Productcoproductpreserving

In category theory, a branch of mathematics, a **productcoproductpreserving** functor is a type of functor that preserves both finite products and finite coproducts. These are fundamental constructions in category theory, representing ways to combine objects in a structured manner. A product of objects in a category is a universal construction that generalizes the notion of intersection or simultaneous solution, while a coproduct generalizes the notion of union or disjoint combination.

A functor \( F: \mathcal{C} \to \mathcal{D} \) between categories \( \mathcal{C} \) and \( \mathcal{D} \) is called productcoproductpreserving if it maps products in \( \mathcal{C} \) to products in \( \mathcal{D} \) and coproducts in \( \mathcal{C} \) to coproducts in \( \mathcal{D} \). This means that for any collection of objects \( \{X_i\}_{i \in I} \) in \( \mathcal{C} \), the functor \( F \) preserves the universal property of both the product \( \prod_{i \in I} X_i \) and the coproduct \( \coprod_{i \in I} X_i \). Specifically, if \( (P, p_i) \) is the product of the \( X_i \) in \( \mathcal{C} \) and \( (C, i_j) \) is the coproduct, then \( (F(P), F(p_i)) \) is the product of the \( F(X_i) \) in \( \mathcal{D} \), and \( (F(C), F(i_j)) \) is the coproduct of the \( F(X_i) \) in \( \mathcal{D} \).

Productcoproductpreserving functors are particularly important in contexts where both products and coproducts play a symmetric role, such as in the study of abelian categories or in the representation theory of algebras. They often arise in the context of exact functors, which preserve both kernels and cokernels, as products and coproducts can be expressed in terms of these constructions. Examples of such functors include the forgetful functor from groups to sets, which preserves both products (direct products) and coproducts (free products), though it does not preserve the universal properties in the same way as a strictly productcoproductpreserving functor in the category-theoretic sense. More precise examples can be found in specific mathematical structures where both product and coproduct are well-behaved and interact nicely with the functor.