tangent category

This page is about the construction of “the tangent category of a category” by abelianization. For categories equipped with an abstract “tangent bundle” construction on their objects, see tangent bundle category.



The notion of tangent category of a category CC 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 T CT_C of CC is effectively the fiberwise abelianization of the codomain fibration cod:[I,C]Ccod : [I,C] \to C:

we may think of it as obtained from the codomain fibration by replacing each overcategory fiber [I,C] A=C/A[I,C]_A = C/A by the corresponding category of abelian group objects and restricting the morphisms such as to respect the abelian group object structure.


Let CC be a category with pullbacks. Then the tangent category T CT_C of CC is the category whose

  • objects are pairs (A,𝒜)(A,\mathcal{A}) with AOb(C)A \in Ob(C) and with 𝒜\mathcal{A} a Beck module over AA, i.e. an abelian group object in the overcategory C/AC/A;

    notice that for B\mathcal{B} \to B an object in the overcategory that is equipped with the structure of an abelian group object – notably with a product prod :× Bprod_{\mathcal{B}} : \mathcal{B} \times_B \mathcal{B} \to \mathcal{B} – and for f:ABf : A \to B any morphism in CC, the pullback f *:=A× Bf^* \mathcal{B} := A \times_B \mathcal{B} in CC is naturally equipped with the structure of an abelian group object in C/AC/A;

  • morphisms (f,𝒻):(A,𝒜)(B,)(f, \mathcal{f}) : (A,\mathcal{A}) \to (B, \mathcal{B}) are commuting squares

    𝒜 𝒻 A f B \array{ \mathcal{A} &\stackrel{\mathcal{f}}{\to}& \mathcal{B} \\ \downarrow && \downarrow \\ A &\stackrel{f}{\to}& B }

    such that the induced morphism 𝒜f *\mathcal{A} \to f^*\mathcal{B} is a morphism of abelian group objects in C/AC/A;

  • composition of morphisms is given in the evident way by (f 2,𝒻 2)(f 1,𝒻 1)=(f 2f 1,𝒻 2𝒻 1)(f_2, \mathcal{f}_2) \circ (f_1, \mathcal{f}_1) = (f_2 \circ f_1, \mathcal{f}_2 \circ \mathcal{f}_1) .



There is an evident functor p:T CCp : T_C \to C, the underlying codomain fibration:

p:(𝒜 𝒻 A f B)(AfB). p \;\;: \;\; \left( \array{ \mathcal{A} &\stackrel{\mathcal{f}}{\to}& \mathcal{B} \\ \downarrow && \downarrow \\ A &\stackrel{f}{\to}& B } \right) \;\; \mapsto \;\; (A \stackrel{f}{\to} B) \,.

The fiber over Id AId_A of this functor is the category of abelian group objects in the overcategory C/AC/A:

(T C) AAb(C/A). (T_C)_A \simeq Ab(C/A) \,.

There is also another functor q:T CCq : T_C \to C, inherited from the domain cofibration

q:(𝒜 𝒻 A f B)(𝒜𝒻), q \;\;: \;\; \left( \array{ \mathcal{A} &\stackrel{\mathcal{f}}{\to}& \mathcal{B} \\ \downarrow && \downarrow \\ A &\stackrel{f}{\to}& B } \right) \;\; \mapsto \;\; (\mathcal{A} \stackrel{\mathcal{f}}{\to} \mathcal{B}) \,,

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 CC).

Of a site

We discuss morphisms of sites from a site to its tangent category.


Let CC be a category with finite limits and let T CCT_C \to C be its tangent category.

There is then the 0-section i:CT Ci : C \to T_C which sends AA to the terminal object Id:AAId : A\to A in the overcategory, equipped, necessarily, with the trivial group structure. This exhibits CC as a retract of T CT_C

(CiT CpC)=Id C. (C \stackrel{i}{\to} T_C \stackrel{p}{\to} C) = Id_C \,.

Assume now that C opC^{op} has pullbacks and is equipped with a coverage, hence with the structure of a site.

Equip (T C) op(T_C)^{op} with the coverage where {f i:U iU}\{f_i : U_i \to U\} is a cover in (T C) op(T_C)^{op} precisely if its image {p(f i):p(U i)p(U)}\{p(f_i) : p(U_i) \to p(U)\} is a cover in C opC^{op}.

Then the 0-section C op(T C) opC^{op} \to (T_C)^{op} preserves covers.

We claim it also preserves limits: i.e. that i:CT Ci : C \to T_C preserves colimits:

let F:KCF : K \to C be a diagram and lim F\lim_\to F its colimit in CC. Then let QQ be any cocone under iFi \circ F in T CT_C. By applying pp to that cocone we find that there is a unique morphism of cocones lim Fp(Q)\lim_\to F \to p(Q) in CC. But any morphism of the form Ap(B)A \to p(B) for ACA \in C and BT CB \in T_C has a unique lift to a morphism i(A)Bi(A) \to B in T CT_C (because the trivial ablian group is initial, so that the morphism in T CT_C is fixed by its underlying morphism in CC).

