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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2010

    created tangent category

    in order to have a place where to keep just details on the purely 1-categorical "shadow" of tangent (infinity,1)-category.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2010

    added to tangent category a new Examples-section titled Modules over smooth algebras where I state the characterization of the tangent category over C C^\infty-rings / smooth algebras and spell out the bulk of its proof.

    • CommentRowNumber3.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 4th 2015

    I added a proof that the left adjoint of the “square-zero extension” functor ModCRingMod \to CRing constructs Kähler differentials.

    • CommentRowNumber4.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 10th 2015

    So, I learned today that a “Beck module” over an object AA in a category 𝒞\mathcal{C} with finite limits is precisely an abelian group object in 𝒞 /A\mathcal{C}_{/ A}. Perhaps it would be worth noting that somewhere on the tangent category page.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 10th 2015

    Didn’t know this. Please add!

    • CommentRowNumber6.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 10th 2015

    I looked around a bit and it seems to me that Beck modules are considered in a more “local” picture – one fixes the base AA – whereas the current page is more concerned with the global picture. So I think I will start a new page instead.

    • CommentRowNumber7.
    • CommentAuthorn.mertes
    • CommentTimeApr 13th 2021

    Regarding the statement “we should be claiming that this functor has a left adjoint which is a section and computes the Kähler differentials of objects in CC.” This is discussed a bit more on the Kähler differential page, but I still can’t find any explicit construction of this left adjoint. What is the explicit construction of the module of Kähler differentials in a general category?

    • CommentRowNumber8.
    • CommentAuthorn.mertes
    • CommentTimeApr 14th 2021

    I suppose I spoke a bit too soon, it looks like the page Beck Module states that this left adjoint exists whenever CC is a locally presentable category. However, there are categories like the (opposite) category of schemes which have Kähler differentials but is not a locally presentable category. Does anyone know of any references that explore for what categories this left adjoint exists?

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