What part of speech is the 「〜たい」 form of a verb?

Question

To say that one wants to do something, one uses the 「〜たい」 form of the verb, for example, 「日本に　行きたい　です」.

In the 「〜たい」 form, the word behaves like an adjective, using similar patterns such as 「〜くない」 for negation and 「〜かった」 for the past tense.

But what is the part of speech of the 「〜たい」 form word? Is it still a verb, albeit one that "acts" like an adjective? Or is it in fact an adjective, and if so, is there a grammatical concept that captures this "adjectivalization"?

Answer 1

(I don't know if this helps but just for your information...) In Japanese schools we are taught this way:

たい is a 形容詞型助動詞 (an auxiliary that conjugates like an i-adjective).

It conjugates to たく, たかっ*, とう in 連用形 (continuative forms).

So 行きたくない consists of:

行き -- 連用形 of the verb 行く
たく -- 連用形 of the auxiliary たい
ない -- 終止形 (terminal form) of the 補助形容詞 (subsidiary adjective) ない

And 行きたかった consists of:

行き -- 連用形 of the verb 行く
たかっ -- 連用形 of the auxiliary たい
た -- 終止形 of the 過去の助動詞 (past tense auxiliary) た

So 行く is a verb and acts as a verb, but 行きたい acts as an i-adjective and conjugates to 行きたく, 行きたかっ, 行きたけれ, etc. (but it's not that 行きたい is an i-adjective.)

_{*たかっ was originally たかり. たかり(+た) changed to たかっ(+た) due to 促音便.}