The problem
If $\mathcal C$ is a category with finite products, we can define an internal ring as an object $R$ equipped with morphisms $m\colon R\times R\to R$ (multiplication), $a\colon R\times R\to R$ (addition), $z\colon 1\to R$ (additive unit), $e\colon 1\to R$ (multiplicative unit), $i\colon R\to R$ (additive inverse) such that appropriate diagrams commute. It is also easy to define a homomorphism between internal rings.
However,
How does one define an internal local ring and local homomorphism between these?
In the "classical" setting (for $\mathcal C=\textsf{Set}$) this is a property – a ring is local iff it has a unique maximal ideal. This can be phrased in several equivalent ways. I am however unsure how to rephrase this in a general category $\mathcal C$. Is this still a property, or does it require some additional structure (e.g. an additional arrows modelling in some way the group of units)?
I guess that the condition:
Ring $R$ is local iff for every $r\in R$ either $r$ or $1-r$ is invertible and $0\neq 1$. $ ~~~~(*)$
can be translated into categorical terms, but I don't know how. (And how to define local homomorphisms).
More context
This question is motivated mainly by the definition of a locally ringed space which is ubiquitous in algebraic geometry – on MO it is claimed that a locally ringed space is an internal local ring in the category of sheaves over $X$. I know how to define a ringed space in this manner (or a sheaf of modules over a ringed space) and I would like to see such an interpretation of the locally ringed space as well.
I am not sure if this is possible for a general category $\mathcal C$ with finite products – the MO answer uses the fact that the category of sheaves is a topos and that one can phrase the condition $(*)$ using the internal language of a topos. But the topos property is a much stronger property that finite products… Also, in a comment Tim Campion points out:
The morphisms of models of the coherent theory of local rings are simply ring homomorphisms, rather than local ring homomorphisms (i.e. homomorphisms preserving the maximal ideal).
I hope that a categorical notion of an internal local ring would come with a canonical notion of a local ring homomorphism, generalizing to the maps between locally ringed space that induce local homomorphisms on stalks.
Best Answer
I asked about this on MO some years ago and the answer I got seems to imply there is no "logical" way of defining local rings so that the homomorphisms are local ring homomorphisms, at least if one wants to have a definition that works in any topos. (There is a definition that works in boolean toposes, however.)
Putting aside the question of homomorphisms, however, there is no way of defining local rings using only limits, let alone using only finite products. If it were possible, then the class of local rings would be closed under limits as computed in $\textbf{Set}$, but this is already false for binary products. (We may think of this as a violation of the HSP theorem, but structures definable by limits is a more general notion than varieties in the sense of universal algebra.)