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