tangent bundle

> This entry is about [[tangent vectors]] on [[differentiable manifolds]] and the [[bundle]] they form. For *[[tangents]]* and the *[[tangent function]]* see there.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _tangent bundle_ $T X \to X$ of a (sufficiently [[differentiation|differentiable]]) [[space]] $X$ is a [[bundle]] over $X$ whose [[fiber]] over a [[point]] $x \in X$ is the [[tangent space]] at that point, namely the collection of [[infinitesimal space|infinitesimal]] curves in $X$ emanating at $x$: "[[tangent vectors]]".

For nice enough [[spaces]] such as [[differentiable manifolds]] or more generally [[microlinear spaces]], the tangent bundle of $X$ is a [[vector bundle]] over $X$.

For example the graphics on the right shows the [[2-sphere]] with one of its [[tangent spaces]]. The tangent bundle of the sphere is the union of all these [[tangent spaces]], regarded as a [[topological space|topological]] [[bundle]] of [[vector space]] (a [[vector bundle]]) over the 2-sphere.

> graphics grabbed from [Hatcher](vector+bundle#Hatcher)

With a notion of tangent bundle comes the following terminology

* A _tangent vector_ on $X$ at $x \in X$ is an element of $T_x X$.

* The _tangent space_ of $X$ at a point $x$ is the [[fiber]] $T_x(X)$ of $T X$ over $x$;.

* A _tangent vector field_ on $X$ is a [[section]] of $T X$.

The precise definition of tangent bundle depends on the nature of the ambient category of [[space | spaces]]. Below we give first the traditional definitions in ordinary [[differential geometry]]. Then we discuss the construction in more general context of [[smooth toposes]] in [[synthetic differential geometry]] and other categories of [[generalized smooth spaces]].

## Definitions

### Traditional definition
 {#TraditionalDefinition}

We discuss the tangent bundle of a differentiable manifold by first defining tangent vectors as equivalence classes
of differentiable curves in the manifold, then analyzing this construction locally over an atlas, and then gluing these
local constructions together via transition functions.

+-- {: .num_defn #TangencyRelationOnSmoothCurves}
###### Definition
**(tangency relation on differentiable curves)**

Let $X$ be a [[differentiable manifold]] of [[dimension]] $n$ and let $x \in X$ be a point. On the set of [[smooth functions]] of the form

$$
  \gamma \;\colon\; \mathbb{R}^1 \longrightarrow X
$$

such that

$$
  \gamma(0) = x
$$

define the [[relations]]

$$
  (\gamma_1 \sim \gamma_2)
   \coloneqq
   \underset{
      {  { \mathbb{R}^n  \underoverset{}{\phi \, \text{chart}}{\to} U_i \subset X } } \atop { U_i \supset \{x\} }
      }{
        \exists
      }
   \left(
      \frac{d}{d t}(\phi^{-1}\circ \gamma_1)(0)
      =
      \frac{d}{d t}(\phi^{-1}\circ \gamma_2)(0)
   \right)
$$

and

$$
  (\gamma_1 \sim' \gamma_2)
   \coloneqq
   \underset{
      {  { \mathbb{R}^n  \underoverset{}{\phi \, \text{chart}}{\to} U_i \subset X } } \atop { U_i \supset \{x\} }
      }{
        \forall
      }
   \left(
      \frac{d}{d t}(\phi^{-1}\circ \gamma_1)(0)
      =
      \frac{d}{d t}(\phi^{-1}\circ \gamma_2)(0)
   \right)
$$

saying that two such functions are related precisely if either there exists a chart around $x$ such that (or else for all charts around $x$ it is true that) the first [[derivative]] of the two functions regarded via the given chart as functions $\mathbb{R}^1 \to \mathbb{R}^n$, coincide at $t = 0$ (with $t$ denoting the canonical [[coordinate]] function on $\mathbb{R}$).

=--

+-- {: .num_lemma #TangencyIsEquivalenceRelation}
###### Lemma
**(tangency is equivalence relation)**

The two relations in def. \ref{TangencyRelationOnSmoothCurves} are [[equivalence relations]] and they coincide.

