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I expanded the Examples-section at petit topos and included a reference to Lawvere’s “Axiomatic cohesion”, which contains some discussion of some aspects of a characterization of “gros” vs “petit” (which I wouldn’t have noticed were it not for a talk by Peter Johnstone).
I am thinking that it should be possible to give more and more formal discussion here, using Lawvere’s article and potentially other articles. But that’s it from me for the time being.
I haven’t had a look at the article by Lawvere, but it did make me think, how does one define the petit topos of an object in an arbitrary site? One thing which is worth pointing out is the definition of a gros topos of a space, namely the sheaves on the induced site $Top/X$, where a covering family of an object $Y \to X$ is one that is a covering family on passing along $Top/X \to Top$. So it is clear what the gros topos of an object in a site $(C,J)$ is, namely $Sh(C/a,J/a)$.
To define the petit topos of an object $a$ in an arbitrary site $(C,J)$, consider the subcategory $J/a$ of $C/a$ with objects $u_0 \to a$ such that $u_0$ is a member of some covering family $U = \{u_i \to a\}$. Given two such objects $u_0 \to a$, $v_0 \to a$, and covering families $U$, $V$ that contain them, there is a covering family $W = UV$ which is the pullback (or at least a weak pullback) of $U$ and $V$ in $C$. There is then some element $w$ of $W$ such that there is a square
$\array{ w & \to & v_0 \\ \downarrow & & \downarrow \\ u_0 &\to & a }$so $J/a$ is ’a bit like’ the category of opens of a space (it’s probably filtered, but I haven’t checked that there are weak equalisers).
Now the morphisms of $J/a$ are those triangles
$\array{ v_0 & \to & u_0 \\ & \searrow& \downarrow \\ & & a }\, .$such that $v_0 \to u_0$ is an element of a covering family of $u_0$, so the arrows $w \to u_0$ and $w \to v_0$ really are morphisms of $J/a$. Then we say a covering family of $u_0\to a$ is a collection of triangles that, when we forget the maps to $a$, form a covering family of $u_0$ in $C$. This is at the very least a coverage, and so we can define sheaves on $J/a$, and this should be the petit topos of $a$.
I’m sure I’m not the first one to write this down, but I haven’t gone looking. If this is what people think is correct, then I’ll put it into the page petit topos, but some discussion first would be good, especially around examples.
I learned something about the difference between petit and gros toposes from this paper.
Thanks for the reference, Mike! Will try to read it at some point. For the moment I just included a pointer in the entry.
David,
yes, this kind of discussion would be very useful at the entry, eventually. It would be good to accompany it a bit by some dscussion properties, something that tells us to which extent an object regarded as an object in one or the other, gros or petit topos, does appear as “the same object”. Or whatever other properties we have.
I’ll be a bit busy with some other things, being on the way to a workshop, so for the moment I just voice wishes here, instead of doing something myself. :-)
Mamuka Jibladze had a Lauvaine thesis on alegbraic structures I think in gros topoi, there is an online link which I supplied in one of the earlier discussions either on cafe or on category list, answering a question of Steve Lack, but by now it all evaporated from my mind.
No worries, Urs. I’ll have a think about the relation between the gros and petit topoi in this framework, and see what I come up with. There is a map of sites $J/a \to C/a$, so this should give us some sort of comparison morphism between $Sh(J/a) = Petit(a)$ and $Sh(C/a) = Gros(a)$.
David,
I should say: you should put in the construction you describe into the entry anyway and now. It certainly gives what one expects to see. But eventually it would be nice if we could add to the intuition of what gros and petit is some concrete characterizations.
Got to run to my fiancée’s flute recital, so it will have to be later :)
Fiancée?
Yes. I was in a hurry (fixed the typo :) Or was that a general question? I’m getting married later this month, if that was the case.
So with the 2 body problem it will be harder to hire David than before. His talent is now out of "our" sight and power (I put the quotation marks as I am not able to hire anybody and I myself have now unsecure job future, what did not look before, nearly serious as now).
For Urs:
abstract of Jibladze's work on homotopy type in gross topoi
Congrats, David!
Zoran,
thanks for these links! I should look at them. could you put them into gros topos for me for the moment. I have to run now…
I done it gladly.
And also the entry Mamuka Jibladze about the most inspiring applied category theorist I ever met.
@Zoran #11
actually it’s not as bad as you think: private music tuition is welcome pretty much anywhere :) I am aiming for Europe in a few years, but we’ll see if it academic or not. Thanks for the vote of confidence!
More at petit topos, pasting my working definition of the petit topos from comment #2 above, and mention of the topos of sheaves on the ’large site’. Actually I think we need a new page (or two), namely large site and small site of an object in an arbitrary site. The former is just the slice category, and the latter is what I describe in #2. The gros topos and petit topos of an object in a site are just sheaves on the large and small sites respectively. Also I put in a comment about how sheaves on the large site are generalised ’spaces’ parameterised by an ’ordinary space’. For example (and maybe I should put this in - but it needs checking), sheaves on the large site $Diff/M$ are smooth spaces of some sort parameterised by the smooth manifold $M$. Perhaps the utility of this comes when mixing things up. Considering higher smooth bundles (2- or even oo-bundles) on a manifold. These are morally oo-sheaves on the site gotten by taking the subcategory of $Diff/M$ where the objects are map $V \to M$ for $V \in CartSp \subset Diff$.
Hmm.. I have an idea: Let $\tau$ be a Grothendieck topology on a site $C$. Let $\tau^x:=\bigcup_{J\in K_x} J$ where $K_x:=\{\text{ covering families of } x\}$. We say that $f\in \tau^x$ is an affine $\tau-\text{open subset}$ of $x$ (affine is not optimal terminology, but representable already means something else). There’s some reasoning (from algebraic geometry) behind this, but I’d rather hear what you have to say about it first, because I don’t really know anything about the more general motivation from topos theory.
Do you mean the union of elements of covering families? Otherwise $\tau^x = K_x$. If I interpret this correctly, in the case of $C = Top$ and $\tau =$ open covers, then $\tau^x$ is the set of opens of $x$? So in the case of schemes, $\tau^x$ would be the collection of affine opens (or affine schemes with an open embedding). And so on. I would even go so far as to call this just a ’$\tau$-affine on (or of) $x$’. There’s some stuff by Ping Xu and collaborators on differentiable stacks where they prove thing about de Rham cohomology on the big and small sites of a manifold, I’ll have to look it up.
The indexing that I may have messed up is unimportant, but yes, you’ve got the gist of it. In the etale topology on Aff, we get etale open sets (which specialize to etale open neighborhoods of a point when a k-point factors through an etale open), etc. I know for sure that there is a more robust way of describing this using the machinery of Grothendieck toposes (I read about the notion in Milne’s notes (not his book) on etale cohomology). There is a general way to go further than affine open sets by extending the topology to more diverse types of sheaves (schemes or algebraic spaces), but I don’t know how to do it.
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