Consider the category $\mathrm{TopMon}$ of topological monoids and continuous monoid homomorphisms. Consider the inclusion $i:\Bbb{R}_{\ge 0}\hookrightarrow \Bbb{R}$, where the spaces are taken with their usual topology and addition as the monoid operation. This is map is a monomorphism. Compare it now with the (still injective) map $j:\Bbb{R}_{\ge 0, D}\hookrightarrow \Bbb{R}$, where $\Bbb{R}_{\ge 0, D}$ is now the monoid $(\Bbb{R}_{\ge 0},+,0)$ equipped with the discrete topology.
I'm looking for a formal (category-theoretical) way to express that $i$ is "nicer" than $j$, since for example $i$ is the inclusion of a subspace, while $j$ is not. However, $i$ is not strong mono, nor extremal mono (since it is epi as well), nor regular mono, et cetera. Is there a class which is less restrictive than strong or extremal mono to which $i$ belongs, but not $j$?
Of course, one possible way would be to take the obvious forgetful functor $U:\mathrm{TopMon}\to\mathrm{Top}$ and say that $Ui$ is extremal mono, while $Uj$ is not. However, I was wondering if there is a way to distinguish the two which is internal to $\mathrm{TopMon}$.
