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\begin{document}

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\section*{tangent category}

\begin{quote}%
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]].


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{of_a_site}{Of a site}\dotfill \pageref*{of_a_site} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{OrdinaryModules}{Modules as tangents to rings}\dotfill \pageref*{OrdinaryModules} \linebreak
\noindent\hyperlink{ModulesOverSmoothAlgebras}{Modules over smooth algebras}\dotfill \pageref*{ModulesOverSmoothAlgebras} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \textbf{tangent category} of a [[category]] $C$ 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 $T_C$ of $C$ is effectively the \emph{fiberwise abelianization} of the [[codomain fibration]] $cod : [I,C] \to C$:

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

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $C$ be a [[category]] with [[pullback]]s. Then the \textbf{tangent category} $T_C$ of $C$ is the category whose

\begin{itemize}%
\item [[object]]s are pairs $(A,\mathcal{A})$ with $A \in Ob(C)$ and with $\mathcal{A}$ a [[Beck module]] over $A$, i.e. an abelian [[group object]] in the [[overcategory]] $C/A$;

notice that for $\mathcal{B} \to B$ an object in the overcategory that is equipped with the structure of an abelian group object -- notably with a product $prod_{\mathcal{B}} : \mathcal{B} \times_B \mathcal{B} \to  \mathcal{B}$ -- and for $f : A \to B$ any morphism in $C$, the [[pullback]] $f^* \mathcal{B} := A \times_B \mathcal{B}$ in $C$ is naturally equipped with the structure of an abelian group object in $C/A$;


\item [[morphism]]s $(f, \mathcal{f}) : (A,\mathcal{A}) \to (B, \mathcal{B})$ are commuting squares

\begin{displaymath}
\itexarray{
    \mathcal{A} &\stackrel{\mathcal{f}}{\to}& \mathcal{B}
    \\
    \downarrow && \downarrow 
    \\
    A &\stackrel{f}{\to}& B
  }
\end{displaymath}
such that the induced morphism $\mathcal{A} \to f^*\mathcal{B}$ is a morphism of abelian group objects in $C/A$;


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



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

There is an evident [[functor]] $p : T_C \to C$, the underlying [[codomain fibration]]:

\begin{displaymath}
p 
  \;\;:
  \;\; 
  \left(
    \itexarray{
      \mathcal{A} &\stackrel{\mathcal{f}}{\to}& \mathcal{B}
      \\
      \downarrow && \downarrow
      \\
      A &\stackrel{f}{\to}& B
    }
  \right)
  \;\;
  \mapsto
  \;\;
  (A \stackrel{f}{\to} B)
  \,.
\end{displaymath}
The fiber over $Id_A$ of this functor is the category of abelian group objects in the [[overcategory]] $C/A$:

\begin{displaymath}
(T_C)_A \simeq Ab(C/A)
  \,.
\end{displaymath}
There is also another functor $q : T_C \to C$, inherited from the [[domain cofibration]]

\begin{displaymath}
q 
  \;\;:
  \;\; 
  \left(
    \itexarray{
      \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})
  \,,
\end{displaymath}
where we first forget the abelian group object structure and then project onto the domains.

\begin{quote}%
(we should be claiming that this functor has a left adjoint which is a section and computes the [[Kähler differentials]] of objects in $C$).


\end{quote}
\hypertarget{of_a_site}{}\subsubsection*{{Of a site}}\label{of_a_site}

We discuss morphisms of [[site]]s from a site to its tangent category.

\begin{quote}%
check


\end{quote}
Let $C$ be a category with [[finite limit]]s and let $T_C \to C$ be its tangent category.

There is then the 0-section $i : C \to T_C$ which sends $A$ to the terminal object $Id : A\to A$ in the overcategory, equipped, necessarily, with the trivial group structure. This exhibits $C$ as a [[retract]] of $T_C$

\begin{displaymath}
(C \stackrel{i}{\to} T_C \stackrel{p}{\to} C) = Id_C
  \,.
\end{displaymath}
Assume now that $C^{op}$ has [[pullback]]s and is equipped with a [[coverage]], hence with the structure of a [[site]].

Equip $(T_C)^{op}$ with the [[coverage]] where $\{f_i : U_i \to U\}$ is a cover in $(T_C)^{op}$ precisely if its image $\{p(f_i) : p(U_i) \to p(U)\}$ is a cover in $C^{op}$.

