Re: A category internal to itself

[Alex Simpson <Alex.Simpson@ed.ac.uk>]

A comment on Zhen Lin's reply to 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 ...

Just to remark that there is no need to add any assumption here. If a
locally cartesian closed category has a "universal fibration" as you
call it (called "generic family" in the reference below), then it is
degenerate. This appears in:

    Pitts, Andrew M.; Taylor, Paul.
    A note on Russell's paradox in locally Cartesian closed categories.
    Studia Logica 48 (1989), no. 3, 377?387. (MR1059248)

With best wishes,
Alex

-- 
Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK
Email: Alex.Simpson@ed.ac.uk             Tel: +44 (0)131 650 5113
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