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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2010
   • PermaLink
   Author: Urs
   Format: Markdowncreated [[tangent category]] in order to have a place where to keep just details on the purely 1-categorical "shadow" of [[tangent (infinity,1)-category]].

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 31st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[tangent category]] a new Examples-section titled <a href="http://ncatlab.org/nlab/show/tangent+category#ModulesOverSmoothAlgebras">Modules over smooth algebras</a> where I state the characterization of the tangent category over $C^\infty$-rings / smooth algebras and spell out the bulk of its proof.

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

3. • CommentRowNumber3.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 4th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexI added a proof that the left adjoint of the "square-zero extension" functor $Mod \to CRing$ constructs Kähler differentials.

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

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 10th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexSo, I learned today that a "Beck module" over an object $A$ in a category $\mathcal{C}$ with finite limits is precisely an abelian group object in $\mathcal{C}_{/ A}$. Perhaps it would be worth noting that somewhere on the [[tangent category]] page.

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

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 10th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexDidn't know this. Please add!

   Didn’t know this. Please add!

6. • CommentRowNumber6.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 10th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexI looked around a bit and it seems to me that Beck modules are considered in a more "local" picture – one fixes the base $A$ – whereas the current page is more concerned with the global picture. So I think I will start a new page instead.

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

7. • CommentRowNumber7.
   • CommentAuthorn.mertes
   • CommentTimeApr 14th 2021
   • PermaLink
   Author: n.mertes
   Format: MarkdownItexRegarding 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 $C$." 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?

   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 $C$.” 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?

8. • CommentRowNumber8.
   • CommentAuthorn.mertes
   • CommentTimeApr 14th 2021
   • PermaLink
   Author: n.mertes
   Format: MarkdownItexI suppose I spoke a bit too soon, it looks like the page Beck Module states that this left adjoint exists whenever $C$ 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?

   I suppose I spoke a bit too soon, it looks like the page Beck Module states that this left adjoint exists whenever $C$ 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?

9. • CommentRowNumber9.
   • CommentAuthornLab edit announcer
   • CommentTimeApr 4th 2024
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   Author: nLab edit announcer
   Format: MarkdownItexFixed typo sjb <a href="https://ncatlab.org/nlab/revision/diff/tangent+category/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/tangent+category/23">v23</a>, <a href="https://ncatlab.org/nlab/show/tangent+category">current</a>

   Fixed typo

   sjb

   diff, v23, current