A condition for functors to reflect orthogonality

A condition for functors to reflect orthogonality

[Zhen Lin Low]

Dear categorists,

I am wondering if the following property of a functor U : C -> D has a name
in the literature:

* For every lifting problem in C and any solution in D to the image under
U, there is a unique solution in C whose image under U is that solution.

More precisely:

* For any morphisms X -> Y and Z -> W in C, the induced commutative diagram

      C(W, X) ------> C(Z, X) \times_{C(Z, Y)} C(W, Y)
         |                            |
         |                            |
         v                            v
     D(UW, UX) --> D(UZ, UX) \times_{D(UZ, UY)} D(UW, UY)

   is a pullback square.

Of course, any fully faithful functor has the property in question; a less
trivial example is the projection from a (co)slice category to its base.
Every functor between groupoids has this property, so they need not be
faithful. One also notes that the class of functors with this property is
closed under composition.

It is not hard to see that if a functor has the above property, then it
reflects both orthogonality and weak orthogonality in the naive sense. The
converse is false. Nonetheless, my inclination is to call these functors
"orthogonality-reflecting".

Best wishes,
--
Zhen Lin


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Re: A condition for functors to reflect orthogonality

[Jean Bénabou]

Dear Zhen Lin,
 
Before I answer your question let me give names to arbitrary maps of C:
f: X --> Y,  g: Z --> W,  x: Z -->X,  y: W --> Y and h: W --> X . Your condition reads:
for every pair (f,g) and every triple (x,y,h') where:
  yg = fx and  h': UW --> UX  satisfies :  h' Ug = Ux  and  Uf h' = Uy,
there exits a unique h: W -->X  such that:  Uh = h',  hg = x  and fh = y . 
One verifies first that this condition is satisfied for all pairs (f,g) iff it is satisfied for f arbitrary and g is an identity.
In that case that means that f is what I call hypercartesian (which in the anglo-american literature is called cartesian). Since f is arbitrary , the condition becomes: 
(i) Every map of C is hyper cartesian.
Let me call U locally full and faithful (lff) iff for every object X of C the obvious functor  
C/X --> D/UX  is full and fathull.
In my mail to Joyal and the catgory list, dated July 28 I already mentioned that (i) is equivalent to
(ii) U is lff.
I also said, it is obvious, that U full and faithful => U is lff . 
I mentioned also the case of groupoids, with a sharper result than the one you stated, namely: If C is a groupoid, every functor U: C --> D, where D is arbitrary, is lff.

Let me add a remark which was not in my mail to Joyal, namely, the previous property characterizes groupoids. More precisely we have:

PROPOSITION 1. Let C be a category. The following are equivalent:
(i) C is a groupoid
(ii) Every functor with domain C is lff
(iii) The unique functor C --> 1 is lff.

There are MANY MORE properties of lff functors which would be too long to give here.
Let me mention a few which are not in my mail to Joyal.
  The following theorem generalizes greatly the previous proposition.

THEOREM. Let  U: C --> D be a fibration. The following are equivalent:
(i) U is lff
(ii) All the fibers of U are groupoids.
(iii) U is conservative (i.e. reflects isomorphisms).

Such fibrations are very important. Because of (ii) they have sometimes been called groupoid fibrations. In particular, it follows from (iii), that for such a fibration if D is a groupoid so is C.

You said that the functors satisfying your condition are stable by composition. This result can be strengthened since we have:

PROPOSITION 2. Let  U and V be functors such that the composite UV is defined. 
If U is lff, then UV is lff iff V is.

The following result is easy to prove but nevertheless important for many theoretical reasons.

THEOREM 2. lff functors are stable by pull back along any functor.

I could add many significant results, in particular about cartesian functors, or orthogonality but this mail is already a bit long, and I apologize for this length.
Thus  there is no need to give a name to the property you mentioned, locally full and faithful describes precisely this property.

Best wishes,
Jean 


Le 4 août 2014 à 20:24, Zhen Lin Low a écrit :

> Dear categorists,
> 
> I am wondering if the following property of a functor U : C -> D has a name
> in the literature:
> 
> * For every lifting problem in C and any solution in D to the image under
> U, there is a unique solution in C whose image under U is that solution.
> 
> More precisely:
> 
> * For any morphisms X -> Y and Z -> W in C, the induced commutative diagram
> 
>      C(W, X) ------> C(Z, X) \times_{C(Z, Y)} C(W, Y)
>         |                            |
>         |                            |
>         v                            v
>     D(UW, UX) --> D(UZ, UX) \times_{D(UZ, UY)} D(UW, UY)
> 
>   is a pullback square.
> 
> Of course, any fully faithful functor has the property in question; a less
> trivial example is the projection from a (co)slice category to its base.
> Every functor between groupoids has this property, so they need not be
> faithful. One also notes that the class of functors with this property is
> closed under composition.
> 
> It is not hard to see that if a functor has the above property, then it
> reflects both orthogonality and weak orthogonality in the naive sense. The
> converse is false. Nonetheless, my inclination is to call these functors
> "orthogonality-reflecting".
> 
> Best wishes,
> --
> Zhen Lin
> 
> 



