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The coproduct in the category of sets is simply the disjoint union with the maps i j being the inclusion maps.Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be ...
The plastic used in plastic shopping bags also started as a by-product of oil refining. [1] By-products are sometimes called co-products to indicate that although they are secondary, they are desired products. For example, hides and leather may be called co-products of beef production. There is no strict distinction between by-products and co ...
For any objects ,, of a category with finite products and coproducts, there is a canonical morphism + (+), where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct X × Y + X × Z {\displaystyle X\times Y+X\times Z} guarantees the existence of unique arrows filling out the following diagram ...
Pushouts are equivalent to coproducts and coequalizers (if there is an initial object) in the sense that: Coproducts are a pushout from the initial object, and the coequalizer of f, g : X → Y is the pushout of [f, g] and [1 X, 1 X], so if there are pushouts (and an initial object), then there are coequalizers and coproducts;
Bicartesian closed categories extend Cartesian closed categories with binary coproducts and an initial object, with products distributing over coproducts. Their equational theory is extended with the following axioms, yielding something similar to Tarski's high school axioms but with a zero: x + y = y + x
Much of chemistry research is focused on the synthesis and characterization of beneficial products, as well as the detection and removal of undesirable products. Synthetic chemists can be subdivided into research chemists who design new chemicals and pioneer new methods for synthesizing chemicals, as well as process chemists who scale up chemical production and make it safer, more ...
In a preadditive category, every finitary product is necessarily a coproduct, and hence a biproduct, and conversely every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). The empty product, is a final object and the empty product in the case of an empty diagram, an initial object. Both ...
For small categories, this is the same as the action on objects of the categorical product in the category Cat. A functor whose domain is a product category is known as a bifunctor. An important example is the Hom functor, which has the product of the opposite of some category with the original category as domain: Hom : C op × C → Set.