My question is inspired by the fact that there is no universal set (at least in ZF). There are many abstract objects such as group, ring, field, vector space, topology, etc. such that we can say about embedding. In a sense, we can interpret $A$ embeds $B$ as $A$ 'contains' $B$. So it seems that there is no such universal objects (universal group/ring/field...) that embeds every possible objects (group/ring/field ..). How can I prove/disprove it?
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There are cardinality limits in these cases, too. That is, there exist groups rings fields topologies (etc.) of arbitrarily large cardinal. So no " universal" ones exist.
But for example there are nice things like: a universal separable metric space: that is, a separable metric space such that any other separable metric space is isometric to a subset of it. Or a universal countable group. And so on.