=--

+-- {: .proof}
###### Proof

First to see that they coincide, we need to show that if the derivatives in question coincide in one chart $\mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X$, that then they coincide also in any other chart $\mathbb{R}^n \underoverset{\simeq}{\phi_j}{\to} U_j \subset X$.

For brevity, write

$$
  U_{i j} \coloneqq U_i \cap U_j
$$

for the intersection of the two charts.

First of all, since the derivative may be computed in any [[open neighbourhood]] around $t = 0$, and since the differentiable functions $\gamma_i$ are in particular [[continuous functions]], we may restrict to the open neighbourhood

$$
  V \coloneqq \gamma_1^{-1}( U_{i j} ) \cap \gamma_2^{-1}(U_{i j})
  \;\subset\; \mathbb{R}
$$

of $0 \in \mathbb{R}$ and consider the derivatives of the functions

$$
  \gamma_n^i
  \;\coloneqq\;
  (\phi_i^{-1}\vert_{U_{i j}} \circ \gamma_n\vert_{V}) \;\colon\; V \longrightarrow \phi_i^{-1}(U_{i j}) \subset \mathbb{R}^n
$$

and

$$
  \gamma_n^j
  \;\coloneqq\;
  (\phi_j^{-1}\vert_{U_{i j}} \circ \gamma_n \vert _{V}) \;\colon\; V \longrightarrow \phi_j^{-1}(U_{i j}) \subset \mathbb{R}^n
  \,.
$$

But then by definition of the differentiable [[atlas]], there is the differentiable gluing function

$$
  \alpha
  \;\coloneqq\;
  \phi_i^{-1}(U_{i j})
    \underoverset{\simeq}{\phi_i}{\longrightarrow}
  U_{i j}
    \underoverset{\simeq}{\phi_j^{-1}}{\longrightarrow}
  \phi_j^{-1}(U_{i j})
$$

such that

$$
  \gamma_n^j = \alpha \circ \gamma_n^i
$$

for $n \in \{1,2\}$. The [[chain rule]] now relates the derivatives of these functions as

$$
  \frac{d}{d t} \gamma_n^j
    \;=\;
  (D \alpha) \circ \left(\frac{d}{d t} \gamma_n^i  \right)
  \,.
$$

Since $\alpha$ is a [[diffeomorphism]] and since derivatives of diffeomorphisms are linear isomorphisms, this says that the derivative of $\gamma_n^j$ is related to that of $\gamma_n^i$ by a linear isomorphism , and hence

$$
  \left(
   \frac{d}{d t} \gamma_1^i  = \frac{d}{d t} \gamma_2^i
  \right)
    \;\Leftrightarrow\;
  \left(
   \frac{d}{d t} \gamma_1^j = \frac{d}{d t} \gamma_2^j
  \right)
  \,.
$$

Finally, that either relation is an equivalence relation is immediate.

=--

+-- {: .num_defn #TangentVector}
###### Definition
**([[tangent vector]])

Let $X$ be a [[differentiable manifold]] and $x \in X$ a point. Then a _tangent vector_ on $X$ at $x$ is an [[equivalence class]] of the the tangency equivalence relation (def. \ref{TangencyRelationOnSmoothCurves}, lemma \ref{TangencyIsEquivalenceRelation}).

The set of all tangent vectors at $x \in X$ is denoted $T_x X$.

=--

+-- {: .num_lemma #LinearTangentSpace}
###### Lemma
**([[real vector space]] structure on [[tangent vectors]])**

For $X$ a [[differentiable manifold]] of [[dimension]] $n$ and $x \in X$ any point,  let $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U \subset X$ be a [[chart]] with $x \in U \subset X$.

