This is maybe only useful to me and a handful of other folks, but I think of category theory sometimes as a generalization of topology. Rather than topologies you have categories and rather than liftings you have functors (this is not a strict, mathematical generalization, but an intuitive generalization of how the picture in my head of how topology works generalizes to how category theory works, so don’t come at me that this is not strictly true).
This is more mathematically justified than you seem to think. Posets are topological spaces and categories, and every space is weak homotopy equivalent to a poset space, which explains why the intuition works so well.
This is maybe only useful to me and a handful of other folks, but I think of category theory sometimes as a generalization of topology. Rather than topologies you have categories and rather than liftings you have functors (this is not a strict, mathematical generalization, but an intuitive generalization of how the picture in my head of how topology works generalizes to how category theory works, so don’t come at me that this is not strictly true).
This is more mathematically justified than you seem to think. Posets are topological spaces and categories, and every space is weak homotopy equivalent to a poset space, which explains why the intuition works so well.