question about discrete op-fibrations

question about discrete op-fibrations

[David Spivak]

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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Re: question about discrete op-fibrations

[Mark Weber]

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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Re: question about discrete op-fibrations

[David Spivak]

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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Re: question about discrete op-fibrations

[Thorsten Palm]

[ To all, moderator in particular, and with apologies to David:
   Please ignore my previous message. It contains a seriously wrong
   piece of information. ]

Dear David,

This time the answer is "yes" for discrete C, "no" otherwise.

If C is discrete, the functor !_C : C-->1 is a discrete
op-fibration, so that the terminal objet (1,!_C) of Cat_{C/}
belongs to DOF(C).

Now let C contain a non-trivial morphism u : c_0-->c_1 and suppose
that (T,z) is terminal in DOF(C). Construct a category C_{c_1} by
freely adding to C an object d and a morphism v : d-->c_1. Then we
have an inclusion j : C-->C_{c_1}, which is a discrete op-fibration.
We obtain two morphisms (C_{c_1},j)-->(T,z) in DOF(C) by applying z on
the subcategory C and sending v to z(u) and the identity of z(c_1),
respectively. By terminality of (T,z) they have to be the same. But
since z is an op-fibration, z(u) cannot be an identity ---
contradiction.

(I hope Mark hasn't beaten me to it again.)

Thorsten


David Spivak hat am 01.02.12 geschrieben:

>
> 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.
>







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Re: question about discrete op-fibrations

[Thorsten Palm]

David Spivak hat am 31.01.12 geschrieben:

>
> 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/}?

Answer: No.

The "op-" is irrelevant; let us look at DF_{C/} instead. There is not
even a weakly terminal object, for the same reason for which there
isn't one in DF itself. (Suppose (T,z) is one. Let A be an arbitrary
small category having a terminal object a. From the object (C+A,incl)
of DF_{C/} we get a discrete fibration f : A-->T. But then for t =
f(a) the slice category T_{/t} is isomorphic to A. There are not
enough t to accommodate all possible A.)

Thorsten Palm




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