FOM: small category theory

Kanovei kanovei at
Wed May 5 03:29:41 EDT 1999

Date: Wed, 05 May 1999 00:04:05 -0700
From: Till Mossakowski <till at>

> Several ideas for 
> the foundation of category theory have been proposed: (...)

1) As for vNBG, why not to consider a type theoretic 
class extension of ZFC, ie, level 0 = sets, level 1=
classes of sets, level 2 = classes of level 1 elements, 
etc, but, to stick to ZFC, still only set quantifiers 
are allowed in Comprehension for all levels. 
Furthermore, one can then try to abolish the levels 
and consider this as a kind of Z extension of the ZFC 
set universe, still with only set quantifiers working. 

2) How about the following Borel-motivated approach ? 

Consider only those categories whose objects and 
morphisms between them are Borel structures in 
separable metric spaces. 

After all, the major body of mathematics can be 
coded by Borel sets, if not by natural numbers. 

Is there an immediate drawback ? 

This is clearly not for those who develops the 
category theory for its own sake, but could it 
work for number-theoretic applications ? 


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