productpreserving

Product-preserving is a term used to describe a functor or, less commonly, a morphism, that preserves product structures under mapping. In category theory, a functor F: C → D is product-preserving if for all objects A and B in C there is a natural isomorphism F(A ×C B) ≅ F(A) ×D F(B), and F sends the terminal object 1C to a terminal object isomorphic to 1D. If the source category C has finite products, this condition means F preserves all finite products.

Equivalently, a functor is product-preserving when it preserves binary products and the terminal object, and the preservation is natural in the arguments A and B. The property is closed under composition: if F and G are product-preserving, then G ∘ F is product-preserving.

Examples include forgetful functors from algebraic categories to Sets, which commonly preserve products. For instance, the forgetful functor U: Grp → Set preserves products, since the product of groups maps to the Cartesian product with componentwise operation, which is the product in Grp; similarly for Mon, Ring, and modules. The identity functor trivially preserves products.

Not all functors are product-preserving; for example, the power-set functor P on Sets is not product-preserving because P(X × Y) is not naturally isomorphic to P(X) × P(Y). In contexts such as topology and topos theory, preserving finite limits (of which finite products are a part) is a common and important property, ensuring that product constructions commute with the functor.

Related notions include finite-limit preservation and the role of product-preserving functors in transferring product structures across categories.