Not signed in (Sign In)

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 adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book 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 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 nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar 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 stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 25th 2010
    • (edited Mar 25th 2010)

    So Terry Tao gave an answer on MO earlier, where he said that you can construct Top using sheaves of sets on Set, but he didn't really know how.

    Is this actually true? Can we construct Top as a stack on Set without giving the axioms for a topological space (hidden or otherwise)?

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeMar 25th 2010

    Reading that I think he may be referring to Grothendieck topologies.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2010
    • (edited Mar 25th 2010)

    So you are thinking of this sentence of Tao's, here:

    I believe that there is some equivalent way to axiomatise topology via continuous functions using the machinery of sheaves, which is in some ways more "natural" than the simple but somewhat arbitrary-looking axioms for open sets,


    This does not mention sheaves on  Set. Did he say something about that elsewhere?

    Yes, so my impression, too, is that he is simply referring to the sheaf topos on the category of open subsets of a topological space.

    But there are of course ways to speak of topologies that are very category-theoretic and don't look like the ordinary definition. For instance a topology on a space is a sub-quantale of the power-set quantale or something like that. (I forget. Maybe sub-*-quantale).

    But I think the royal road to topological spaces in terms of just category theory is regarding them as localic toposes. We can say "localic topos" using only very natural category-theoretic language that never looks ad hoc.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeMar 25th 2010