Re: A category internal to itself

[Zhen Lin Low <zll22@cam.ac.uk>]

Dear Andrej,

To begin, consider a category C with finite limits. Suppose C has an
internal category U such that the externalisation of U as a C-indexed
category (or category fibred over C) is equivalent to the
self-indexing of C. Since U is locally small as a C-indexed category,
the self-indexing of C has the same property, so we deduce that C is
locally cartesian closed.

We have a universal fibration el U -> ob U (by restricting the
fibration mor U -> ob U x ob U), so it follows that every object X
admits a monomorphism X -> el U. Now, if we add the assumption that C
(or U) is well-powered as a C-indexed category, then C must be an
elementary topos. But then the existence of el U implies that the
internal logic of C is inconsistent, so C must be the degenerate
topos.

It appears we need to relax the notion of "internal" to get something
more reasonable. Here is one idea: instead of taking just one internal
category, we take a (large) filtered diagram of them. More precisely,
let U be a diagram of shape J in the category of internal categories
in C, where J is filtered and the transition functors are (internally)
fully faithful, and define a C-indexed category whose fibre over X is
the (external) category colim Hom(X, U). When C is an elementary
topos, there exists a diagram U such that this construction yields a
C-indexed category that is equivalent to the self-indexing of C: take
J to be the poset of all finite subsets of ob C, and take as the
internal category at a finite set {X_1, ..., X_n} of objects of C to
be the internal full subcategory whose objects are the subobjects of
the power object P(X_1 + ... + X_n). In the converse direction, if
such a diagram of internal categories exists, then one can still
deduce (from the condition on transition functors) that the
self-indexing of C is locally small.

But perhaps there is a strange locally cartesian closed category out
there that is self-internal in the naive sense.

Best regards,
--
Zhen Lin

On 4 September 2013 10:23, 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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