@AlexanderCampbell more generally this is true when p is just an inner fibration, see htt.2.4.1.6 (the proof is spread out over the preceding couple pages)
Assuming you define orbifold as stacks, it should totally be part of the data
But I seem to remember that the "correct" definition of a map of orbifolds has been in a state of flux for a while, until people got the "right" definition
Yeah, I understand that :). The reason I always fall back to stacks is because they tend to make the "correct" notions clear, but I guess that's a language much more common among those that come from algebraic geometry
I have a preprint using so called "complete" orbifold mappings of Borzellino and Brunsden, but I am a bit worried about this now. I realized that all the examples I worked out were having isolated singularities, where these problems don't show up.
I'm just saying, if you believe orbifolds ought to be special stacks, then it follows that the lifts need to be part of the data. For example a map X→[M//G] ought to be a principal G-bundle P→X and a G-equivariant map P→M
But I think people might be using different definitions of maps in different papers, so one must be very careful
Ok, so we have a couple of atlases $\{U_i→X\}_{i∊I}$, $\{V_j→Y\}_{j∊J}$. Then what you would like a map to be defined by a function $\varphi:I→J$ and a collection of maps $f_i:U_i→V_{\varphi i}$ satisfying some compatibilities
Only, now you cannot talk about "overlaps" anymore since the maps are not embeddings
In AG we solve this by picking maps on the fiber products $U_i×_X U_{i'}→ V_{\varphi_i}×_Y V_{\varphi i'}$ that satisfy a cocycle condition and that are compatible with the projections
Only you need to be a bit careful because those fiber products are taken in a "derived" way that remembers the stabilizers. Hold on I need to think a bit on how to say this without using the words "sheaf of groupoids" :)
Morally you want $U_i×_X U_j$ to be the set of triples $(x_i,x_j,g)$ where $x_i∊U_i$, $x_j∊U_j$, with the same image in $X$, and $g$ is an element of the stabilizer of the common image in $X$
In the orbifold case, if two charts "overlap" (there images in the space do), there is a third chart that injects in both of them as embeddings. Isn't it sufficient that I have the obvious commutativity wrt all those embeddings?
Sorry, I was thinking too abstract -- it wasn't completely obvious to me that you could cover the intersection of charts with charts, but of course it's possible
I am in the effective orbifold setting. But that is what I thought: But that still does not explain why people talk about orbifold mappings without these compatibility conditions.
To be honest? I think that people are still shackled to the idea that a map of orbifolds is determined by the underlying map of sets. Which it really doesn't want to be
Let O be the orbifold given as S^1//Z_2 by complex conjugation. Then any equivariant map $S^1\rightarrow S^1$ should induce an orbifold map. But the identity $S^1\rightarrow S^1$ or (x,y)\mapsto (x,|y|) have completely different properties
@AlexanderCampbell @asdq Going back to the question from the other day about universality of colimits in $RTop$, the question got me wondering -- are colimits universal in the category of locales (and if not, does this imply the same for toposes?)? Certainly they're not universal in the category of topological spaces.
I think the categories $Loc$ of locales and $RTop$ of toposes are at least extensive.
i.e. coproducts are disjoint and universal
If I remember right, the quotient map $\mathbb Q \to \mathbb Q / \mathbb Z$ (where the quotient is taken as spaces, not as groups!) is an explicit example which fails to be preserved by some product in $Top$ (= topological spaces) -- maybe with $\mathbb Q$?. Does this quotient also fail to be preserved by this product in $Loc$?
@CharlesRezk Hrmm.. I have mixed feelings regarding that article. I have the impression they are trying to create an image of controversy where there isn't one anymore (are people still suggesting that the literature using ∞-cats is still significantly less rigorous than the one not using it?).But maybe that is only my impression. I particularly object to the sentence "relatively few have read Lurie's text in their entirety".I mean,that's not false.But how many people have read EGA in its entirety?
@TimCampion This is what I had in mind, but I was away from my Elephant at the time, so I didn't think about it any further. Rather than do so now, I might just ask Richard today if I see him.
@CharlesRezk Great, thanks. I see we became stuck at the same point: 2.4.2.8 (1)=>(3).
@TimCampion Is the category Loc of locales total(ly cocomplete)? If so, the adjoint functor theorem for total categories will imply that colimits aren't universal in Loc, since only the locally compact locales are exponentiable.
