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 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 limit 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 subobject 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.
    • 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 14th 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?

  1. Fixed typo


    diff, v23, current