Then there is induced a [[bijection]] of sets

$$
  \mathbb{R}^n \overset{\simeq}{\longrightarrow}  T_x X
$$

from the $n$-dimensional [[Cartesian space]] to the set of tangent vectors at $x$ (def. \ref{TangentVector}) given by sending $\vec v \in \mathbb{R}^n$ to the equivalence class of the following differentiable curve:

$$
  \array{
    \gamma^\phi_{\vec v}
    \colon
    &
    \mathbb{R}^1
      &\overset{ \phi^{-1}(x) + (-)\cdot \vec v }{\longrightarrow}&
   \mathbb{R}^n
      &\underoverset{\simeq}{\phi}{\longrightarrow}&
    U_i
    \subset X
    \\
    &
    t
      &\overset{\phantom{AAA}}{\mapsto}&
    \phi^{-1}(x) +  t \vec v
      &\overset{\phantom{AAA}}{\mapsto}&
    \phi(\phi^{-1}(x) + t \vec v)
  }
  \,.
$$

For $\mathbb{R}^n \underoverset{\simeq}{\phi'}{\longrightarrow} U' \subset X$
another chart with $x \in U' \subset X$, then the linear isomorphism relating these two identifications is
the [[derivative]]

$$
  d \left((\phi')^{-1} \circ \phi  \right)_{ \phi^{-1}(x) }
  \in
  GL(n,\mathbb{R})
$$

of the [[gluing function]] of the two charts at the point $x$:

$$
  \array{
    \mathbb{R}^n
    &&
    \overset{
       d \left((\phi')^{-1} \circ \phi\right)_{\phi^{-1}(x)}}{\longrightarrow} &&
    \mathbb{R}^n
    \\
    & {}_{\mathllap{\simeq}}\searrow && \swarrow_{\mathrlap{\simeq}}
    \\
    && T_x X
  }
  \,.
$$

This is also called the _[[transition function]]_ between the two local identifications of the tangent space.

If $\left\{ \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X \right\}_{i \in I}$
is an [[atlas]] of the [[differentiable manifold]] $X$, then the transition functions

$$
  \left\{
    g_{i j}
    \coloneqq
    d( \phi_j^{-1} \circ \phi_i )_{\phi_i^{-1}(-)}
    \colon
    U_i \cap U_j
      \longrightarrow
    GL(n,\mathbb{R})
  \right\}_{i,j \in I}
$$

defined this way satisfy the following [[Cech cohomology|Cech cocycle]] conditions for all $i,j \in I$, $x \in U_i \cap U_j$

1. $g_{i i}(x) = id_{\mathbb{R}^n}$;

1. $g_{j k}\circ g_{i j}(x) = g_{i k}(x)$.

=--

+-- {: .proof}
###### Proof

The bijectivity of the map is immediate from the fact that the first derivative of $\phi^{-1}\circ \gamma^\phi_{\vec v}$
at $\phi^{-1}(x)$ is $\vec v$.

The formula for the transition function now follows with the [[chain rule]]:

$$
  d \left( (\phi')^{-1} \circ \phi( \phi^{-1}(x) + (-) \vec v )   \right)_0
  =
  d \left( (\phi')^{-1} \circ \phi \right)_{\phi^{-1}(x)} \circ \underset{ =(-)\vec v }{\underbrace{ d ( \phi^{-1}(x) +(-)\vec v )_0 }}
  \,.
$$

Similarly the Cech cocycle condition follows by the [[chain rule]]:

$$
  \begin{aligned}
    g_{j k} \circ g_{i j}(x)
     & =
    d( \phi_k^{-1} \circ \phi_j )_{\phi_j^{-1}(x)}
    \circ
    d( \phi_j^{-1} \circ \phi_i )_{\phi_i^{-1}(x)}
    \\
    & =
    d( \phi_k^{-1} \circ \phi_j \circ \phi_j^{-1} \circ \phi_i )_{\phi_i^{-1}(x)}
    \\
    & =
    d( \phi_k^{-1} \circ \phi_i )_{\phi_i^{-1}(x)}
    \\
    &=
    g_{i k}(x)
  \end{aligned}
  \,.
$$

=--

+-- {: .num_defn #TangentSpace}
###### Definition
**([[tangent space]])**

For $X$ a [[differentiable manifold]] and $x \in X$ a point, then the _[[tangent space]]_ of $X$ at $x$ is the set $T_x X$ of [[tangent vectors]] at $x$ (def. \ref{TangentVector}) regarded as a [[real vector space]] via lemma \ref{LinearTangentSpace}.

