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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)?

user1770201
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2 Answers2

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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.

Patrick Stevens
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  • Formally speaking, when you say it's an "object", do you mean its an object in a comma category (or slice/coslice category)? That is, does the "object" encode both an object and a morphism, or just an object? – user1770201 Mar 19 '17 at 09:02
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    It need not be in a comma category: for example, a universal construction of an initial object can be stated as "the object $0$ such that for every object $A$, there is a unique arrow $0 \to A$", which doesn't involve a comma category explicitly. – Patrick Stevens Mar 19 '17 at 09:16
  • In the setting of the initial object as a universal construction, what is the underlying slice/coslice/comma category? That is, I'm assuming what you are saying equates to $(0, f)$ is a universal arrow in some $(A, \mathcal{C})$ for some object $A$ in some category $\mathcal{C}$. If this is true, then what is $\mathcal{C}$ and $A$? What is $f$? Or am I totally confused? :) – user1770201 Mar 19 '17 at 09:30
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    I ended up here after reading the paragraph about universal constructions from "Category Theory for Programmers" by Bartosz Milewski. This answer helped me a lot, but I still miss a lot of contexts to fully understand it. This resource clarified to me the meaning of "universal property" and its role in the universal construction thing. https://ncatlab.org/nlab/show/universal+construction Hope it helps someone. – Ilario Pierbattista May 19 '21 at 06:31
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The term "universal construction" isn't that widely used in my experience, but it's pretty clearly a construction that is characterized by a universal property. In general, as Patrick Stevens and the nLab page states, you actually have to make a real construction to show that something satisfies the characterization. In some formal/logical/type theoretic contexts, you can view this as providing rules for completely syntactic objects. In effect, you are axiomatically assuming the existence of an object that satisfies the universal property. In this case though, the exercise is to find a universal property which characterizes the polynomial ring construction. A bit more colorfully, the exercise is to find the "essence" of the polynomial ring construction and present it in the form of a universal property.

Derek Elkins left SE
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