Suppose you take mathematical structures which have axioms based on sets and their subsets and you replace this with objects and subobjects, for example:
Let a very general topological space T be an object A together with a collection T of subobjects of A satisfying:
1) The base object and A are in T
2) If G and H are in T then so is the greatest common subobject of G and H
3) For any subobjects Oi in T the lowest common superobject of the Oi is also in T
Where greatest common subobject and lowest common superobject exist and can be uniquely defined, and base object B is defined to be the unique object, where such exists, such that greatest common subobject(B,A)=B for all A and lowest common superobject(B,A)=A for all A, depending on some specified composition/decomposition for these objects.
Between topological spaces, continuity of maps at a point x could be rephrased as continuity on the subset {x} in order to generalise to continuity at a subobject G.
Has this sort of thing been done before ?
Edit:
Also what about very general sigma-algebras: the complement of subobj G in A would be the lcm of all the subobjs of A that don't have G as a subobj. Or very general matroids ? : M is an object E with a collection I of independent subobjects such that: 1) base object is in I, 2) If A is in I, then every subobj of A is in I, 3) For A and B in I, if A is in some way larger than B then there is a subobj C of A that is not a subobj of B such that lcm (B,C) is in I.