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I want to talk about several categories. Each category has only one URL.
Which one is correct and why?
- Category URLs
- Categories URL
- Categories URLs
It would be "category URLs"; you're using the noun category as an adjective, and when you do this you use the singular form in most cases, indeed even with words that are normally singular (hence "trouser press").
You could also use "categories' URLs", that is using the genitive to say "the URLs of the categories".
If you wanted talk about one URL that was covering several categories, only this second form would be clear, "categories' URL".
I'd also favour URI over URL in most cases today, though that's more a technical matter than an English one.
That's entirely dependent on the context: The most probable is "Category URLs", where "Category" is a specifier for the noun URL, and the noun takes the plural, because there are many instances of "Category URL". An example would be something like "I have created the categroy URLs, now all we need to do is fill in the information". If the term used is "Categories URL" then it implies that there is an URL named "Categories". If the term used is "Categories URLs" then it implies each URL in a set is called "Categories", which is possible, but for specific usage only. An example would be a set "Categories A-De", "Categories Df-Ol" and "Categories Om-Zz" for a set of URLs that split a number of categories according to alphabetic naming. Possible, but specific.