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nLab > Latest Changes: Grothendieck topology

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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 20th 2021

    This entry defines Grothendieck topologies using sieves.

    However, in the original definition (Michael Artin’s seminar notes “Grothendieck topologies”), a Grothendieck topology on a category CC is defined as a set TT of coverings.

    More precisely (to cite from Artin’s notes), a Grothendieck topology is defined as families of maps {ϕ i:U i→U} i∈I\{\phi_i\colon U_i\to U\}_{i\in I} such that

    • for any isomorphism ϕ\phi we have {ϕ}∈T\{\phi\}\in T;

    • if {U i→U}∈T\{U_i\to U\}\in T and {V i,j→U i}∈T\{V_{i,j}\to U_i\}\in T for each ii, then {V i,j→U}∈T\{V_{i,j}\to U\}\in T;

    • if {U i→U}∈T\{U_i\to U\}\in T and V→UV\to U is a morphism, then U i× UVU_i\times_U V exist and {U i× UV→V}∈T\{U_i\times_U V\to V\}\in T.

    This is almost identical to the current definition of Grothendieck pretopology, except that in Artin’s definition only the relevant pullbacks are required to exist.

    It seems to me that the original definition by Artin is the one used most often in algebraic geometry.

    diff, v43, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 20th 2021

    Moved the content from historical note on Grothendieck topology here.

    diff, v43, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 20th 2021

    Added the original definition.

    diff, v43, current

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 20th 2021

    Added the original reference.

    diff, v43, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 21st 2021

    Added names for the axioms. Added a reference to SGA 4.

    I removed the following axiom from the definition of a Grothendieck topology, since it is redundant (i.e., implied by the other axioms).

    1. Two sieves F,GF, G of cc cover cc if and only if their intersection F∩GF \cap G covers cc. (Here the saturation condition is important.)

    diff, v44, current

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeAug 30th 2022
    • (edited Aug 30th 2022)

    Pullback sieve is used in the definition, but not defined in the entry so I added it.

    If g:d→cg:d\to c is a morphism in a category CC and F⊂C(−,c)F\subset C(-,c) a sieve on cc then

    g *F={h:dom(h)→d|g∘h∈F}⊂C(−,d) g^* F = \{ h: dom(h)\to d | g\circ h \in F\}\subset C(-,d)

    is a sieve on dd, the pullback sieve of FF along gg.

    diff, v46, current

    • CommentRowNumber7.
    • CommentAuthorperezl.alonso
    • CommentTimeApr 19th 2024

    pointer

    • Ana Luiza Tenório, Hugo Luiz Mariano. Grothendieck prelopologies: towards a closed monoidal sheaf category (2024). (arXiv:2404.12313).

    diff, v48, current

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