So for any coverage on C opC^{op} and the above induced coverage on (T C) op(T_C)^{op}, the 0-section i:C op(T C) opi : C^{op} \to (T_C)^{op} is a morphism of sites.

Accordingly, we obtain a geometric morphism of sheaf toposes

Sh((T C) op)Sh(C op). Sh((T_C)^{op}) \stackrel{\leftarrow}{\underset{}{\to}} Sh(C^{op}) \,.


Modules as tangents to rings


For C=C = CRing we have T CT_C \simeq Mod.

(More generally for C=C = Ring then T CT_C is the category of bimodules, see at Beck module – Over associative algebras).


Consider the functor

F:ModT CRing F: Mod \to T_{CRing}

that sends an RR-module NN to the square-0 extension ring RNp 1RR \oplus N \stackrel{p_1}{\to} R, regarded as an abelian group object in CRing/RCRing/R.

The action on morphisms is given as follows: if (R 1,N 1)(R_1,N_1) and (R 2,N 2)(R_2,N_2) are two objects in Mod, then a morphism between them is a pair (f:R 1R 2,f *:N 1f *N 2)(f : R_1 \to R_2, f_*:N_1 \to f^* N_2) consisting of a ring homorphism and a morphism of R 1R_1 modules from N 1N_1 to R 1 fN 2R_1 \otimes_f N_2; the corresponding morphism of rings R 1N 1R 2N 2R_1\oplus N_1\to R_2\oplus N_2 is (r 1,n 1)(f(r 1),f *(n 2))(r_1,n_1)\mapsto (f(r_1),f_*(n_2)). The induced morphism of rings R 1N 1R 1× R 2(R 2N 2)R 1f *N 2R_1\oplus N_1\to R_1\times_{R_2}(R_2\oplus N_2)\cong R_1\oplus f^*N_2 is explicitly given by (r 1,n 1)(r 1,f *(n 1))(r_1,n_1)\mapsto (r_1,f_*(n_1)) and is easily checked to be a morphism of abelian group objects over R 1R_1.

Moreover, by the natural isomorphism R 1× R 2(R 2N 2)R 1f *N 2R_1\times_{R_2}(R_2\oplus N_2)\cong R_1\oplus f^*N_2 in Ab(CRing/R)Ab(CRing/R), showing that F:ModT CRingF:Mod\to T_{CRing} is an equivalence is reduced to showing that FF is a fibrewise equivalence, i.e., that that for any fixed ring RR,

F R:Mod RAb(CRing/R) F_R: Mod_R \to Ab(CRing/R)

is an equivalence of categories. This is shown at module.


The domain projection ModCRingMod \to CRing has a left adjoint, namely the functor assigning to each commutative ring AA the pair (A,Ω A)(A, \Omega_A), where Ω A\Omega_A is the AA-module of Kähler differentials.


Let AA and BB be commutative rings, let MM be a BB-module, and consider BMB \oplus M as a ring as in the previous proof. Then, to give a ring homomorphism f:ABMf : A \to B \oplus M is the same as giving a ring homomorphism f 0:ABf_0 : A \to B and an additive homomorphism f 1:AMf_1 : A \to M such that

f 1(xy)=f 0(x)f 1(y)+f 0(y)f 1(x)f_1 (x y) = f_0 (x) f_1 (y) + f_0 (y) f_1 (x)

for all xx and yy in AA. But by the universal property of Ω A\Omega_A, this is the same as giving a morphism (A,Ω A)(B,M)(A, \Omega_A) \to (B, M) in ModMod.

Modules over smooth algebras

Let SmoothAlgSmoothAlg (or C RingC^\infty Ring) be the category of smooth algebras. Notice that there is a canonical forgetful functor

U:SmoothAlgRing U : SmoothAlg \to Ring

to the underlying ordinary rings.


There is an equivalence of categories

T SmoothAlgSmoothAlg× RingT Ring, T_{SmoothAlg} \stackrel{\simeq}{\to} SmoothAlg \times_{Ring} T_{Ring} \,,

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.


We may regard an object in T SmoothAlgT_{SmoothAlg} as a module over a smooth algebra. The above says in particular that modules over smooth algebras are just modules over the underlying ordinary rings. However, the category structure on T SmoothAlgT_{SmoothAlg} does reflect that modules over smooth algebras have a different nature than just bare modules, notably in that the left adjoint to the projection T SmoothAlgSmoothAlgT_{SmoothAlg} \to SmoothAlg produces the correct C C^\infty-derivations and C C^\infty-Kähler differentials (see there) as opposed to the purely algebraic ones.


For any category SS we have that

(T SmoothAlg) SSmoothAlg S× Ring S(T Ring) S. (T_{SmoothAlg})^S \simeq SmoothAlg^S \times_{Ring^S} (T_{Ring})^S \,.

So in particular for S=ΔS = \Delta 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.


