Can you link to the lecture?

@LSpice Thanks! I edited it in.

Hear, hear to the second. That's a trick I use when refereeing: look for those "clearly"s.

I'd say that $(\infty, n)$-category theory is still pretty young, and you shouldn't expect it as of today to solve problems explainable to high-school students that are beyond the reach of old established fields like algebraic number theory and algebraic geometry. Let's be clear not to mythologize higher category theory as blowing other powerful fields out of the water, at least not yet.

More search terms: Gabriel topology on an abelian category, universal closure operator.

I don't have time at the moment to give much more than a cursory reply, but a good search term might be "torsion theory". The nLab article ncatlab.org/nlab/show/torsion+theory has some material and references to get you started.

I started teaching again, and being part of an institution makes certain aspects of doing research a lot easier (I found).

I think it would be a very nice gesture to inform the reader of what you already know about this problem, to avoid having people waste time. For instance, you are mentioned in a footnote here, arxiv.org/pdf/2005.03255.pdf, which recalls the finite case covered by results of Duffus and Wille.

An excellent answer. I hope the following works as an explicit example of the phenomenon: let $Y$ be an uncountable set, and let $X$ be countably infinite. Let $I$ be the directed set of finite subsets $F$ of $Y$, ordered by inclusion. For each finite subset $F$, let $E_F$ be the set of injective functions $\phi: F \to X$. For $F' \subseteq F$, the transition map $E_F \to E_{F'}$ is given by restriction. These maps are easily seen to be surjective. But if there were an element $(\phi_F)$ of the inverse limit, then we could manufacture an injective map $f:Y \to X$ by $f(y) = \phi_{\{y\}}(y)$.

@PrinceM I'm impressed! Stay strong.

It would be common only for complete noobs to functional analysis.

@SamuelAdrianAntz Here's what I have. For any given rational $q$, suppose $r$ is a rational root of $x^3 - 5x -q$. After a long division, we conclude $x^3 - 5x - q = (x-r)(x^2 + rx + (r^2 - 5))$. The discriminant of the quadratic factor is $D = r^2 - 4(r^2-5) = 20 - 3r^2$, and another rational root of $x^3 - 5x - q$ exists only if $D$ is a rational square. So it suffices to show that there are no rational solutions $(r, s)$ to $3r^2 + s^2 = 20$. For this, show there are no integer solutions to $3r^2 + s^2 = 5t^2$ (except $(r,s,t)=(0,0,0)$). This boils down to simple modular arithmetic.

It sounds like you want to say in your first sentence "left adjoint of the nerve functor from the category of (small) categories to the category of simplicial sets". Yes, the fact that the counit of the first adjunction is an isomorphism is correct; that follows from full faithfulness of the nerve functor, as you point out. The singularization functor from topological spaces to simplicial sets is not fully faithful, so the counit of that adjunction will not be an isomorphism.

In answer to Mike's question in his comment, and following on Dap's comment on cofinality, I'd mention König’s theorem ncatlab.org/nlab/show/K%C3%B6nig%27s+theorem.