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Re: A condition for functors to reflect orthogonality

[Zhen Lin Low]

Dear Jean,

Thank you for your reply, but I'm afraid I do not understand your
first paragraph. The condition I'm interested in is self-dual, but the
notion of locally fully faithful functors is not. For instance, the
projection from a coslice category to its base is not locally fully
faithful. It is an interesting suggestion though. This seems to be the
correct version:

* U : C -> D is a functor such that, for every object A in C, the
induced functor A/C -> UA/D is locally fully faithful.

Thus one might say "colocally locally fully faithful". Expanding the
definitions a bit, this reduces to the condition on "categories of
factorisations" that Steve Lack mentioned in his reply.

If I am not mistaken, locally fully faithful functors are in
particular colocally locally fully faithful. Thus, by what you said in
the following paragraphs, discrete fibrations (resp. discrete
opfibrations) are examples of colocally locally fully faithful
functors generalising the projection from a slice (resp. coslice)
category to its base. This may be useful for transporting (weakly)
orthogonal factorisation systems.

Best wishes,
--
Zhen Lin


On 6 August 2014 04:08, Jean Bénabou <jean.benabou@wanadoo.fr> wrote:
>
> Dear Zhen Lin,
>
> Before I answer your question let me give names to arbitrary maps of C:
> f: X --> Y,  g: Z --> W,  x: Z -->X,  y: W --> Y and h: W --> X . Your condition reads:
> for every pair (f,g) and every triple (x,y,h') where:
>  yg = fx and  h': UW --> UX  satisfies :  h' Ug = Ux  and  Uf h' = Uy,
> there exits a unique h: W -->X  such that:  Uh = h',  hg = x  and fh = y .
> One verifies first that this condition is satisfied for all pairs (f,g) iff it is satisfied for f arbitrary and g is an identity.
> In that case that means that f is what I call hypercartesian (which in the anglo-american literature is called cartesian). Since f is arbitrary , the condition becomes:
> (i) Every map of C is hyper cartesian.
> Let me call U locally full and faithful (lff) iff for every object X of C  the obvious functor
> C/X --> D/UX  is full and fathull.
> In my mail to Joyal and the catgory list, dated July 28 I already mentioned that (i) is equivalent to
> (ii) U is lff.
> I also said, it is obvious, that U full and faithful => U is lff .
> I mentioned also the case of groupoids, with a sharper result than the one you stated, namely: If C is a groupoid, every functor U: C --> D, where D  is arbitrary, is lff.
>
> Let me add a remark which was not in my mail to Joyal, namely, the previous property characterizes groupoids. More precisely we have:
>
> PROPOSITION 1. Let C be a category. The following are equivalent:
> (i) C is a groupoid
> (ii) Every functor with domain C is lff
> (iii) The unique functor C --> 1 is lff.
>
> There are MANY MORE properties of lff functors which would be too long to  give here.
> Let me mention a few which are not in my mail to Joyal.
>  The following theorem generalizes greatly the previous proposition.
>
> THEOREM. Let  U: C --> D be a fibration. The following are equivalent:
> (i) U is lff
> (ii) All the fibers of U are groupoids.
> (iii) U is conservative (i.e. reflects isomorphisms).
>
> Such fibrations are very important. Because of (ii) they have sometimes been called groupoid fibrations. In particular, it follows from (iii), that for such a fibration if D is a groupoid so is C.
>
> You said that the functors satisfying your condition are stable by composition. This result can be strengthened since we have:
>
> PROPOSITION 2. Let  U and V be functors such that the composite UV is defined.
> If U is lff, then UV is lff iff V is.
>
> The following result is easy to prove but nevertheless important for many  theoretical reasons.
>
> THEOREM 2. lff functors are stable by pull back along any functor.
>
> I could add many significant results, in particular about cartesian functors, or orthogonality but this mail is already a bit long, and I apologize for this length.
> Thus  there is no need to give a name to the property you mentioned, locally full and faithful describes precisely this property.
>
> Best wishes,
> Jean
>
>

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