Proving by Universal Properties

Intro to Proofs · Axiom Academy

LESSON Proving by Universal Properties Define an object not by what it is inside, but by the one map every other object must send through it — and uniqueness comes for free. A universal property describes an object U by a job: for every object X that comes with the right data, there is exactly one map — written — that makes the diagram commute (both routes around it land on the same arrow). the unique map that makes the triangle commute The product isn't just ordered pairs. It is the object carrying two projections with this property: given any X with maps and , there is a unique with and . Concretely . Any X with a pair of maps , — one for each factor. Reading h then recovers f ; h then recovers g . Any other object with two projections satisfying the same property accepts a unique map to and gives one back; the two maps compose to identities, so the two objects are isomorphic . The property — not the bracket notation — is what makes a product a product. A quotient has its own universal property through the quotient map : any that is constant on classes factors as a unique with . Watch collapse the integers, colored by remainder mod 3 , onto the three classes [0],[1],[2] . The concrete instance (checked) Take and . Since , f kills , so it is constant on each class and descends to , :

This is the written version of the interactive lesson above. See the full Intro to Proofs course.