What 'canonical' actually means
A natural transformation assigns to each object a morphism between the values of two functors, commuting with everything in sight. It is the formalisation of a construction that is defined without arbitrary choices. In homological algebra it does real work: the connecting homomorphism of a long exact sequence is natural, and that naturality is what allows a map of short exact sequences to induce a map of long exact sequences.
Learning objectives
- Define a natural transformation and its naturality square.
- Distinguish a natural isomorphism from a pointwise one.
- State why naturality of the connecting map matters.
- Describe the functor category and the Yoneda lemma.
Section 01The definition
For functors S, T: C → D, a natural transformation τ: S → T is a family τA: SA → TA such that for every f: A → B the square
commutes. If every τA is an isomorphism, τ is a natural isomorphism and the two functors are indistinguishable.
A finite-dimensional vector space is isomorphic to its dual, but not naturally — the isomorphism requires a choice of basis. It is naturally isomorphic to its double dual. This example is the reason naturality was invented, and it is worth carrying as the test case.
Section 02Naturality at work
The connecting homomorphism of the long exact sequence is natural in the short exact sequence. Concretely, given a map of short exact sequences, the diagram relating the two connecting maps commutes.
- Take a morphism of short exact sequences — three vertical maps making two squares commute.
- Each row yields a long exact sequence of derived functors.
- Naturality says the vertical maps assemble into a map of long exact sequences, including at every connecting homomorphism.
- Now apply the five lemma to conclude that if two of every three vertical maps are isomorphisms, so is the third. This is the standard comparison argument.
Constructing a connecting map is routine; proving it natural is the part that makes it useful. When a text says a sequence is natural, it is signalling that comparison arguments are available.
Section 03Functor categories and Yoneda
Functors C → D with natural transformations form a category. When D is abelian so is the functor category, which is how diagrams, chain complexes and presheaves all become abelian categories in their own right.
The Yoneda lemma says a natural transformation out of a representable functor is exactly an element of the target at the representing object. Its immediate consequence is that an object is determined up to isomorphism by the functor it represents — the formal justification for defining objects by universal properties.
A chain complex is a functor from a suitable index category, so complexes inherit abelian structure pointwise. This is what lets homological algebra be done on complexes themselves, which is the starting point for derived categories.
ReferenceFrequently asked questions
Is every family of maps a natural transformation?
No — the naturality square is a genuine condition and usually fails for arbitrary choices. That is exactly its value: it distinguishes canonical constructions from ones depending on choices.
What does it mean for Ext to be a functor of two variables?
It is a bifunctor: contravariant in the first argument, covariant in the second, with the two actions commuting. Naturality in each variable separately is what licenses the two long exact sequences.
Does the Yoneda lemma have a computational use here?
Indirectly but importantly. It underlies the claim that universal properties determine objects uniquely, which is what allows tensor products, kernels and limits to be defined by mapping properties rather than by construction.
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