* question about discrete op-fibrations
@ 2012-02-01 0:03 David Spivak
2012-02-01 23:29 ` Mark Weber
` (2 more replies)
0 siblings, 3 replies; 5+ messages in thread
From: David Spivak @ 2012-02-01 0:03 UTC (permalink / raw)
To: categories list
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: question about discrete op-fibrations
2012-02-01 0:03 question about discrete op-fibrations David Spivak
@ 2012-02-01 23:29 ` Mark Weber
[not found] ` <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com>
2012-02-02 10:22 ` Thorsten Palm
2 siblings, 0 replies; 5+ messages in thread
From: Mark Weber @ 2012-02-01 23:29 UTC (permalink / raw)
To: David Spivak; +Cc: categories list
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
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: question about discrete op-fibrations
[not found] ` <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com>
@ 2012-02-02 5:51 ` David Spivak
2012-02-03 14:07 ` Thorsten Palm
0 siblings, 1 reply; 5+ messages in thread
From: David Spivak @ 2012-02-02 5:51 UTC (permalink / raw)
To: Mark Weber; +Cc: categories list
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
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: question about discrete op-fibrations
2012-02-01 0:03 question about discrete op-fibrations David Spivak
2012-02-01 23:29 ` Mark Weber
[not found] ` <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com>
@ 2012-02-02 10:22 ` Thorsten Palm
2 siblings, 0 replies; 5+ messages in thread
From: Thorsten Palm @ 2012-02-02 10:22 UTC (permalink / raw)
To: David Spivak; +Cc: categories list
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: question about discrete op-fibrations
2012-02-02 5:51 ` David Spivak
@ 2012-02-03 14:07 ` Thorsten Palm
0 siblings, 0 replies; 5+ messages in thread
From: Thorsten Palm @ 2012-02-03 14:07 UTC (permalink / raw)
To: David Spivak; +Cc: Mark Weber, categories list
[ 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.
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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2012-02-01 0:03 question about discrete op-fibrations David Spivak
2012-02-01 23:29 ` Mark Weber
[not found] ` <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com>
2012-02-02 5:51 ` David Spivak
2012-02-03 14:07 ` Thorsten Palm
2012-02-02 10:22 ` Thorsten Palm
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