Set systems and concrete categories

The standard way of relating a category $C$ to sets is through a faithful set-valued functor, which turns $C$ into a concrete category. In that case, we can get a set from any object. Alternatively, we can get a set from the image of a function, which is a subset of the output set of a function. In particular, we can construct set systems from the families of images of monomorphisms of a concrete category. This causes posets of subobjects of concrete categories to have underlying set systems.

While an object system of a concrete category can be used to construct a family of sets, the underlying set of an object is not an invariant up to isomorphism so this isn't that interesting of a construction. A family of common output monomorphisms on the other hand produces a family of subsets of some underlying set which can be classified as a hypergraph up to isomorphism. Therefore, we will construct set systems by morphism systems rather then object systems.

Sets of elements:
If we have a concrete topos, then an element of an object $X$ is defined by a monomorphism of the form $1 \to X$. Then in a topos we have that the elements of the object of the topos are a subset of the underlying set of elements. Consider the particlar example of a topos of monoid actions. There we see that there are fixed points which are a subset of the underlying set. The set system produced by families of element morphisms are unary families, which are determined entirely by their unions.

Set systems:
In the more general context we can by any family of morphisms get an underlying family of images. In particular, since the subobjects of an object in a category are determined by the preorder on monomorphisms, we can get a set system corresponding to the poset of subobjects of an object. In the particular case of monoid actions, we see that for any M-set $X$ we have that $Sub(X)$ is an Alexandrov topology because action closed sets are completely union closed.

For most algebraic strutures such as a semigroup $S$ then $Sub(S)$ is merely a Moore family because it doesn't have union closure. In the special case of a group $G$ then $Sub(G)$ is in fact a union-free Moore family because it has the opposite condition which is that there are non-trivial unions in the family of all subgroups of a group. By these sort of constructions on concrete categories, we see that category theory can be related back to set theory so that the two subjects don't have to be treated separately.