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We just set up a non-profit for robotics in the county, the corporation was set up with the State of Arizona, and the non-profit status (501c3) was conferred by the federal IRS. You might want to get the IRS form 1023 and instructions and spend a lot of time reading it and talking about it, it's a great way to get to know others who will be in this ordeal with you
Reading the IRS stuff, it looked to me like one would set up a private foundation, but we got Public Charity status because the IRS agent suggested that would be more appropriate.
We set up an Association, with a mission of supporting technical and science education and awareness, by supporting robotics teams and clubs. We are rather limited in what we do, which is to buy robot parts and give them to robotics clubs, pay for entry fees to competitions, and pay for travel and lodging for students at competitions.
The whole point is to get the non-profit status so that large corporations will be willing to contribute money, as they can write it off.