In this question When can adjectives be used with the particle "no"? I was taught that "next" in Japanese is a noun. In English, the definition of adjective is "A describing word which qualifies a noun, a noun phrase or a pronoun". As far as I can tell, "next" in the phrase "tsugi no sekai" is qualifying a noun, so it seems either there are some other different definitions of adjective in English, or the definition of adjective in Japanese is different than the English one. Which is the definition of adjective in Japanese?
The definition of adjective varies from linguist to linguist, from textbook to textbook. First, not every "describing word which qualifies a noun" is called an adjective, even in English. Just because you can say "credit card" and "phone number" doesn't mean "credit" and "phone" are English adjectives.
In traditional Japanese grammar taught in Japanese schools, 次 is a noun, and の is an independent linking particle similar to "of" in English. As a native speaker of Japanese, I had never thought 次 could be an adjective before I started visiting this site. In other words, 次の世界 is structured more like "book of chemistry" rather than "next book". (Can you call "of chemistry" an adjective even though it modifies a noun?)
However, many people who learn Japanese as a second language like to treat 次の as an adjective, calling it a no-adjective. The concept of no-adjective is actually very "foreign" to native Japanese speakers. Nevertheless, jisho.org categorizes 次 as a no-adjective, and you can say so, too. Some color names such as 緑 are other typical examples of no-adjectives.
Moreover, you will soon come across articles that say "Na-adjectives are not adjectives but actually nouns" or even "There are no true adjectives in Japanese; only verbs and relative clauses exist" (like this). Certainly confusing at first, but this post from a structural linguist is worth taking a look at. You don't have to wonder why だ is not used with i-adjectives once you accept they are actually verbs.
- i-adjectives (赤い, 美しい): Safely considered as adjectives both by average English and Japanese speakers. However a few experts say it's technically a verb. 形容詞.
- na-adjectives (簡単, ハンサム): As a Japanese learner you can usually safely call them adjectives. But some say it's technically a noun. 形容動詞.
- no-adjectives (緑, 次): As a Japanese leaner you may call them "so-called no-adjectives", but most Japanese people believe it's a noun. Even some western people dislike this concept.
- attributives (大きな, 色んな): They definitely modify nouns, but don't call them adjectives if you want to avoid confusion. 連体詞.