FOM: Vaughan Pratt's continua

[Vaughan Pratt]

From: cxm7 at po.cwru.edu (Colin McLarty)

>Vaughan goes on to describe categories *as* structures. ... I am
interested in foundational uses of categories as structures.  And I think
a lot of categorists are,
...
>But I have been defending the perspective of categories *of*
structures: the category of sets, the category of differentiable spaces, et c.

Actually it's neither "as" nor "of" but "with": categories *with*
structure.  A topos is a cartesian closed category with a specified
suboject classifier.  This is how you've been developing categories *of*
structures within the framework of categories *as* (possibly large)
structures, whose structure needs to be augmented as part of the program
of adequately approximating ZF in a categorical setting.

Any sharp demarcation between "as", "of", and "with" in this context
muddies the foundational claims of category theory.  These three
prepositions are not alternatives but facets of a single viewpoint:
start from the continuum end of the picture as simple and basic and add
structure and axioms as needed to build towards the more complex and
comprehensive collection end.  Set theory does all this in reverse: it
starts from collections as simple and basic and adds structure and axioms
as needed to build up to the more complex and comprehensive continuum end.

Vaughan Pratt