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Identity and Categorification April 21, 2008

Posted by Alexandre Borovik in Uncategorized.
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A paper by  Andrei Rodin, arXiv:math/0509596 [pdf]. Appears to be relevant to our project.

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1. dcorfield - April 22, 2008

There’s quite a literature on this topic. See, e.g., Barry Mazur’s When is one thing equal to some other thing.

2. Alexandre Borovik - April 22, 2008

David — thanks for a reference to Mazur. Can you suggest more?

3. dcorfield - April 22, 2008

Three good accounts of the rationale for higher categories are An introduction to n-categories, From Finite Sets to Feynman Diagrams and Categorification.

Yuri Manin says in Georg Cantor and his heritage p. 8:

…the slow emergence of the following hierarchical picture. Categories themselves form objects of a larger category Cat morphisms in which are functors, or “natural constructions” like a (co)homology theory of topological spaces. However, functors do not form simply a set or a class: they also form objects of a category. Axiomatizing this situation we get a notion of 2–category whose prototype is Cat. Treating 2–categories in the same way, we get 3–categories etc.

The following view of mathematical objects is encoded in this hierarchy: there is no equality of mathematical objects, only equivalences. And since an equivalence is also a mathematical object, there is no equality between them, only the next order equivalence etc., ad infinitim (sic).

This vision, due initially to Grothendieck, extends the boundaries of classical mathematics, especially algebraic geometry, and exactly in those developments where it interacts with modern theoretical physics.


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