> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/07-constructions/universal-construction.md).

# Universal construction

<https://en.wikipedia.org/wiki/Universal_construction>

In category theory, a **universal property** is a property which is satisfied by a *universal morphism*.

Universal morphisms can also be thought of more abstractly as *initial* or *terminal objects* of a *comma category*.

Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated. Universal properties may be used in other areas of mathematics implicitly, but the abstract and more precise definition of it can be studied in category theory.

* **Universal construction** is a common construction in category theory for defining objects in terms of their relations. One way to do it, is to pick a *pattern* (i.e. a particular shape constructed from objects and morphisms) and find all occurrences of this pattern in the category. For a common pattern (and a large category) there will be lots of "hits". The problem is then how to rank them; we need to establish some kind of ranking that will help us discover the best candidate (Google uses PageRank technology for the same problem).