There are nice small overviews of classes of mathematical structures like this overview of group-like structures that organize different classes of mathematical structures by the data and axioms that they have. Similarly here is a list of possible added axioms to a topological space, and here the relation between inner-product spaces and so forth.
However, these are all small specialized lists. Is there a resource that attempts to catalog a vast number of mathematical objects (including relatively specialized ones) in an easily navigable format, such as a tree or DAG of increasingly restrictive signatures and axioms?