This is a few years old, but it's a good rundown of global cities:
http://www.lboro.ac.uk/gawc/world2012t.html
The list breaks down global cities into categories of alpha/beta/gamma. Then on the right there are sufficiency cities that are not global cities as such, but are big and developed enough to stand on their own to some extent. Winnipeg is in that category, and that sounds about right... Winnipeg is a city of major importance regionally, perhaps secondary importance nationally, and is a bit of a footnote globally, i.e. people know we're here, but Winnipeg is obviously not a city on the global radar the way that places like Moscow, Paris or Shanghai are.
Of course, you could always drill down into specific sub-categories where Winnipeg (or any city) has a massive sphere of influence... when it comes to the worlds of Inuit art, human rights museums or cities with extreme climates, we're world class baby.