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• CommentRowNumber1.
• CommentAuthorEric
• CommentTimeOct 17th 2010

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 $C/A$.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 17th 2010

Right, thanks for fixing that!

• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeOct 17th 2010
• CommentRowNumber4.
• CommentAuthorEric
• CommentTimeOct 18th 2010
• (edited Oct 18th 2010)

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

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 18th 2010
• (edited Oct 18th 2010)

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