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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 14th 2010
   • (edited Oct 14th 2010)
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   Author: DavidRoberts
   Format: MarkdownItexPeople may have noticed my question on MO and on the categories mailing list, so I'm not going to labour the point too much. This thread is really to record my thoughts and welcome input from the peanut gallery. ---- Note $Cat$ is the 1-/2-category of small categories. The functor $Obj:Cat \to Set$ is a fibration (amongst other things), because one can perform 'change of base' given a category $C$ and a function $f:D \to C_0$, we have a cartesian arrow $F:C[f] \to C$. Given a natural isomorphism between $F,G$ arising from $f,g:D\to C_0$, we have a canonical isomorphism between $C[f]$ and $C[g]$ over $C$. I think that there is a more general behaviour for non-invertible 2-arrows, and for 2-arrows from and/or to $F$, but I will think about this later (busy, busy at work). Now there is a notion of 2-fibration given by Hermida, and one given by Bakovic. I'll examine whether $Obj$ is a 2-fibration by either of these definitions.

   People may have noticed my question on MO and on the categories mailing list, so I’m not going to labour the point too much. This thread is really to record my thoughts and welcome input from the peanut gallery.

   ---

   Note $Cat$ is the 1-/2-category of small categories.

   The functor $Obj:Cat \to Set$ is a fibration (amongst other things), because one can perform ’change of base’ given a category $C$ and a function $f:D \to C_0$, we have a cartesian arrow $F:C[f] \to C$. Given a natural isomorphism between $F,G$ arising from $f,g:D\to C_0$, we have a canonical isomorphism between $C[f]$ and $C[g]$ over $C$. I think that there is a more general behaviour for non-invertible 2-arrows, and for 2-arrows from and/or to $F$, but I will think about this later (busy, busy at work).

   Now there is a notion of 2-fibration given by Hermida, and one given by Bakovic. I’ll examine whether $Obj$ is a 2-fibration by either of these definitions.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 14th 2010
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   Author: Mike Shulman
   Format: MarkdownItexIf we make Cat into a 2-category in the usual way, and Set into a 2-category with only identity 2-cells, then Obj is not a 2-functor. So if it is to be a 2-fibration, some cleverness must be involved. I also don't quite understand the putative pseudofunctor $Set \to 2 Cat$. What 2-category is to be associated to a set A? The pseudofunctor $Set^{op} \to Cat$ coming from the 1-fibration-ness of Obj sends A to the category whose objects are "categories with object set A" and whose morphisms are functors that are the identity on objects. Do you want to enhance that category to a 2-category somehow?

   If we make Cat into a 2-category in the usual way, and Set into a 2-category with only identity 2-cells, then Obj is not a 2-functor. So if it is to be a 2-fibration, some cleverness must be involved.

   I also don’t quite understand the putative pseudofunctor $Set \to 2 Cat$. What 2-category is to be associated to a set A? The pseudofunctor $Set^{op} \to Cat$ coming from the 1-fibration-ness of Obj sends A to the category whose objects are “categories with object set A” and whose morphisms are functors that are the identity on objects. Do you want to enhance that category to a 2-category somehow?

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 14th 2010
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   Author: DavidRoberts
   Format: MarkdownItexIf one restricts to the 2-arrows of Cat that make it a 2-category over Set (these are natural transformations with component map landing in endomorphisms of the codomain) then this precludes the sort of phenomenon I'm considering. I really am guessing here, I don't know what this means if anything. But this seems to me to be at least a curio that no one else has bothered to look at, which generalises to internal categories nicely, and is a useful fact.

   If one restricts to the 2-arrows of Cat that make it a 2-category over Set (these are natural transformations with component map landing in endomorphisms of the codomain) then this precludes the sort of phenomenon I’m considering. I really am guessing here, I don’t know what this means if anything. But this seems to me to be at least a curio that no one else has bothered to look at, which generalises to internal categories nicely, and is a useful fact.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 14th 2010
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   Author: Mike Shulman
   Format: MarkdownItex> natural transformations with component map landing in endomorphisms of the codomain I don't understand what that means, can you clarify?

   natural transformations with component map landing in endomorphisms of the codomain

   I don’t understand what that means, can you clarify?