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

We claim it also preserves limits: i.e. that $i : C \to T_C$ preserves [[colimit]]s:

let $F : K \to C$ be a diagram and $\lim_\to F$ its colimit in $C$. Then let $Q$ be any cocone under $i \circ F$ in $T_C$. By applying $p$ to that cocone we find that there is a unique morphism of cocones $\lim_\to F \to p(Q)$ in $C$. But any morphism of the form $A \to p(B)$ for $A \in C$ and $B \in T_C$ has a unique lift to a morphism $i(A) \to B$ in $T_C$ (because the trivial ablian group is initial, so that the morphism in $T_C$ is fixed by its underlying morphism in $C$).

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

Accordingly, we obtain a [[geometric morphism]] of [[category of sheaves|sheaf toposes]]

\begin{displaymath}
Sh((T_C)^{op}) \stackrel{\leftarrow}{\underset{}{\to}}
  Sh(C^{op})
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{OrdinaryModules}{}\subsubsection*{{Modules as tangents to rings}}\label{OrdinaryModules}

\begin{prop}
\label{}\hypertarget{}{}
For $C =$ [[CRing]] we have $T_C \simeq$ [[Mod]].

\end{prop}
(More generally for $C =$ [[Ring]] then $T_C$ is the category of [[bimodules]], see at \emph{\href{Beck+module#OverAssociativeAlgebras}{Beck module -- Over associative algebras}}).

\begin{proof}
Consider the functor

\begin{displaymath}
F: Mod \to T_{CRing}
\end{displaymath}
that sends an $R$-module $N$ to the [[square-0 extension]] ring $R \oplus N \stackrel{p_1}{\to} R$, regarded as an abelian group object in $CRing/R$.

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

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

\begin{displaymath}
F_R:  Mod_R \to Ab(CRing/R)
\end{displaymath}
is an [[equivalence of categories]]. This is shown at [[module]].

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
The domain projection $Mod \to CRing$ has a left adjoint, namely the functor assigning to each commutative ring $A$ the pair $(A, \Omega_A)$, where $\Omega_A$ is the $A$-module of [[Kähler differentials]].

\end{prop}
\begin{proof}
Let $A$ and $B$ be commutative rings, let $M$ be a $B$-module, and consider $B \oplus M$ as a ring as in the previous proof. Then, to give a ring homomorphism $f : A \to B \oplus M$ is the same as giving a ring homomorphism $f_0 : A \to B$ and an additive homomorphism $f_1 : A \to M$ such that

\begin{displaymath}
f_1 (x y) = f_0 (x) f_1 (y) + f_0 (y) f_1 (x)
\end{displaymath}
for all $x$ and $y$ in $A$. But by the universal property of $\Omega_A$, this is the same as giving a morphism $(A, \Omega_A) \to (B, M)$ in $Mod$.

\end{proof}
\hypertarget{ModulesOverSmoothAlgebras}{}\subsubsection*{{Modules over smooth algebras}}\label{ModulesOverSmoothAlgebras}

Let $SmoothAlg$ (or $C^\infty Ring$) be the category of [[smooth algebra]]s. Notice that there is a canonical [[forgetful functor]]

\begin{displaymath}
U : SmoothAlg \to Ring
\end{displaymath}
to the underlying ordinary [[ring]]s.

\begin{theorem}
\label{}\hypertarget{}{}
There is an [[equivalence of categories]]

\begin{displaymath}
T_{SmoothAlg} \stackrel{\simeq}{\to}
  SmoothAlg \times_{Ring} T_{Ring}
  \,,
\end{displaymath}
where on the right we have the strict [[pullback]] (i.e. taken in the 1-category [[Cat]]).

\end{theorem}
We give the proof below. First some remarks and corollaries.

\begin{remark}
\label{}\hypertarget{}{}
We may regard an object in $T_{SmoothAlg}$ 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 $T_{SmoothAlg}$ does reflect that modules over smooth algebras have a different nature than just bare modules, notably in that the left adjoint to the projection $T_{SmoothAlg} \to SmoothAlg$ produces the correct $C^\infty$-[[derivation]]s and $C^\infty$-[[Kähler differential]]s (see there) as opposed to the purely algebraic ones.

\end{remark}
\begin{cor}
\label{}\hypertarget{}{}
For any category $S$ we have that

\begin{displaymath}
(T_{SmoothAlg})^S \simeq
  SmoothAlg^S \times_{Ring^S} (T_{Ring})^S
  \,.
\end{displaymath}
So in particular for $S = \Delta$ the [[simplex category]] we have that [[simplicial object|simplicial]] modules over simplicial smooth algebras are as objects just ordinary simplicial modules over the underlying [[simplicial ring]]s.

