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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