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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