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 O_{i} in *T* the lowest common superobject of the O_{i} 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.