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1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeOct 17th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexI presume that >## Definition >Let $C$ be a [[category]] with [[pullback]]s. Then the **tangent category** $T_C$ of $C$ is the category whose >* [[object]]s are pairs $(A,\mathcal{A})$ with $A \in Ob(C)$ and with $\mathcal{A}$ an abelian [[group object]] in the [[overcategory]] $C/B$; contained a minor typo, so I replaced $C/B$ with $C/A$.

   I presume that

   Definition

   Let $C$ be a category with pullbacks. Then the tangent category $T_C$ of $C$ is the category whose

   • objects are pairs $(A,\mathcal{A})$ with $A \in Ob(C)$ and with $\mathcal{A}$ an abelian group object in the overcategory $C/B$;

   contained a minor typo, so I replaced $C/B$ with $C/A$.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 17th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexRight, thanks for fixing that!

   Right, thanks for fixing that!

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeOct 17th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexLink: [[tangent category]]

   Link: tangent category

4. • CommentRowNumber4.
   • CommentAuthorEric
   • CommentTimeOct 18th 2010
   • (edited Oct 18th 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexTracing references took me to [[tangent (infinity,1)-category]] and from there to [[Deformation Theory]] and from there to the [arxiv](http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.3091v2.pdf). I'm starting see a blurry image appear, but an image nontheless. One question... Why does Lurie begin with [[Kähler differentials]]? They've always seemed a bit kinky to me (but in a cool way). That is, they never seemed 100% natural due to the implied commutative of functions and differentials. Much more natural, in my opinion, are [[universal differential envelopes]] and their quotients. What if, instead of starting with Kähler differentials, you started with simply differential modules (without the implied $a (d b) = (d b) a$) and then followed Lurie's prescriptions? Is there a noncommutative version of [[stabilization]] or something? PS: In Lurie's paper on [page 5](http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.3091v2.pdf#Page=5), he says >here $Stab$ denotes the stabilization construction introduced in §S.8. Silly question. What is he referring to? §S.8? There are only 3 sections in the paper :o

   Tracing references took me to tangent (infinity,1)-category and from there to Deformation Theory and from there to the arxiv.

   I’m starting see a blurry image appear, but an image nontheless.

   One question…

   Why does Lurie begin with Kähler differentials? They’ve always seemed a bit kinky to me (but in a cool way). That is, they never seemed 100% natural due to the implied commutative of functions and differentials.

   Much more natural, in my opinion, are universal differential envelopes and their quotients. What if, instead of starting with Kähler differentials, you started with simply differential modules (without the implied $a (d b) = (d b) a$) and then followed Lurie’s prescriptions?

   Is there a noncommutative version of stabilization or something?

   PS: In Lurie’s paper on page 5, he says

   here $Stab$ denotes the stabilization construction introduced in §S.8.

   Silly question. What is he referring to? §S.8? There are only 3 sections in the paper :o

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2010
   • (edited Oct 18th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex"S" stands for his article on [[stable (infinity,1)-categories]], and the stabilization he refers to is [[stabilization]]

   “S” stands for his article on stable (infinity,1)-categories, and the stabilization he refers to is stabilization