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Categorification
arxiv.org/abs/math/9802029v1
Abstract
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 Ω k S k . We conclude by describing a definition of weak n-categories based on the theory of operads.
Abstract
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 Ω k S k . We conclude by describing a definition of weak n-categories based on the theory of operads.
51 pages, ebook
First published February 5, 1998
2 people want to read
About the author
John C. Baez
15 books30 followershttp://en.wikipedia.org/wiki/John_Baez
JOHN BAEZ is a mathematical physicist working on quantum gravity using the techniques of "higher-dimensional algebra". A professor of mathematics at the University of California, Riverside, he enjoys answering physics questions on the usenet newsgroup sci.physics.research, and also writes a regular column entitled "This Week's Finds in Mathematical Physics". (The Third Culture: http://www.edge.org/3rd_culture/bios/... )
JOHN BAEZ is a mathematical physicist working on quantum gravity using the techniques of "higher-dimensional algebra". A professor of mathematics at the University of California, Riverside, he enjoys answering physics questions on the usenet newsgroup sci.physics.research, and also writes a regular column entitled "This Week's Finds in Mathematical Physics". (The Third Culture: http://www.edge.org/3rd_culture/bios/... )
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