Subject
Category Theory
Objects, arrows, and the laws that decide when a mapping between them preserves structure. A category is a composition satisfying associativity and identity; a functor is a mapping satisfying two further laws, and those laws are what the word structural means. A mapping can send objects sensibly and give every arrow the right endpoints while failing them. Naturality is a square that must commute at every arrow rather than a diagram that can be drawn.
Learning paths
Category Theory
Verifying the laws that define categories, functors and natural transformations, and locating the failure when a mapping does not satisfy them.
What this subject develops
Decide whether a mapping preserves structure
Check a proposed category against its axioms, verify both functor laws on a mapping between small categories, and evaluate a naturality square arrow by arrow. The competency exists because the decisive checks are the ones most easily skipped: a mapping that sends objects sensibly and gives every arrow correct endpoints can still fail the composition law, and a naturality square can always be drawn whether or not its two routes agree. What distinguishes the skill is treating the laws as equations to be evaluated rather than as descriptions to be recognised.
See detailed outcomes
Decide whether a mapping preserves structure
- Given a small category and a proposed functor between two of them, the learner can check associativity and the identity laws on every composable case, and check both functor laws on every composable pair.
- Given a mapping that is not a functor, or a family of arrows that is not natural, the learner can identify the specific law and the specific composite that breaks it, and distinguish a law failure from a mapping whose arrows have the wrong endpoints.