FIrst of all, I can't give you any MN-specific information, as I'm in MA.
However, I went ahead to post an answer because there is federal-law protection available to you that I am conversant in.
The term you need to use is "emotional support animal." The ADA does not protect you here, but the federal Fair Housing Act does. Under the Fair Housing Act, if you have a prescription for an emotional support animal, the landlord may not keep you from having the cat or charge you any extra fee or deposit for it. They can require a letter from your doctor, and they can charge you for any actual damage caused by the animal.
You can read a memo on this from the US Department of Housing and Urban Development at http://servicedogcentral.org/content/files/HUD%20FHEO%202-17-2011%20Assistance%20Animal%20Memo.pdf
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Emotional support animals and companion animals are not given the same level of protection as service animals. I suggest you re-tag/re-post your question with disability law related headings to get a further answer from an ADA attorney.
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Hello. Two laws involved in this issue are the Minnesota Fair Housing Act and American with Disabilities Act. Please confer with an attorney because the issues you describe are not clearcut in statute and case law and need some analysis.
HOUSING LAW ATTORNEY
I agree with Mr. Cowan's response. The Fair Housing Act provides that emotional support animals are a reasonable accomodation. Of course, that does not mean you can have any kind of animal and that the landlord cannot ask for an additional deposit. You should consult with a Minnesota attorney. There may be a local fair housing center in your community that could provide some assistance.