  Categorification is the process of finding category-theoretic analogs of
set-theoretic concepts by replacing sets with categories, functions with
functors, and equations between functions by natural isomorphisms between
functors, which in turn should satisfy certain equations of their own, called
`coherence laws'. Iterating this process requires a theory of `n-categories',
algebraic structures having objects, morphisms between objects, 2-morphisms
between morphisms and so on up to n-morphisms. After a brief introduction to
n-categories and their relation to homotopy theory, we discuss algebraic
structures that can be seen as iterated categorifications of the natural
numbers and integers. These include tangle n-categories, cobordism
n-categories, and the homotopy n-types of the loop spaces Omega^k S^k. We
conclude by describing a definition of weak n-categories based on the theory of
operads.

Quantum Algebra;Category Theory