Re: Equalisers and coequalisers in categories with a \dag-involution

["Jamie Vicary" <jamievicary@gmail.com>]

Peter,

    Thank you for that detailed response! So it seems that if these
dagger-subobjects do exist, then then will have good properties. But
existence is tricky; in particular, there does not seem to be an
elegant property (analagous to having finite limits and colimits) that
will guarantee that all of this works.

    Could we make the following definition: a dagger-category has
'finite bilimits' if any finite diagram D in the category has an
'isometric cone', a cone for which all the associated morphisms to the
objects of D are isometries, along with some sort of condition that
the isometries are orthogonal in the correct way. It is interesting to
consider this in the case of products and equalisers: for products
AxB, so it seems, the isometries to A and B will generally be
_projectors_, but for equalisers E-e->A=f,g=>B, the isometry e will
generally be an _injector_! So we cannot ask for the cone morphisms to
be isometric projectors, or isometric injectors. But perhaps this is
OK, and we can just require them to be isometries. This definition of
bilimit has the 'local flavour' of the definition of biproducts, but
cooking up a generally-applicable orthogonality condition on the
isometries seems tricky.

    Of course, in the light of
              http://www.arxiv.org/abs/math.CT/0604542   ,
perhaps we only need require that our dagger-category has products and
equalizers in order for it to have 'finite bilimits'! In remark 2.6 of
[2] cited in your email below, you write that if a dagger-category has
products then it must of course have coproducts, but it need not have
biproducts. Presumably, math.CT/0604542 proves you wrong here?

                     Jamie.