Is there a name for functors between concrete categories over Set that preserve '$\subset$'?
Example, for a functor $F:\mathbf {Top}\to \mathbf {Grp}$, if $Y$ is a subspace of $X$ then $F(Y)$ is a subgroup of $F(X)$, truly just not isomorphic to a subgroup.
Factual functors as $\mathcal P(\cdot)$ and $(\cdot)^I$, $\mathbf {Set}\to\mathbf {Set}$, factually preserve subsets.
My interest of this comes from my striving to create general classes of concrete categories over $\mathbf {Set}$ (considering such different structures as groups and topological spaces to be comparable). However, I want to restrict to structures in which very natural functions, as inclusions and Cartesian projections, are forced to be morphisms.
Perhaps is it (under certain circumstances) possible to define canonical injections, not being factual inclusions but perhaps significantly more than just monomorphisms or coretractions?