Coproduct in the context of Module (mathematics)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.

Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.

The simply typed lambda calculus (⁠${\displaystyle \lambda ^{\to }}$⁠), a formof type theory, is a typed interpretation of the lambda calculus with only one type constructor (⁠${\displaystyle \to }$⁠) that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus.

The term simple type is also used to refer to extensions of the simply typed lambda calculus with constructs such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems that introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The simple types, except for full recursion, are still considered simple because the Church encodings of such structures can be done using only ${\displaystyle \to }$ and suitable type variables, while polymorphism and dependency cannot.