Probably a diaeresis, a diacritic which indicates that two vowels aren't to be read as a digraph or diphthong. It's fancy and pretentious, and frequently seen in New Yorker articles.

I went through a phase of this because it made sense and I didn’t realize that it wasn’t standard English.

It is standard English, just somewhat antiquated at this point. Still, you see it quite a bit with "naïve".

> Why do you write 'cooperation' with an umlaut?

I grew up speaking French, English and a variant of German and have trouble not reading the double-o as a long vowel. After that autocorrect picked up on it and it doesn't bother me enough to change it.

In English words it's a diaeresis.