It is a search to which I've dedicated my writing.
Isn't it supposed to be the?
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Isn't it supposed to be "the"? Not necessarily. Using the indefinite article in this context is the most common usage. Examples:
This cause is special to me. It is a cause to which I've dedicated my whole life.
There are many great things about this idea. It is an idea which I'm prepared to die for.
Now logically, you might think that as a person has only one life, if they dedicate their whole life to something, it can only be to that one thing. It would then follow that this would be the thing that they dedicated their lives to.
But in reality, when people use this phrasing it is only for dramatic effect, and it is not intended to be exclusive. In other words, the intended meaning is not that this is the only thing the speaker dedicated their life to. So, the indefinite article, a, is in fact more appropriate.
When you say
It is the cause I've dedicated my book to.
You seem to be indicating that when looking for a cause to dedicate your book to, you found this specific one and figured it was a good cause to write a book about. Seemingly you wanted to write a book (e.g. about easily-preventable child diseases) and then discovered that MSF has programs with that exact goal, so you decide to dedicate your book to their cause.
When you say
It is a cause I've dedicated my book to.
I would feel you have been involved with that cause before you decided to write a book about it. It sounds more like "I found this cause a very important one, and to exemplify that importance, I tell you I dedicated a book to it / my life to it / etc.".
So actually, I would think that the form with the indefinite article transmits a stronger expression of importance. It's not just any search / cause / whatever, it is a very special one.