TITLE: Why is it difficult to define n-category?
QUESTION [10 upvotes]: Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally.
If we have a definition for 1-categories (category) and a definition for 2-categories then surely we can abstract this to 3-categories, 4… all the way to infinity. From what I understand the main obstacle in defining the n-category is that two definitions of a category are equivalent if the category of said categories is equivalent, and so we are stuck in a loop. (I may not be communicating what I mean properly here..) But is it not the case that the infinite-category contains the n-category for all n, allowing us to stop the loop of cyclically defining categories? So my question is, what are the main obstacles in creating a definition for the n-category? Please note that I have had limit exposure to category theory (adjunction's, Yoneda…) and much less exposure to higher category theory, so a not overly technical explanation would be appreciated.

REPLY [3 votes]: As Qiaochu said, it's all about the coherence diagrams. The first non-trivial one is the pentagon axiom in the definition of a monoidal category, which also appears in the definition of a (weak) $2$-category. An excellent introduction into higher categories is given in

John Baez, James Dolan, Categorification, http://arxiv.org/abs/math/9802029

If you want to dive more into the subject, you might want to consult

Tom Leinster, Higher operads, Higher categories, http://arxiv.org/abs/math/0305049