Part of the conservative opposition to the Islamic Center near Ground Zero being proposed by the Cordoba Initiative is the presumption that the name itself, Cordoba, is an indication of the group's intentions to impose Sharia law on the United States. For example, here's Newt Gingrich:
The true intentions of Rauf are also revealed by the name initially proposed for the Ground Zero mosque --“Cordoba House” -- which is named for a city in Spain where a conquering Muslim army replaced a church with a mosque. This name is a very direct historical indication that the Ground Zero mosque is all about conquest and thus an assertion of Islamist triumphalism which we should not tolerate.
Of course, this is the exact opposite of the reason the Cordoba Initiative claims it named itself "Cordoba Initiative":
Cordoba Initiative aims to achieve a tipping point in Muslim-West relations within the next decade, bringing back the atmosphere of interfaith tolerance and respect that we have longed for since Muslims, Christians and Jews lived together in harmony and prosperity eight hundred years ago.
So the notion that the use of Cordoba is meant to celebrate a period of "Islamist triumphalism" rather than a symbol of interfaith cooperation is in the eye of the beholder, but the organization says it's the latter. Whether or not you think that's an accurate description of that particular historical period, to think it's the former you have to ignore entirely the organization's stated intentions, and imagine that they deliberately named themselves Cordoba in order to celebrate Muslim domination over the West thinking no one would figure it out and expose their dastardly scheme, which is cartoon-villain stupid. It seems particularly silly given that the head of the organization has a record of doing exactly what he claims his organization is meant to do.