For a fixed smooth algebra RR, the forgetful functor

U:Ab(SmoothAlg/R)Ab(Ring/U(R)) U : Ab(SmoothAlg/R) \to Ab(Ring/U(R))

is an equivalence of categories.


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 SmoothAlg/RSmoothAlg/R its smooth algebra structure on the underlying ring structure is the unique such smooth algebra that makes it an abelian group object over RR. Whith this it is then easy to see that UU is in fact an isomorphism of categories.

The crucial property underlying this statement is that the Lawvere theory T=T = CartSp over wich smooth algebras are TT-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 kk \in \mathbb{N} and every smooth function f: kf : \mathbb{R}^k \to \mathbb{R} there are smooth functions {h i,jC ( k× k,)} i,j=1 n\{h_{i,j} \in C^\infty(\mathbb{R}^k \times \mathbb{R}^k, \mathbb{R})\}_{i,j = 1}^n such that the function

f+: k× k f \circ + : \mathbb{R}^k \times \mathbb{R}^k \to \mathbb{R}

has an expansion given for all p,w kp, w \in \mathbb{R}^k by

f(p+w)=f(p)+ l=1 kw lfx l(p)+ i,jw iw jh i,j(p,w). f(p+w) = f(p) + \sum_{l = 1}^k w_l \cdot \frac{\partial f}{\partial x_l}(p) + \sum_{i,j} w_i \cdot w_j h_{i,j}(p,w) \,.

We now use that any smooth algebra AA regarded as a product-preserving functor A:CartSpSetA : CartSp \to Set reflects these relations in that for all r,eA(k)=U(A) kr, e \in A(k) = U(A)^k we have that

A(f)(r+e)=A(f)(r)+ l=1 kw lA(fx l)(r)+ i,je ie jA(h i,j)(r,e). A(f)(r+e) = A(f)(r) + \sum_{l = 1}^k w_l \cdot A\left(\frac{\partial f}{\partial x_l}\right)(r) + \sum_{i,j} e_i \cdot e_j A(h_{i,j})(r,e) \,.

Now if RSmoothAlgR \in SmoothAlg and AA is an object in Ab(SmoothAlgebra/R)Ab(SmoothAlgebra/R) then in particular its underlying ring will be an object in Ab(Ring/U(R))Ab(Ring/U(R)). By the above theorem this means that the underlying ring is a square-zero extension U(R)NU(R) \oplus N by some NU(R)ModN \in U(R) Mod.

So it follows every element of A(1)A(1) is of the form (r,ϵ)(r, \epsilon) with ϵN\epsilon \in N and we can always write it as

(r,0)+(0,ϵ). (r,0) + (0, \epsilon) \,.

Moreover, since AA is by assumption a group object over RR, it follows that for all fC ( k,)f \in C^\infty(\mathbb{R}^k , \mathbb{R}) and for all rR(1)r \in R(1) we have

A(f)(r)=R(f)(r). A(f)(r) = R(f)(r) \,.

So we only need to know how AA 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 A(f)(r)A(f)(r) and to ordinary multiplication and addition, so it actually fixes A(f)A(f) from the restriction of A(f)A(f) to elements of the form (r,0)(r,0) and the module structure on NN:

A(f)(r+ϵ)=A(f)(r)+ l=1 kw lA(fx l)(r)+0 A(f)(r +\epsilon) = A(f)(r) + \sum_{l = 1}^k w_l \cdot A\left(\frac{\partial f}{\partial x_l}\right)(r) + 0

since ϵ iϵ j=0\epsilon_i \cdot \epsilon_j = 0 in the underlying square-0 extension of AA and hence also in AA.

In summary this shows that the forgetful functor UU 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 T SmoothAlgSmoothAlg× RingT RingT_{SmoothAlg} \simeq SmoothAlg \times_{Ring} T_{Ring} is a bijection on objects.

Since the pullbacks that are involved in the definition of the tangent category T SmoothAlgT_{SmoothAlg} are preserved by the right adjoint forgetful functor U:SmoothAlgRingU : SmoothAlg \to Ring (a special case of the general facts about Relative free T-algebra adjunctions), checking bijection on hom-sets

T SmoothAlg(AR 1,BR 2)SmoothAlg× RingT Ring((U(A),R 1),(U(B),R 2)) T_{SmoothAlg}(A\to R_1,B \to R_2) \to SmoothAlg \times_{Ring} T_{Ring} ((U(A), R_1), (U(B), R_2))

amounts to checking for each f:R 1R 2f : R_1 \to R_2 bijections of hom-sets of abelian group objects

Ab(SmoothAlg/R 1)(A,f *B)Ab(Ring/U(R 1))(U(A),f *U(B)). Ab(SmoothAlg/R_1)(A, f^* B) \to Ab(Ring/U(R_1))(U(A), f^* U(B)) \,.

That this is a bijection is the statement of the above lemma.


The original observation that T RingModT_{Ring} \simeq Mod is due to

A discusson of T SmoothAlgT_{SmoothAlg} is in

Revised on October 17, 2017 02:12:55 by Mike Shulman (