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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 ...
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 ...
The limit L of F is called the product of these objects. The cone φ consists of a family of morphisms φ X : L → F(X) called the projections of the product. In the category of sets, for instance, the products are given by Cartesian products and the projections are just the natural projections onto the various factors. Powers.
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 ...
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;
If, furthermore, the category has all finite products and coproducts, it is called an additive category. If all morphisms have a kernel and a cokernel, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an abelian category. A typical example of an abelian category is the category of abelian groups.
Coproducts, fibred coproducts, coequalizers, and cokernels are all examples of the categorical notion of a colimit. Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object.
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 ...