\end{cor}
For proving the \hyperlink{TangentOfSmoothAlgTheorem}{above theorem} the main step is the following lemma.

\begin{lemma}
\label{}\hypertarget{}{}
For a fixed smooth algebra $R$, the [[forgetful functor]]

\begin{displaymath}
U : Ab(SmoothAlg/R) \to Ab(Ring/U(R))
\end{displaymath}
is an [[equivalence of categories]].

\end{lemma}
\begin{proof}
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 (\hyperlink{Stel}{Stel}).

We discuss in detail that the functor is injective on objects, in that for an any abelian group object in $SmoothAlg/R$ its smooth algebra structure on the underlying ring structure is the \emph{unique} such smooth algebra that makes it an abelian group object over $R$. Whith this it is then easy to see that $U$ is in fact an isomorphism of categories.

The crucial property underlying this statement is that the [[Lawvere theory]] $T =$ [[CartSp]] over wich smooth algebras are $T$-algebras is in fact a [[Fermat theory]] in that [[Hadamard's lemma]] holds for [[smooth function]]s in particular on [[Cartesian space]]s.

This implies that for every $k \in \mathbb{N}$ and every smooth function $f : \mathbb{R}^k \to \mathbb{R}$ there are [[smooth function]]s $\{h_{i,j} \in C^\infty(\mathbb{R}^k \times \mathbb{R}^k, \mathbb{R})\}_{i,j = 1}^n$ such that the function

\begin{displaymath}
f \circ + : \mathbb{R}^k \times \mathbb{R}^k \to \mathbb{R}
\end{displaymath}
has an expansion given for all $p, w \in \mathbb{R}^k$ by

\begin{displaymath}
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)
  \,.
\end{displaymath}
We now use that any [[smooth algebra]] $A$ regarded as a product-preserving functor $A : CartSp \to Set$ reflects these relations in that for all $r, e \in A(k) = U(A)^k$ we have that

\begin{displaymath}
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)
  \,.
\end{displaymath}
Now if $R \in SmoothAlg$ and $A$ is an object in $Ab(SmoothAlgebra/R)$ then in particular its underlying ring will be an object in $Ab(Ring/U(R))$. By the \hyperlink{OrdinaryModules}{above theorem} this means that the underlying ring is a [[square-zero extension]] $U(R) \oplus N$ by some $N \in U(R) Mod$.

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

\begin{displaymath}
(r,0) + (0, \epsilon) 
  \,.
\end{displaymath}
Moreover, since $A$ is by assumption a group object over $R$, it follows that for all $f \in C^\infty(\mathbb{R}^k , \mathbb{R})$ and for all $r \in R(1)$ we have

\begin{displaymath}
A(f)(r) = R(f)(r)
  \,.
\end{displaymath}
So we only need to know how $A$ 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 $A(f)(r)$ and to ordinary multiplication and addition, so it actually fixes $A(f)$ from the restriction of $A(f)$ to elements of the form $(r,0)$ and the module structure on $N$:

\begin{displaymath}
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
\end{displaymath}
since $\epsilon_i \cdot \epsilon_j = 0$ in the underlying [[square-0 extension]] of $A$ and hence also in $A$.

In summary this shows that the forgetful functor $U$ 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]].

\end{proof}
Finally we come to the proof of the \hyperlink{TangentOfSmoothAlgTheorem}{full theorem above}

\begin{proof}
The above lemma shows that $T_{SmoothAlg} \simeq SmoothAlg \times_{Ring} T_{Ring}$ is a bijection on objects.

Since the [[pullback]]s that are involved in the definition of the tangent category $T_{SmoothAlg}$ are preserved by the [[right adjoint]] forgetful functor $U : SmoothAlg \to Ring$ (a special case of the general facts about ), checking bijection on [[hom-set]]s

\begin{displaymath}
T_{SmoothAlg}(A\to R_1,B \to R_2) \to SmoothAlg \times_{Ring} T_{Ring} ((U(A), R_1), (U(B), R_2))
\end{displaymath}
amounts to checking for each $f : R_1 \to R_2$ bijections of hom-sets of abelian group objects

\begin{displaymath}
Ab(SmoothAlg/R_1)(A, f^* B) 
  \to 
  Ab(Ring/U(R_1))(U(A), f^* U(B))
  \,.
\end{displaymath}
That this is a bijection is the statement of the above lemma.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[tangent (∞,1)-category]].

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original observation that $T_{Ring} \simeq Mod$ is due to

\begin{itemize}%
\item [[Daniel Quillen]], \ldots{}

\end{itemize}
A discusson of $T_{SmoothAlg}$ is in

\begin{itemize}%
\item [[Herman Stel]], \emph{[[schreiber:master thesis Stel|∞-Stacks and their Function Algebras]]}

\end{itemize}
[[!redirects tangent categories]]



\end{document}