does the writing system for inuktitut fall under syllabary or something else?
Canadian Aboriginal syllabics, is the writing system used for Inuktitut and other Inuit dialects, basically all dialects of Cree, and sometimes Ojibwe. Despite the name, they are an abugida, not technically a syllabary, since the symbols for each syllable are not totally independent of each other (like with Japanese kana, a true syllabary), but rather you modify them predictably depending on the vowel sound.
They were actually inspired by another abugida, Devanagari, as well as the Cherokee syllabary (which is a true syllabary). By the way, the story of Sequoyah’s invention of the Cherokee syllabary is great and you should read it too.
Here’s a demonstration of one form of the syllabics from Wikipedia, which I’m going to screenshot so that the formatting doesn’t get messed up, but which you can also view in text there:
Also, here’s an example of Inuktitut syllabics on a Tim Horton’s sign in Iqualuit (source):