Maybe there is no simple definition of $n$-category understandable for a physicist. Then I would like to know what are the trivial $0$-category, trivial $1$-category, trivial $2$-category, etc. How to obtain trivial $1$-category from trivial $0$-category? How to obtain trivial $2$-category from trivial $1$-category? etc
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After that I would like to know the definition of 0-category that allows us to obtain all the 0-categories, and the definition of 1-category that allows us to obtain all the 1-categories etc.
I am looking for a particular kind of n-categories. Based on the answer of André Henriques, it appears that the trivial 0-category that I am looking for is an one-dimensional vector space. A less trivial 0-category is a n-dimensional vector space, which is a composition of n trivial 0-categories. The trivial 1-category is a category of vector spaces, with only one simple object, which is a trivial object corresponding to an one-dimensional vector space, and the composition of several simple objects gives us a composite object which corresponds to a finite dimensional vector space.
I also guess that the trivial 0-category (ie an one-dimensional vector space) and the composite 0-categories (ie n-dimensional vector spaces) are all the 0-categories (of this type). The category of all vectors spaces is the "trivial" 1-category. All possible 1-categories (of this type) are fusion categories. Along this line, the "trivial" 2-category is the collection of all fusion categories. Then what are the most general 2-categories. We need a definition here.
The purpose of this question is not to find out an answer, but to find out a proper way to ask the question. I hope after some exchanges, I know what really is the question that I want to ask.