coproductpreserving

In category theory, a branch of mathematics, a **coproduct-preserving** functor is a type of functor between categories that respects coproducts. Coproducts, also known as disjoint unions, are a fundamental construction in category theory that generalize the notion of direct sums or disjoint unions of objects. For a functor \( F \): \( \mathcal{C} \rightarrow \mathcal{D} \) between categories \( \mathcal{C} \) and \( \mathcal{D} \), coproduct-preserving means that \( F \) maps coproducts in \( \mathcal{C} \) to coproducts in \( \mathcal{D} \).

Formally, if \( \{A_i\}_{i \in I} \) is a family of objects in \( \mathcal{C} \) with coproduct \( A = \coprod_{i \in I} A_i \), then \( F \) preserves this coproduct if \( F(A) \) is isomorphic to the coproduct \( \coprod_{i \in I} F(A_i) \) in \( \mathcal{D} \). This means that \( F \) commutes with the coproduct operation, ensuring that the structure of coproducts is preserved under the functor.

Coproduct-preserving functors are particularly important in contexts where coproducts play a significant role, such as in the study of algebraic structures or in the theory of enriched categories. For example, in the category of sets, coproducts correspond to disjoint unions, and a coproduct-preserving functor would map these unions to coproducts in the target category, often preserving the disjointness property.

A functor that preserves both products and coproducts is called a **product-coproduct-preserving functor**. Such functors are well-behaved and often arise in the study of limits and colimits. Coproduct-preserving functors are also related to the concept of **left adjoints**, as left adjoints preserve colimits, including coproducts, by the adjoint functor theorem. This connection highlights their role in establishing relationships between different categories and their structures.