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Created a page on the set-theoretical meaning of the “class function”.
If here in a comment edit box you type
[[class function (set theory)]]
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@Urs: Thanks for pointing out.
Added two basic references to
I tweaked a phrase to read, “Class functions are an important concept when formalizing category theory on set-theoretic foundations…”. The point is that there can be many ways of laying formal foundations for category theory: Mac Lane took recourse in a set-theoretic framework back in the early 1970’s, but today there are various options.
I suggest to add an Idea-section that provides some context, maybe such as the following:
Recall that in set theory a function $f$ from a set $S_1$ to a set $S_2$ may be encoded in terms of a relation on the Cartesian product $S_1 \times S_2$ of the two sets, namely the subset $R \subset S^1 \times S^2$ with $R = \{ (x,y) \vert y = f(x)\}$.
This concept has an evident generalization to the case where $S_1$ and $S_2$ are allowed to be proper classes. In this case one speaks of class functions.
I clarified even further: even when using set theory, you can avoid proper classes entirely by using universes.
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