Re: question about discrete op-fibrations

[David Spivak <dspivak@gmail.com>]

Hi Mark,

Nice work; thank you for the simple answer and good explanation.

I hope this isn't annoying, but what if I change the problem somewhat
and take DOF(C) to be the full subcategory of Cat_{C/} spanned by the
discrete opfibrations C-->D? Again I want to know whether DOF(C) has a
terminal object. Under this definition, by setting C=empty-category we
get DOF(C)=Cat, which does have a terminal object.

Thanks,
David

On Wed, Feb 1, 2012 at 5:29 PM, Mark Weber
<mark.weber.math@googlemail.com> wrote:
> Dear David
>
> I'll assume by the coslice DOF_{C/} you mean the category whose objects are discrete opfibrations C --> D, and whose arrows are strictly commuting triangles under C.
>
> In that case the answer to your question is no. When C is empty, DOF_{C/}  is just your category DOF, of small categories and discrete opfibrations between them, and DOF lacks a terminal object. For suppose that D is terminal in DOF. Then for any set X, there is a discrete opfibration I(X) --> D, where I(X) is the category obtained from X by freely adding an initial object. That is, the objects of I(X) are the elements of X together with one additional object 0, and one has a unique arrow 0 --> x for all x in X.
>
> If F:I(X) --> D is a discrete opfibration, then F(0) is an object of D such that the cardinality of the set of all arrows with source F(0) is that of X. Thus since D is terminal in DOF, for any set X there is an object x of  D such that the cardinality of the set of all arrows with source x is that  of X. This contradicts the smallness of D.
>
> With best regards,
>
> Mark Weber
>

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