Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories accessible adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nlab noncommutative noncommutative-geometry number number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2010

    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.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 2nd 2010
    • (edited Jun 2nd 2010)

    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/XTop/X, where a covering family of an object YXY \to X is one that is a covering family on passing along Top/XTopTop/X \to Top. So it is clear what the gros topos of an object in a site (C,J)(C,J) is, namely Sh(C/a,J/a)Sh(C/a,J/a).

    To define the petit topos of an object aa in an arbitrary site (C,J)(C,J), consider the subcategory J/aJ/a of C/aC/a with objects u 0au_0 \to a such that u 0u_0 is a member of some covering family U={u ia}U = \{u_i \to a\}. Given two such objects u 0au_0 \to a, v 0av_0 \to a, and covering families UU, VV that contain them, there is a covering family W=UVW = UV which is the pullback (or at least a weak pullback) of UU and VV in CC. There is then some element ww of WW such that there is a square

    w v 0 u 0 a\array{ w & \to & v_0 \\ \downarrow & & \downarrow \\ u_0 &\to & a }

    so J/aJ/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/aJ/a are those triangles

    v 0 u 0 a.\array{ v_0 & \to & u_0 \\ & \searrow& \downarrow \\ & & a }\, .

    such that v 0u 0v_0 \to u_0 is an element of a covering family of u 0u_0, so the arrows wu 0w \to u_0 and wv 0w \to v_0 really are morphisms of J/aJ/a. Then we say a covering family of u 0au_0\to a is a collection of triangles that, when we forget the maps to aa, form a covering family of u 0u_0 in CC. This is at the very least a coverage, and so we can define sheaves on J/aJ/a, and this should be the petit topos of aa.

    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.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 2nd 2010

    I learned something about the difference between petit and gros toposes from this paper.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2010

    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. :-)

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJun 2nd 2010

    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.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 2nd 2010

    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/aC/aJ/a \to C/a, so this should give us some sort of comparison morphism between Sh(J/a)=Petit(a)Sh(J/a) = Petit(a) and Sh(C/a)=Gros(a)Sh(C/a) = Gros(a).

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2010
    • (edited Jun 2nd 2010)

    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.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 2nd 2010
    • (edited Jun 2nd 2010)

    Got to run to my fiancée’s flute recital, so it will have to be later :)

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 2nd 2010

    Fiancée?

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 2nd 2010

    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.

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJun 2nd 2010

    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:

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 2nd 2010

    Congrats, David!

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2010

    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…

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeJun 2nd 2010

    I done it gladly.

    • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeJun 2nd 2010

    And also the entry Mamuka Jibladze about the most inspiring applied category theorist I ever met.

    • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 3rd 2010

    @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!

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 3rd 2010

    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/MDiff/M are smooth spaces of some sort parameterised by the smooth manifold MM. 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/MDiff/M where the objects are map VMV \to M for VCartSpDiffV \in CartSp \subset Diff.

    • CommentRowNumber18.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 3rd 2010

    Hmm.. I have an idea: Let τ\tau be a Grothendieck topology on a site CC. Let τ x:= JK xJ\tau^x:=\bigcup_{J\in K_x} J where K x:={ covering families of x}K_x:=\{\text{ covering families of } x\}. We say that fτ xf\in \tau^x is an affine τopen subset\tau-\text{open subset} of xx (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.

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 3rd 2010

    Do you mean the union of elements of covering families? Otherwise τ x=K x\tau^x = K_x. If I interpret this correctly, in the case of C=TopC = Top and τ=\tau = open covers, then τ x\tau^x is the set of opens of xx? So in the case of schemes, τ x\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 ’τ\tau-affine on (or of) xx’. 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.

    • CommentRowNumber20.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 3rd 2010
    • (edited Jun 3rd 2010)

    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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)