Editing Universal property (section)

== Motivation ==

Before giving a formal definition of universal properties, we offer some motivation for studying such constructions.

* The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construction is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details. For example, the [[tensor algebra]] of a [[vector space]] is slightly complicated to construct, but much easier to deal with by its universal property.
* Universal properties define objects uniquely up to a unique [[isomorphism]].<ref>Jacobson (2009), Proposition 1.6, p. 44.</ref> Therefore, one strategy to prove that two objects are isomorphic is to show that they satisfy the same universal property.
* Universal constructions are functorial in nature: if one can carry out the construction for every object in a category ''C'' then one obtains a [[functor]] on ''C''. Furthermore, this functor is a [[adjoint functors|right or left adjoint]] to the functor ''U'' used in the definition of the universal property.<ref>See for example, Polcino & Sehgal (2002), p. 133. exercise 1, about the universal property of [[group ring]]s.</ref>
* Universal properties occur everywhere in mathematics. By understanding their abstract properties, one obtains information about all these constructions and can avoid repeating the same analysis for each individual instance.