Re: Name for a concept

[Marco Grandis <grandis@dima.unige.it>]

I do not know the original problem of M. Barr, may be he is really
interested in getting an epimorphism onto the pullback.

However - I apologise for insisting - I think that the important
notion for such squares should be a natural self-dual generalisation
of pullbacks and pushouts, based on commutative squares and nothing
else - so that, in particular, it cannot depend on the variation of
monos or epis one is interested in.

The one I have proposed in that old paper (under the name of
"semicartesian square") is of this kind:

- the square  (f,g; h,k)  commutes, and for every span  (f',g')
which commutes with the cospan  (h,k)  and every cospan  (h',k')
which commutes with the span  (f,g),  the new span and cospan form a
commutative square.

All this comes from the obvious Galois connection between sets of
spans and cospans (in an arbitrary category), derived from the
commutativity relation.

Explicitly, let us start with two fixed objects A, B.  Let  S  be the
set of spans from A to B:

x = (f: C -> A,  g: C -> B)    (for arbitrary C)

and  C  the set of cospans

y = (h: A -> D, k: B  -> D)   (for arbitrary D).

Take their set of parts,  PS  and  PC,  ordered by inclusion, and the
following (contravariant) Galois connection between them (X in PS, Y
in PC):

R(X) = set of cospans which commute with all the spans in X,
L(Y) = set of spans which commute with all the cospans in Y.

Now, a square  (x, y)  (span/cospan) commutes iff  {x} is contained
in  L({y})  iff  {y} is contained in R({x}).
A square  (x, y)  is "semicartesian" (or "exact") iff it satisfies
the stronger, equivalent conditions:

1.  R{x} = RL({y})
2.  L{y} = LR({x}).

Marco Grandis

-------------------------
On 1 Dec 2005, at 02:48, Michael Barr wrote:

> Is there a standard name for a square
> A ----> B
> |       |
> |       |
> |       |
> v       v
> C ----> D
> in which the canonical map A ---> B x_D C is epic?  I had always
> called it
> a weak pullback, but Peter Freyd claims that that phrase is
> reserved for
> the case that it satisfies the existence, but not necessarily the
> uniqueness of the definition of pullback.  In fact, he claims it means
> that Hom(E,-) converts it to the kind of square I am talking about.
> What is interesting is that in an abelian category, it satisfies
> this condition iff it satisfies the dual condition iff the evident
> sequence A ---> B x C ---> D is exact.  Putting a zero at the left end
> characterizes a genuine pullback and at the other end a pushout.
>
> Michael
>
>
>
>
>