempty X, there are distinct compatible maps, so P fails the product requirement. Extra hidden choice breaks the specification even though all requested outputs can be produced.
- Suppose P and Q both satisfy the requirement. P's projections produce a unique compatible map P โ Q; Q's produce one Q โ P. The composite P โ Q โ P has the same projections as the identity on P. Uniqueness therefore forces the composite to be that identity, and the reverse composite is the identity on Q. The two constructions are isomorphic because they fulfill the same role.
- The isomorphism is unique when it must preserve the chosen projections. On {0, 1} ร {0, 1}, swapping coordinates is another automorphism of the underlying set, but it exchanges the projections. Forgetting the specified maps turns a correct uniqueness statement into a false one.
Relationships retain information only when they compose
- For an object A, Hom(โ, A) records the maps into A. It also records how a map X โ A changes when preceded by a map Y โ X. That action on maps is essential: a table of how many arrows exist would discard how the arrows fit together.
- A map u: A โ B turns each f: X โ A into u โ f: X โ B. This gives a natural transformation from Hom(โ, A) to Hom(โ, B). Naturality requires that changing the incoming object and then applying the comparison agree with applying the comparison and then changing the incoming object.
- Yoneda supplies the converse. Given such a natural transformation, apply its component at A to the identity on A; the result is a map u: A โ B. Naturality then forces every other component to act by composition with u. The entire comparison is determined by what it does to one identity, because compatibility propagates that choice through every incoming map.
- This is a theorem about the complete representable functor in a locally small category. It does not establish that a few measurements or selected relationships determine an arbitrary real-world system. The force of the result comes from its quantifiers and composition law; an analogy that drops them has lost the guarantee.