Re: A category internal to itself

[Colin McLarty <colin.mclarty@case.edu>]

In the set theory New Foundations (NF)  using Quine's type-level pairing (so
a pair has the same type in a stratification as its components) you can
define small categories and small functors the usual way.  Then, just as
there is a set of all sets, there is a small category of all small
categories.  This is not a tautology.  You have to verify a few things.

Notably, in this context there is a set of all small functors because there
is a set of all functions (yet the category of sets is not cartesian
closed, because it lacks evaluation functions).   Since a function is
stratified at the same level as its domain and codomain sets there is no
problem defining domain, codomain, composition, and identity-assigning
functors for this category.

This category is internal to itself.  This example is even left exact.  But
it is not cartesian closed.

Of course the consistency of NF is not settled.  But I think everyone
supposes it is equiconsistent with some more usual set theory (likely with
ETCS).

best, Colin




On Wed, Sep 4, 2013 at 5:23 AM, Andrej Bauer <andrej.bauer@andrej.com>wrote:

> Chatting at a conference, the question came up why there is no
> (non-trivial) category which is "internal to itself" (interpret this
> in some sensible sense). And over coffee we thought this must be well
> known, but not to us. Can somene shed some light on the matter?
>
> With kind regards,
>
> Andrej
>

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