The notion of tangent category of a category is an approximation to the notion of tangent (∞,1)-category in ordinary category theory.
For the moment see there for further motivation.
The tangent category of is effectively the fiberwise abelianization of the codomain fibration :
we may think of it as obtained from the codomain fibration by replacing each overcategory fiber by the corresponding category of abelian group objects and restricting the morphisms such as to respect the abelian group object structure.
Let be a category with pullbacks. Then the tangent category of is the category whose
objects are pairs with and with a Beck module over , i.e. an abelian group object in the overcategory ;
notice that for an object in the overcategory that is equipped with the structure of an abelian group object – notably with a product – and for any morphism in , the pullback in is naturally equipped with the structure of an abelian group object in ;
morphisms are commuting squares
such that the induced morphism is a morphism of abelian group objects in ;
composition of morphisms is given in the evident way by .
There is an evident functor , the underlying codomain fibration:
The fiber over of this functor is the category of abelian group objects in the overcategory :
There is also another functor , inherited from the domain cofibration
where we first forget the abelian group object structure and then project onto the domains.
(we should be claiming that this functor has a left adjoint which is a section and computes the Kähler differentials of objects in ).
Of a site
We discuss morphisms of sites from a site to its tangent category.
Let be a category with finite limits and let be its tangent category.
There is then the 0-section which sends to the terminal object in the overcategory, equipped, necessarily, with the trivial group structure. This exhibits as a retract of
Assume now that has pullbacks and is equipped with a coverage, hence with the structure of a site.
Equip with the coverage where is a cover in precisely if its image is a cover in .
Then the 0-section preserves covers.
We claim it also preserves limits: i.e. that preserves colimits:
let be a diagram and its colimit in . Then let be any cocone under in . By applying to that cocone we find that there is a unique morphism of cocones in . But any morphism of the form for and has a unique lift to a morphism in (because the trivial ablian group is initial, so that the morphism in is fixed by its underlying morphism in ).
So for any coverage on and the above induced coverage on , the 0-section is a morphism of sites.
Accordingly, we obtain a geometric morphism of sheaf toposes
Modules as tangents to rings
For CRing we have Mod.
(More generally for Ring then is the category of bimodules, see at Beck module – Over associative algebras).
Consider the functor
that sends an -module to the square-0 extension ring , regarded as an abelian group object in .
The action on morphisms is given as follows: if and are two objects in Mod, then a morphism between them is a pair consisting of a ring homorphism and a morphism of modules from to ; the corresponding morphism of rings is . The induced morphism of rings is explicitly given by and is easily checked to be a morphism of abelian group objects over .
Moreover, by the natural isomorphism in , showing that is an equivalence is reduced to showing that is a fibrewise equivalence, i.e., that that for any fixed ring ,
is an equivalence of categories. This is shown at module.
The domain projection has a left adjoint, namely the functor assigning to each commutative ring the pair , where is the -module of Kähler differentials.
Let and be commutative rings, let be a -module, and consider as a ring as in the previous proof. Then, to give a ring homomorphism is the same as giving a ring homomorphism and an additive homomorphism such that
for all and in . But by the universal property of , this is the same as giving a morphism in .
Modules over smooth algebras
Let (or ) be the category of smooth algebras. Notice that there is a canonical forgetful functor
to the underlying ordinary rings.
There is an equivalence of categories
where on the right we have the strict pullback (i.e. taken in the 1-category Cat).
We give the proof below. First some remarks and corollaries.
For any category we have that
So in particular for the simplex category we have that simplicial modules over simplicial smooth algebras are as objects just ordinary simplicial modules over the underlying simplicial rings.
For proving the above theorem the main step is the following lemma.
The statement was suggested at some point by Thomas Nikolaus in discussion with Urs Schreiber, who then asked Herman Stel to prove it. A writeup is in (Stel).
We discuss in detail that the functor is injective on objects, in that for an any abelian group object in its smooth algebra structure on the underlying ring structure is the unique such smooth algebra that makes it an abelian group object over . Whith this it is then easy to see that is in fact an isomorphism of categories.
The crucial property underlying this statement is that the Lawvere theory CartSp over wich smooth algebras are -algebras is in fact a Fermat theory in that Hadamard's lemma holds for smooth functions in particular on Cartesian spaces.
This implies that for every and every smooth function there are smooth functions such that the function
has an expansion given for all by
We now use that any smooth algebra regarded as a product-preserving functor reflects these relations in that for all we have that
Now if and is an object in then in particular its underlying ring will be an object in . By the above theorem this means that the underlying ring is a square-zero extension by some .
So it follows every element of is of the form with and we can always write it as
Moreover, since is by assumption a group object over , it follows that for all and for all we have
So we only need to know how acts on mixed terms. The point now is that the above Hadamard-quotient formula reduces the action of any smooth function to just operations of this form and to ordinary multiplication and addition, so it actually fixes from the restriction of to elements of the form and the module structure on :
since in the underlying square-0 extension of and hence also in .
In summary this shows that the forgetful functor is injective on objects. The above formula also directly implies, conversely, that the functor is surjective on objects, hence an isomorphism on objects, and moreover that it is a full and faithful functor.
Finally we come to the proof of the full theorem above
The above lemma shows that is a bijection on objects.
Since the pullbacks that are involved in the definition of the tangent category are preserved by the right adjoint forgetful functor (a special case of the general facts about Relative free T-algebra adjunctions), checking bijection on hom-sets
amounts to checking for each bijections of hom-sets of abelian group objects
That this is a bijection is the statement of the above lemma.
The original observation that is due to
A discusson of is in