categories: Re: injective modules in a topos

[Paul Taylor <pt@dcs.qmw.ac.uk>]

Mike Barr asks
> Does anyone know if an injective is internally injective?

I cannot contribute anything to the question about rings and modules,
but I have recently had to think about this problem in the context
of (my re-axiomatisation of) topological spaces.

We want any subspace to have the "subspace topology".  This is the
same as saying that the Sierpinski space is injective with respect
to subspace inclusions (regular monos, if you please).

In my setting I think of the topology (the lattice of open sets)
not as a lattice over the category of sets, but as a space with
the Scott topology.                

The "internal injectivity" property in this situation is therefore
that we have a retraction

               i
      U  >----------->  X

                I
         U  >-------->      X
    Sigma              Sigma
            <<--------
                   i
              Sigma

but if the map I is "internal" then this is a Scott-continuous map,
and we only have certain kinds of subspaces.   

Particularly annoyingly, we cannot form the intersection of two
such subspaces.

This is what I am currently writing up, as the successor to the
paper "Sober Spaces and Continuations" that I advertised on Saturday.

Now, if you interpret all of this in the traditional axiomatisation
of topology or locale theory,  all of this only makes sense for
locally compact spaces anyway.  My "monadic" axiomatisation
does this more abstractly, but with locally compact spaces as the
motivating model.

The intersection problem is clearly an undesirable feature of this
theory, and I believe that the "internal" injectivity is the flaw.

Paul
http://www.dcs.qmul.ac.uk/~pt/ASD