The term “cardinalate” does not appear to be a standard word in the English language and may not have a widely recognized definition. It might relate to the concept of “cardinal” in a specific context, such as in mathematics (referring to cardinal numbers) or in the context of the Catholic Church (referring to cardinals in the hierarchy). However, without further context or usage, it is difficult to provide a precise definition. If you have a specific context in mind, please provide it, and I can offer more tailored information.