Is it possible to have a monoid with a proper class as the collection of elements? My question if it possible for us to have, for example, a monoid(any algebraic structure really) with a class as the collection of the elements, instead of a set.
An example would be the class of sets and set union as addition. Clearly, it is closed under the operation, is associative and it has an identity element, so it is a monoid. Is it possible to say so?
 A: Just as with groups, rings, topological spaces, etc., the answer is "technically no but sorta yes."


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*Technically no: Well, a monoid (for example) is defined to be a set together with a binary operation such that [stuff]. So, sethood is built right in.

*Sorta yes: But we can clearly talk$^*$ about "class-monoids" - namely, classes with a binary class operation such that [stuff]. And the "basic theory" of these objects will be unchanged.


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*$^*$Well, actually usual set theory can't talk directly about classes. So you either have to upgrade your foundations a bit or avoid quantifying over classes (e.g. you can't say "all class monoids ...") and restrict attention to more-or-less specific examples (essentially, "treat the 'definable' class monoids on a case-by-case basis"). But this is a more minor point for now, so ignore it.



The rub of course is my claim that "the "basic theory" of these objects will be unchanged" - what precisely constitutes the "basic theory?" Or, phrased another way, when can we conclude that a fact about set-sized objects also holds for class-sized objects? This starts to take us a bit into set theory and category theory, and I won't try to give an answer here, except to say that $(i)$ there is genuinely content around this point, so it's not a stupid distinction (and in particular the right answer to your question is "no"), but $(ii)$ you'll rarely run into an actual point of set/class difference if you're not actively searching for it. This isn't too satisfying, but it's the situation.
