To gain a big picture of (pre-categorical) mathematics, is it correct to divide mathematical theories resp. structures in two big families?
universal algebra: classes of objects with arbitrary functions and relations/properties defined on them (basically first order)
universal topology: classes of objects with arbitrary functions and relations/properties defined on sets of those objects (basically second order)
The distinction is not as sharp as it seems, because there is e.g. pointless topology, and because every theory/structure can be defined in set theory, which is first order.
What is missing in this "big picture"? Are there (pre-categorical) theories/structures that fall in none of these two families?
(I emphasize "pre-categorical" because for example "algebraic topology" is a genuine theory but seems to be about something completely different than "classes of objects with functions and relations defined on both themselves and sets of them".)