From pg. 59 of Categories for the Working Mathematician:
Show that the construction of the polynomial ring $K[x]$ in an indeterminate $x$ over a commutative ring $K$ is a universal construction.
Question: What does the author mean by this bolded term?
For context: up to this point in the book, the author has already defined the notions of universal arrow, universal element, and universal property.
Is the author therefore just using the term universal construction as a synonym for the term universal property (or universal X, where X could be either element or arrow)?
Best Answer
A universal construction is simply a definition of an object as "the unique-up-to-isomorphism object satisfying a certain universal property". The name "construction" is a little misleading, because it's a definition: one still needs to show that the object actually exists, and that usually has to be done non-category-theoretically.