Categorification

Details Categorification Categorification

1998-02-05

arxiv.org/abs/math/9802029v1

In Higher Category Theory, eds. Ezra Getzler and Mikhail Kapranov, Contemp. Math. 230, American Mathematical Society, Providen

Mathematics

Quantum Algebra

51 pages LaTeX with 9 encapsulated Postscript figures

Scientific paper

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.

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