## Several things part 3

• Here’s a suggestive hallucination, useful to me for keeping some things straight, but maybe not useful to anyone else:
Let $G$ be a p-adic reductive group, $C$ an algebraically closed field of characteristic zero (or maybe just of characteristic $\neq p$). The set $X_G$ of isom. classes of smooth irreducible $C$-representations of $G(\mathbf{Q}_p)$ really wants to be an algebraic variety, but it’s not. However, $X_G$ has a canonical best approximation by an (ind-)algebraic variety, the Bernstein variety $Z_G$. The canonical map $X_G \to Z_G$ is “quasifinite and birational”.  The algebraic functions on $X_G$ are given by trace forms, i.e. by functions of the form $\pi \mapsto \mathrm{tr}(f| \pi)$ for some arbitrary $f \in C_c(G,C)$. There is also a canonical second-best approximation of $X_G$ by an algebraic variety, the spectral Bernstein variety $Z_{G}^{\mathrm{spec}}$, i.e. the coarse quotient of the stack parametrizing ($G$-relevant) L-parameters $W_{\mathbf{Q}_p} \to \phantom{}^L G$. It is second-best in the sense that there is a quasifinite map $Z_G \to Z_{G}^{\mathrm{spec}}$. The composite map $X_G \to Z_{G}^{\mathrm{spec}}$ sends $\pi$ to its semisimple L-parameter.

Should I post more hallucinations like this?

• I have to admit that I struggle psychologically with things related to foundations, especially subtleties arising from “big” constructions and the usual prophylactics involving universes or cutoff cardinals or whatever. For one thing, I don’t really care. But more significantly, the idea that ZFC (or something like it) should be accepted as the “standard foundations” of mathematics is absolutely revolting and nonsensical to me. The fact that everything in ZFC is a set makes it a complete non-starter for me as a reflection of how mathematics really operates. In some sense, I don’t really believe in “naked” sets.
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Anyway, I was never able to articulate my thoughts about this stuff very precisely. It was thus something of a revelation when I read this article at the Xena project, and realized that type theory is what I’ve been craving all along. I also strongly recommend this article by Todd Trimble which articulates my problems with ZFC much more eloquently than I can. (I don’t really understand ETCS yet, but it also seems like it would satisfy me.)
• Is the twitter account @GeoMoChi08 a parody? I would dearly love to know what’s going on with this account (and with @math_jin).