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I scribbled some thoughts on the Yoneda lemma here. Feedback welcome.
The best it can do to describe Set Theory is topos.
Well, this is not at all true (see Algebraic Set Theory for example), but mainly what I’d be worried about in this discussion is “attitude”, as in the use of phrases like “state of sin” and “the best it can do”. There are many people who frequent the nLab who understand (pace von Neumann) and who could explain the Yoneda lemma, but there may not be many who want to engage if it looks like it’s going to be an uphill climb. We’re kind of busy.
The subobject classifier is the analog of truth values – the defining property of $\Omega$ is the correspondence between two ways of describing a property on elements of a set:
I actually find the “finite limits” + “cartesian closed” + “truth values” description to be a very appealing characterization of toposes, since that basically says outright “a topos is a place where you can do basic logic and function manipulation in the familiar way”.
Any category, really, has a notion of set membership coming from the idea of generalized elements. However, for the purposes of describing toposes you could also work from the idea of “power object”, since (bound) set membership can be described as a property on $PX \times X$ for each $X$.
There are various levels at which one can understand the Yoneda lemma. The proof is somewhat trivial. But one gradually appreciates its importance, eventually overwhelmingly so, when one encounters it every day of one’s life, at varying levels both conceptual and calculational. It truly is THE central result of category theory.
Surely the Yoneda lemma article can be much improved. I’m a little dumbstruck myself because it’s so deep in my blood and bones and soul that it’s very hard ever to feel I can do it proper justice. Perhaps I should jot down a note every time I use it, and accumulate these notes over time, and eventually cull the best ones and put them down as examples in the article. Not sure I’ll ever do that, though.
Actually, the way that the internal logic of a topos gets built up by the interplay between the external (natural operations on subobject lattices) and internal (operations on $\Omega$) is already an excellent illustration of the Yoneda lemma in action. But that’s just one of a billion examples.
For what it’s worth, subobject classifiers and power objects are not necessary to describe sets categorically. Most of the familiar properties of sets are already represented by a well-pointed pretopos; adding power objects then corresponds to assuming the powerset axiom in ZFC.
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