@JamesEHanson Fair enough. I didn’t mean to be reductive, just that higher category theory allows for Hom-spaces that have homotopy groups in degrees $>0$

@JamesEHanson I just mean roughly that the category of sets is a full subcategory of infinity groupoids

@PanMrož One could always work with categories enriched over some nice category of topological spaces, and work up to homeomorphism instead of homotopy. However, I think homeomorphism is too strict of a notion for most applications people have in mind. For example, categories of spectra are supposed to be related to cohomology theories and those are often homotopy invariant. Additionally, one would like to recover derived categories as truncations of certain infinity categories, and necessarily a notion of homotopy enters the picture here again. It just depends on what you want to do

Because homotopy theory enhances set theory

@wlad Fixed, thanks!

@Gabriel Maybe it depends on your definition of closed embedding, but it’s not a closed embedding for the silly reason that $D$ is not a scheme. That said, it is indeed an “ind-closed” embedding in the sense that it’s induced by an inductive system of closed immersions to the curve

See Lemma 106.3.1 of the stacks project. It shows in particular that any algebraic stack with a cover by an irreducible scheme is irreducible

@LSpice Agreed! Your comment is why I responded again to Theo Johnson-Freyd amending the reason I thought the characteristic zero case was different. The existence of these homomorphisms can't on its own explain the discrepancy for the reason you point out

@TheoJohnson-Freyd Actually, reading the counterexample a little more carefully, the characteristic zero assumption seems to be used in an essential way when showing that if an extension splits fppf locally then it splits etale locally

@LSpice My apologies for being unclear, I just meant for a non-reduced $\mathbb{Q}$-algebra $A$ we have the homomorphisms $exp(rT)$ where $r$ is a nilpotent. A map like this seems to be important in the construction of the counterexample

@TheoJohnson-Freyd To me it’s very counterintuitive that the bad behavior occurs is characteristic 0 in this case. However, I think it’s essentially due to the existence of nontrivial homomorphisms from the additive group to the multiplicative group