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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.
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.
I added a proof that the left adjoint of the “square-zero extension” functor $Mod \to CRing$ constructs Kähler differentials.
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.
Didn’t know this. Please add!
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.
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?
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?
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