It really depends on the context of your case:
- Is it for a “power user” tool?
- Is it for single-visit context?
- Is it a technical website/store (like industry-specific store or something)?
Depending on the answers, you might want to keep your global search as-is, or adjust its behavior.
Narrowing down is normal
When you are looking for something on a website, you often start by visiting around to better understand what the website proposes.
It’s part of the human brain to see global before narrowing down information to reach a more precise goal.
Searching is not different: you search global with some keywords, then you want to adjust regarding the categories and filters the interface/store proposes.
That is for the usual behavior, it might not be the right solution for power-users or really technical domain related website/store.
Advanced filters for power-users
You global search might propose advanced keyword or syntax like Google does.
For instance, when you want result for a specific website, Google allows you to limit your search to a specific domain with this syntax:
site:stackexchange.com faceted filter site wide
You can also use modifier to change the meaning of your research, like:
-site:stackexchange.com "faceted filter" site wide
This one means “not in this site”, and I look for the exact term “faceted filter” into the results.
You could totally imagine keywords for your categories. Let’s take a shoes store example:
brand:martens size:44 color:black winter shoes
Could lead me to pre-filtered search for “winter shoes” of a brand containing “Martens” with a 44 size and black color. Your choice then to allow a bit of fuzziness around the results by providing with 44.5 and 43.5, black and dark blue shoes, for instance.
It doesn’t mean that the global research must only be the only way you can filter things, but it could be a good start to bring a powerful tool for your power-users, and keep the sidebar with filters for beginners, after a first more global search.
Of course, that’s only my opinion and point of view. Best is to analyse your context and ask to users 😊