Re: A condition for functors to reflect orthogonality

["Jean Bénabou" <jean.benabou@wanadoo.fr>]

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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