Who would have the courage to tell the truth, that the sentence is wrong because words are misused, and it ultimately does not make sense? Is it just meaningless rubbish? "Interrogate" is transitive - it needs an object. The object is normally either a person or a computer file. It is meaningless to write "interrogate your place in the city". We can negotiate a hairpin bend or some tricky problems, but we cannot "negotiate our place in the city". If the writer wishes to say, "Art helps you to think about what the city means to you", then he should just write it like that. I don't think art is a good excuse for bad writing. When people are reading bad English, we should have the courage and kindness to tell them, so that they may find some correct English to read.

I think they're used a bit more abstractly in this context, especially since it sounds like an article related to art. Interrogate may be being used in the sense of "asking questions of". Perhaps something like looking at the art in the city and using it to determine the culture of the city and the people there, and the kind of things they are trying to express. Interrogate might be being used instead of investigate in order to emphasize the thoroughness of this inquiry. I think in this sense negotiate might mean trying to find, change, or establish your place in the city, in the sense of making a name for yourself and/or finding where you fit in to the life and culture there. In my experience, articles written about art, and authored by artistically minded folks, tend to use more abstract meanings of words that are hard to nail down to a concise definition. They seem to rely more heavily on context and the reader's familiarity with the art world and the tendencies of the people in it.

We do not negotiate with a terrorist, but we can interrogate one.

In away..but interrogate here would mean to ask questions more aggressively, about the city that stems from art. The definition of "Negotiate" here would mean "to find your way through something," this case in a city. So you could say: Art is the medium which is asking questions and allowing you to find your way through the city. That is at least how I would interpret the meaning in this case. Perhaps others could give other suggestions. Hope that helps!

These words are not the same. To interrogate is what the police do to a suspect when they think he has committed a crime and they want more information. To negotiate is to discuss when you are trying to reach an agreement, like in trade, or between two countries, and each country tries to get a better deal. At the end of the negotiation you sign an agreement.