Re: A category internal to itself

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

Perhaps this is not really what you want, though.  What I would like to
know, related to this, is can there be a non-trivial category internally
fibred over itself?

This question, like your question, leaves some room to ask just what it
means.

If we just try to Grothendieck-fiber this category of all categories in NF
over itself we meet type problems.  A type-level notion of pair will not
let us form the set of all pairs of a category and an object in it.  Maybe
a routine NF trick gets around this, maybe not.  I wanted to share this
thought, and clarify it for myself, before plunging into the world of NF
tricks.

best, Colin


On Wed, Sep 4, 2013 at 6:11 PM, Colin McLarty <colin.mclarty@case.edu>wrote:

>
> 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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