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

[selinger@mathstat.dal.ca (Peter Selinger)]

Jamie Vicary wrote:
>
> Dear all,
>
>     Consider the following straightforward coequaliser (e,E) formed by
> f,g:A-->B and e:B-->E, with e.f=e.g. I am working in a category with
> biproducts, and with a contravariant involutive endofunctor (--)^\dag
> on the category which is compatible with the biproducts; i.e.
>                 (projection)^\dag = injection
> for all projections and injections making up a part of a biproduct. In
> such a category, it is natural to consider the coequaliser object E to
> be the subspace of B on which the morphisms f and g agree. It is
> therefore natural to require e.(e^\dag) = id_E; this sort of condition
> is similar to the sorts of conditions that form part of the definition
> of the biproduct.
>
>     I'm asking whether there exists a natural framework generalising
> the theory of biproducts, which is analagous to the way that
> (co)limits generalise (co)products, within which I can safely assume
> that e.(e^\dag) = id_E. Biproducts seem quite different from products
> and coproducts, though, so I don't know how it would work.
>
>                Jamie Vicary.

Dear Jamie,

your equation e.(e^\dag) = id_E only makes sense if your functor
(--)^\dag is the identity on objects. In this case, you are dealing
with a dagger category in the sense of [1]. Dagger categories are
important in quantum physics; an important example is the category
Hilb of Hilbert spaces and linear operators, with dagger being the
adjoint of an operator.

I will dualize your question to make it a question about equalizers,
or more generally, monomorphisms. Monomorphisms with the property
(e^\dag).e = id_E are investigated in [2], where they are called
dagger-subobjects. (Both papers also deal with biproducts of the kind
you asked about).

Your question raises a basic problem, which is that it is not
well-defined. Specifically, while equalizers are only defined "up to
isomorphism", the property

 (e^\dag).e = id_E   (*)

is not invariant under isomorphisms of E. As a simple example, the two
morphisms f,g: C -> C^2 in Hilb, defined by f(x) = (x,0) and g(x) =
(2x,0), define isomorphic subobjects, yet f satisfies (*), whereas g
does not. Therefore one cannot ask whether "the" equalizer of two maps
satisfies (*).

The fundamental issue is that in a dagger-category, there is a
distinguished subclass of isomorphisms: the unitary ones. An
isomorphism f: E -> E' is called unitary if f^\dag = f^{-1}. And
although the property (*) is not invariant under arbitrary
isomorphisms, it is invariant under unitary isomorphisms.

To many category theorists, it may seem strange that some important
categorical property is not invariant under isomorphism. But actually,
this is quite natural. With every notion of structure comes a notion
of structure-preserving isomorphism, and one expects properties
related to the structure to be preserved only by the
structure-preserving isomorphisms, not by arbitrary isomorphisms.
Dagger is such a structure, whose structure-preserving morphisms are
exactly the unitary ones.

Now to get back to your question: Consider equalizers (or more
generally, subobjects) in a dagger category. Of the many maps e: E ->
A representing a given subobject (or equalizing a given pair of
arrows), some may not be unitarily isomorphic to some others, so they
fall into equivalence classes modulo unitary isomophism. One may ask
whether any of these equivalence classes is distinguished, i.e.,
whether one of them deserves to be called "the" equalizer or "the"
subobject, and would be unique up to unitary isomorphism. It turns out
that, provided it exists at all, there is indeed a distinguished
choice of such a subobject, and it is the one satisfying (*). So we
can call a monomorphism satisfying (*) a "dagger-subobject", and an
equalizer satisfying (*) a "dagger-equalizer", and so forth. (In the
literature, particularly on Hilbert spaces, a morphism satisfying (*)
is also called an "isometry").

Fortunately, if any representative of a subobject satisfies (*), then
that representative is unique up to unitary isomorphism. So it really
makes sense to speak of "the" dagger-subobject etc.

While uniqueness is easy, existence is a tricky matter. There
certainly are examples of subobjects that are not isomorphic to any
dagger-subobject. One such example is in the category of integer
matrices (objects are arities, composition is matrix multiplication,
and dagger is transpose). The morphism (2) (as a 1x1-matrix) is monic,
but not isomorphic (as a subobject) to any isometries. However, it is
also not an equalizer. To get an example with equalizers, consider the
two morphisms f = (1,0) and g = (0,1) (as 1x2-matrices). Their
equalizer is the 2x1-matrix e = (1,1)^\dag. However, this is not
isomorphic to any isometry, and hence not to any dagger-equalizer.  So
in general, dagger-equalizers don't exist even if equalizers do.  (For
this example, it is important that the scalars are integers. If real
numbers were allowed, then e/sqrt{2} would be the dagger-equalizer.)

On the other hand, it is proved in [2] (Proposition 5.6) that under
some relatively mild condition, every subobject is isomorphic to a
dagger-subobject.

I hope this answers part of your question!

Let me close with some speculation: if e : E -> A is a monomorphism
such that (e^\dag).e is invertible, then it is probably relatively
easy to add a representative e' : E' -> A freely, and fully
faithfully, to the category such that e,e' are isomorphic (as
subobjects) and e' satisfies (*). [Clearly if (e^\dag).e is not
invertible, then this is never possible]. On the other hand, it is not
clear whether (e^\dag).e is invertible in general, or whether at least
this is the case when e is an equalizer.

-- Peter

[1] P. Selinger. Dagger compact closed categories and completely
positive maps. To appear in Proceedings of the 3rd International
Workshop on Quantum Programming Languages, Chicago, June 30 - July 1,
2005. ENTCS, 23 pages.
http://www.mathstat.dal.ca/~selinger/papers.html#dagger

[2] P. Selinger. Idempotents in dagger categories. To appear in
Proceedings of the 4th International Workshop on Quantum Programming
Languages, Oxford, July 17-19, 2006. ENTCS, 15 pages.
http://www.mathstat.dal.ca/~selinger/papers.html#idem