# Reference request: does the notion of equivalence relation extend to proper classes?

Note: the answer to this question addresses my concerns (I think), but does not contain any references. Hence I believe my question is not a duplicate. This question is related (I think).

Question: To what extent can we extend the notion of equivalence relation, or of a relation in general, to classes from sets (i.e. including proper classes)?

If it is a meaningful restriction, I am interested primarily in the case when the class is the objects of a category.

A reference will suffice for an answer and would be preferred as well -- this seems like it must be a standard topic of set theory, but since I am completely unfamiliar with the set theory literature, I do not know where to begin to look for clarification about (equivalence) relations for classes.

Motivation: Consider a category $\mathscr{C}$. We get transitivity and reflexivity of the objects for free from the axioms of a category. To force symmetry I believe we can just restrict to the subcategory with the same objects but which contains only the invertible morphisms. This subcategory is a groupoid or setoid if I'm not mistaken. However, the definition on Wikipedia says that a groupoid has to be a small category, which is more restricitive than what I have in mind. If I'm not mistaken, what I am describing above is just classifying the objects in a category up to equivalence.

• Use Groethendieck universes and you will never again ask yourself these kinds of questions. – Stefan Perko Dec 9 '16 at 9:35
• @StefanPerko Where can I learn about those? Maybe an algebraic geometry book? – Chill2Macht Dec 9 '16 at 12:35
• @William The original reference is SGA4: fabrice.orgogozo.perso.math.cnrs.fr/SGA4 – user144221 Dec 9 '16 at 13:19
• @William "Handbook of Categorical Algebra 1" by Borceux and "Categories for the Working Mathematican" by Mac Lane both have a section on universes in the first chapter. – Stefan Perko Dec 9 '16 at 13:57
• I would be inclined to say that relations are logic -- they apply natively to classes, and that the nontrivial thing is the other way around: that we can capture a useful analog of relations within a universe of sets. – user14972 Dec 10 '16 at 11:37