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 comma 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 finite foundation foundations functional-analysis functor 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 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.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 21st 2021

    Replaced the requirement of existence of all pullbacks with the requirement that pullbacks of covering families exist, to match SGA 4.

    Added a reference to SGA 4.

    diff, v21, current

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 23rd 2021

    I have no idea why the stronger definition was there for so long. It’s just not true in applications, most particularly the pretopology consisting of surjective submersions on the category of manifolds.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 21st 2022
    • (edited May 10th 2022)

    Need to fix this false statement:

    Notice that a basis for the topology of X is not a Grothendieck pretopology on Op(X) (since it is in general not closed under pullback, which here is restriction) but is a coverage on Op(X).

    The definition of a pretopology does not require morphisms that belong to some covering family to be closed under pullbacks. Rather, it requires that base changes of morphisms in covering families always exist.

    In particular, base changes of inclusions whose domains belong to the base always exist, even though the resulting inclusion has as its domain an open subset that need not belong to the base.

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 21st 2022

    As discussed in this answer

    https://mathoverflow.net/questions/416691/when-is-a-basis-of-a-topological-space-a-grothendieck-pretopology

    there are at least 3 different sites one can produce from a base of a topological space, and only one of them (arguably the least natural one) gives a pretopology.

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 10th 2022

    Corrected a mistake and added another reference.

    diff, v23, current