Talk:Universal property

Relation to adjoint functors
The C & D categories are the opposite way around to the definition on the adjoint functors page. This is a bit confusing. 51.6.165.9 (talk) 12:22, 3 October 2018 (UTC)

Should the name of this article be changed to "Universal Morphisms"?
I think that this article name should be changed.
 * The term "universal property" is a colloquial term used casually in passing conversations regarding category theory. It's not an agreed upon term; for example, there is also "universal construction." So some people don't even call this concept "universal property."
 * Aside from the arbitrary nomenclature, to even talk about a universal property, you need to talk about a universal morphism. When anyone uses this concept rigorously, it's always in the context of a universal morphism. People don't just talk about universal properties without talking about universal morphisms unless they're doing so with the assumption that the audience understands what that universal morphism would be.
 * The term "universal properties" aren't even the meat of what's going on. The meat of what's going on is the concept of a universal morphism, its relation to comma categories, the solution set condition which guarantees the existence of initial objects in categories, the natural lead to adjoint functors, etc. The focus should be on universal morphisms themselves, and not just the concept of a universal property.
 * According to google, nobody is looking up "Universal Property," and then clicking on this wikipedia page. In fact, nobody is googling "universal property" and then clicking on anything math related at all. This is because when one googles "universal property," a bunch of things unrelated to math show up. Based on google's page rank algorithm, this wouldn't be the case if people were googling "universal property" with the intention of researching math websites.
 * Also, the fact that nobody is googling "Universal Property" with the intentions of researching math is a possible indicator that nobody in math is consistently utilizing this term. Hence there's no point in naming it this, when we could name it something more relevant.
 * On the other hand, people ARE googling "Universal Morphism" and clicking on math related things. In fact, the top google result for "Universal Morphism" is literally the "Universal Property" wikipedia page (isn't that hilarious?). All search results are math related in this case. Therefore, it is clear that people are encountering the word "universal morphism" more than "universal property," and that when people are wanting to know more about universal concepts in category theory, they use the search key words "universal morphism."

Basically, as this stands, this wikipedia article is sitting as like the 10th result for the google search result "Universal Property" (the rest of the results are not even math related). Meanwhile, it is the first search result for "Universal Morphism." In addition, what one truly cares about are the universal morphisms themselves; not just the fact that they have a cool property. — Preceding unsigned comment added by Ltrujello (talk • contribs) 08:11, 20 January 2020 (UTC)

Adding an informal definition in the lede
I've added an informal description of the concept in the lede, to follow MOS:LEAD suggestion to make the introduction "stand on its own as a concise overview of the article's topic". With the previous version, one was expected to read and understand several other theories as well as this own article definitions before getting to know what the concept is about.

This description is my current understanding of what universal morphisms and properties are used for. Can someone check that this informal description is not too far off, and that it doesn't make any false statement? Diego (talk) 19:27, 11 March 2022 (UTC)

Adding an informal definition in the lede
I've added an informal description of the concept in the lede, following MOS:LEAD suggestion to make the introduction "stand on its own as a concise overview of the article's topic". With the previous version, one was expected to read and understand several other theories as well as this own article definitions before getting to know what the concept is about.

This description is my current understanding of what universal morphisms and properties are used for. Can someone check that this informal description is not too far off, and that it doesn't make any false statement? Diego (talk) 19:27, 11 March 2022 (UTC)