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

["Jamie Vicary" <jamie.vicary@imperial.ac.uk>]

>     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.

Fred Linton has pointed out to me that my terminology here is not
standard. By "isometric injector", I mean a morphism which is unitary
on its range, i.e., one-to-one and norm-preserving in the case of
Hilbert spaces; I believe this is usually simply referred to as an
isometry. By "isometric projector", I mean a morphism which is unitary
on the complement of its kernel; some people prefer to call this a
"partial isometry". I was then using the terms "isometric" and
"isometry" to mean "isometric projector or isometric injector".

    Anyway, the simple prescription I give for a bilimit cannot work,
as it is easy to find diagrams in the category of finite-dimensional
Hilbert spaces, our canonical example of a strongly compact-closed
category with biproducts, for which the colimit and limit are not
isomorphic. A diagram f:A-->B for non-iso A and B is the simplest
example. However, if we restrict to diagrams F:D-->FdHilb such that D
admits a dagger-operation compatible with the dagger on FdHilb, then I
believe the conjecture becomes plausible.

            Regards,
                 Jamie Vicary.