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 14th 2010
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   Author: DavidRoberts
   Format: MarkdownItexTwo functors $F,G:C \to D$ that have the same object-component map $F_0=G_0=f$ can have a natural transformation $a$ between them without being equal (I was trying to be brief in my parenthetical remark, but it backfired). We need $a_c:G(c) \to F(c)$ which is just an endomorphism of $f(c)$. Hence as a map $C_0 \to D_1$ it lands in the arrows of the subcategory $Endo(D) \hookrightarrow D$ consisting of all objects and their endomorphisms. But as I said, this precludes what I started with, namely two _different_ maps on objects. It needs a bit of cogitatin' on my behalf....

   Two functors $F,G:C \to D$ that have the same object-component map $F_0=G_0=f$ can have a natural transformation $a$ between them without being equal (I was trying to be brief in my parenthetical remark, but it backfired). We need $a_c:G(c) \to F(c)$ which is just an endomorphism of $f(c)$. Hence as a map $C_0 \to D_1$ it lands in the arrows of the subcategory $Endo(D) \hookrightarrow D$ consisting of all objects and their endomorphisms. But as I said, this precludes what I started with, namely two different maps on objects. It needs a bit of cogitatin’ on my behalf….

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeOct 15th 2010
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   Author: TobyBartels
   Format: MarkdownItex>Two functors $F,G:C \to D$ that have the same object-component map $F_0=G_0=f$ can have a natural transformation $a$ between them without being equal The simple familiar case of this is when $C,D$ are (deloopings of) monoids. Two monoid homomorphisms can have a natural transformation, even a natural isomorphism, between them without being equal. We can fix this by defining a monoid to be, not a one-object (or connected) category, but such a category equipped with an object, requiring our functors (trivially) and our natural transformations (not so trivially) to preserve this object. (I think that this idea is due to Mike.) A *pointed* (in the sense of preserving the chosen object) natural transformation between monoid homomorphisms must be the identity transformation between equal homomorphisms. Like Mike, I don't really understand what you're thinking David, but maybe you can do something with that. That is, specify some sort of behaviour on objects first, and fix that while varying other things.

   Two functors $F,G:C \to D$ that have the same object-component map $F_0=G_0=f$ can have a natural transformation $a$ between them without being equal

   The simple familiar case of this is when $C,D$ are (deloopings of) monoids. Two monoid homomorphisms can have a natural transformation, even a natural isomorphism, between them without being equal.

   We can fix this by defining a monoid to be, not a one-object (or connected) category, but such a category equipped with an object, requiring our functors (trivially) and our natural transformations (not so trivially) to preserve this object. (I think that this idea is due to Mike.) A pointed (in the sense of preserving the chosen object) natural transformation between monoid homomorphisms must be the identity transformation between equal homomorphisms.

   Like Mike, I don’t really understand what you’re thinking David, but maybe you can do something with that. That is, specify some sort of behaviour on objects first, and fix that while varying other things.

7. • CommentRowNumber7.
   • CommentAuthorRodMcGuire
   • CommentTimeOct 15th 2010
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   Author: RodMcGuire
   Format: MarkdownItex@DavidRoberts #1 >People may have noticed my question on MO and on the categories mailing list, If you are going to mention something on MO you need to link to it so people directly know (now and in the future) what you are talking about. <a href="http://mathoverflow.net/questions/41383/obj-is-a-grothendieck-fibration-but-cat-is-a-2-category">Obj is a Grothendieck fibration, but Cat is a 2-category…</a>

   @DavidRoberts #1

   People may have noticed my question on MO and on the categories mailing list,

   If you are going to mention something on MO you need to link to it so people directly know (now and in the future) what you are talking about.

   Obj is a Grothendieck fibration, but Cat is a 2-category…