Subject: categories: prone and supine

From: Thomas Streicher <[email-masked]>
Date: Wed, 4 Jan 2006 19:10:46 +0100 (CET)

Thanks to Toby Bartels I have got now an idea how to translate "prone arrow"
and "supine arrow" to German, namely as "Pfeil in Bauchlage" and
"Pfeil in Rueckenlage" (in Engl. "arrow lying on the belly" and "arrow lying
on the back"). At least in German such terminology would be "frowned upon" and
actually it sounds very strange. I haven't got a feeling how strange it sounds
in English (but I guess it does!).
The "linguistic" problem I have with this terminology is that I don't know
what is the "belly" or "back" of an arrow.

The intention of the suggested terminology (besides sounding funny) seems
to be to replace vertical/cartesian by vertical/horizontal. But then
one is sort of forced to call cocartesian arrows "cohorizontal" which sounds
a bit like "vertical", isn't it?
But actually, the "co" here refers to the fact that there are 2 ways of
being horizontal to all vertical arrows (namely a left and a right one
because the orthogonality relation is not symmetric).
So one might be inclined to call "cartesian" "right horizontal" and
cocartesian "left horizontal". For prefibrations and precofibrations one
knows that right horizontal coincides with cartesian and left horizontal
with cocartesian, respectively. But, unfortunately, for defining
pre(co)fibrations one rather needs the notion of pre(co)cartesian arrow which
cannot be characterized in terms of orthogonality conditions.

As pointed out by Jean for arbitrary functors (already prefoliating ones)
horizontal arrows need not be even precartesian (e.g. when all fibres are
discrete).

Finally, if one prefers to call pullbacks cartesian squares then one might
be inclined to prefer the terminology "cartesian" because the cartesian
arrows of the fundamental (also often called codomain) fibration are just
the pullbacks, i.e. the cartesian squares.

I rather have a different problem with "traditional" terminology, namely
that sometimes what Jean in his mail called

       cartesian        and     precartesian

is called

       hypercartesian   and     cartesian

Usually, i.e. when studying fibrations, this is not a problem
because all notions coincide. But if one considers prefibrations the
terminology cartesian/precartesian has the advantage that one speaks
about existence of cartesian liftings when defining fibrations and
about precartesian liftings when speaking about prefibrations.

Thomas Streicher