Universal property

A universal property characterizes an object through its relationships with every object of a specified kind, often determining it uniquely up to isomorphism.

Category theory: The mathematical study of objects and structure-preserving maps between them. Universal properties are commonly expressed as patterns of objects and morphisms in categories.
Product (category theory): An object equipped with projections that uniquely represent pairs of maps into its factors. Its projection maps and unique factorization define the universal property of a product.
Tensor product: An algebraic object that represents bilinear maps from a pair of modules or vector spaces. Its defining property replaces bilinear maps with unique linear maps from one object.
Elementwise definition: A definition that specifies an object by describing its elements and their properties. Universal properties characterize an object by maps, rather than by a chosen description of its elements.
Morphism: A structure-preserving map between objects in a mathematical structure such as a category. Universal properties specify how morphisms enter or leave the characterized object.
Coproduct: An object equipped with maps from its components that uniquely represent maps out of those components. Its injections characterize the object through unique maps from each summand.
Quotient group: A group formed by identifying elements that differ by members of a normal subgroup. Its canonical map is universal among homomorphisms that send the normal subgroup to the identity.
Presentation (mathematics): A description of an algebraic structure by generators and relations. Presentations provide explicit construction data, while universal properties express behavior independently of presentation.
Isomorphism: A structure-preserving map with an inverse that preserves the same structure. Uniqueness in a universal property usually means uniqueness up to this kind of equivalence.
Equalizer: A map capturing the largest part of an object on which two given maps agree. Its universal condition factors every map on which the two composites agree.

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