Re: question about discrete op-fibrations

[Mark Weber <mark.weber.math@googlemail.com>]

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



On 01/02/2012, at 11:03 AM, David Spivak <dspivak@gmail.com> wrote:

> Hi all,
> 
> Here's a quick question perhaps someone here can answer easily.
> 
> Let DOF denote the category whose objects are small categories C,D,
> etc. and in which Hom(C,D) is the set of discrete op-fibrations C-->D.
> For a category C, let DOF_{C/} denote the coslice over C.
> 
> Question: Does there exist a terminal object in DOF_{C/}?
> 
> Thanks!
> David
> 


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