CATProduct

CATProduct refers to the categorical product in category theory, a construction that combines a family of objects into a single object equipped with projection arrows. Formally, in a category C, a product of a family {A_i} consists of an object P and morphisms p_i: P → A_i for each i, such that for any object X with a family of morphisms f_i: X → A_i there exists a unique morphism f: X → P with p_i ∘ f = f_i for all i. This universal property characterizes the product up to isomorphism and makes it a limit: the limiting cone over the discrete diagram indexed by the set of i.

In concrete categories, products take familiar forms. In the category of sets, the product is the Cartesian product with projection maps to each factor. In groups, the product is the direct product with componentwise operations. In topological spaces, the product is endowed with the product topology. In the category Cat of small categories, the product C × D has objects pairs (c, d) and morphisms pairs (f, g), with projection functors to C and D.

Key properties include that products are preserved by functors that preserve limits, and that products are associative and commutative up to canonical isomorphism. The product over an empty index set yields a terminal object, while the product over a singleton is isomorphic to the sole factor. As a central construction, CATProduct is used to form families of objects and to study how structures distribute over multiple components.

See also: product (category theory), limit, terminal object, functor, projection.