The concept is called a coproduct. The coproduct of two sets is the disjoint union. The coproduct of two abelian groups is also called the direct sum [edited], and the category of general groups does not have coproducts. The coproduct of two types in the category of types is exactly the union type. The idea of an Abelian category (basically, a "nice" category) requires the existence of coproducts, which I think is widely used enough in universal algebra to be considered a key concept.
I wrote a long (and unfinished) series about how to interpret category theory with programs on my blog [0], and in [1] I cover universal properties and the coproduct, which describes how you would formulate a "union type" in any category.
I wrote a long (and unfinished) series about how to interpret category theory with programs on my blog [0], and in [1] I cover universal properties and the coproduct, which describes how you would formulate a "union type" in any category.
[0]: http://jeremykun.com/ [1]: http://jeremykun.com/2013/05/24/universal-properties/