=--

+-- {: .num_example #EuclideanSpaceTangentBundle}
###### Example
**([[tangent bundle]] of [[Euclidean space]])

If $X = \mathbb{R}^n$ is itself a [[Euclidean space]], then
for any two points $x,y \in X$ the [[tangent spaces]] $T_x X$ and $T_y X$ (def. \ref{TangentSpace})
are canonically identified with each other:

Using the [[vector space]] (or just [[affine space]]) structure of $X = \mathbb{R}^n$ we may send every
smooth function $\gamma \colon \mathbb{R} \to X$ to the smooth function

$$
  \gamma' \colon t \mapsto \gamma(t) + (x-y)
  \,.
$$

This gives a linear bijection

$$
  \phi_{x,y} \colon T_x X \overset{\simeq}{\longrightarrow} T_y X
$$

and these linear bijections are compatible, in that for $x,y,z \in \mathbb{R}^n$ any three points,
then

$$
  \phi_{y,z} \circ \phi_{x,y} = \phi_{x,z} \;\colon\; T_x X \longrightarrow T_y Y
  \,.
$$

Moreover, by lemma \ref{LinearTangentSpace}, each tangent space is identified with $\mathbb{R}^n$ itself,
and this identification in turn is compatible with all the above identifications:

$$
  \array{
    && \mathbb{R}^n
    \\
    & {}^{\mathllap{\simeq}}\swarrow && \searrow^{\mathrlap{\simeq}}
    \\
    T_x X && \underoverset{\phi_{x,y}}{\simeq}{\longrightarrow} && T_y Y
  }
  \,.
$$

Therefore it makes sense to canonically identify all the tangent spaces of Euclidean space with that Euclidean space
itself. As a result, the collection of all the tangent spaces of Euclidean space is naturally identified
with the [[Cartesian product]]

$$
  T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n
$$

equipped with the [[projection]] on the first factor

$$
  \array{
    T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n
    \\
    \downarrow^{\mathrlap{\pi = pr_1}}
    \\
    \mathbb{R}^n
  }
  \,,
$$

because then the [[pre-image]] of a [[singleton]] $\{x\} \subset \mathbb{R}^n$ under this projection are
canonically identified with the above tangent spaces:

$$
  \pi^{-1}(\{x\}) \simeq T_x \mathbb{R}^n
  \,.
$$

This way, if we equip $T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n$ with the [[product space|product space topology]],
then  $T \mathbb{R}^n \overset{\pi}{\longrightarrow} \mathbb{R}^n$ becomes a [[trivial vector bundle|trivial]] [[topological vector bundle]].

This is called the _[[tangent bundle]]_ of the [[Euclidean space]] $\mathbb{R}^n$ regarded as a [[differentiable manifold]].

=--

+-- {: .num_remark #EuclideanTangentSpaceFunctoriality}
###### Remark
**([[chain rule]] is [[functor|functoriality]] of [[tangent space]] construction on [[Euclidean spaces]])**

Consider the assignment that sends

1. every [[Euclidean space]] $\mathbb{R}^n$ to its [[tangent bundle]] $T \mathbb{R}^n$ according to def. \ref{EuclideanSpaceTangentBundle};

1. every [[differentiable function]] $f \colon \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$ to the
   function on [[tangent vectors]] (def. \ref{TangentVector}) induced by postcomposition with $f$

   $$
     \array{
       T \mathbb{R}^{n_1}
         &\overset{f \circ (-)}{\longrightarrow}&
       T \mathbb{R}^{n_2}
       \\
       \left[
         \mathbb{R}^1
           \overset{\gamma}{\longrightarrow}
         \mathbb{R}^{n_1}
       \right]
         &\mapsto&
       \left[
         \mathbb{R}^1
           \overset{ f \circ \gamma}{\longrightarrow}
         \mathbb{R}^{n_2}
       \right]
     }
   $$

By the [[chain rule]] we have that the [[derivative]] of the composite curve $f \circ \gamma$ is

$$
  d (f \circ \gamma)_t
  =
  (d f_{\gamma(x)}) \circ d\gamma
$$

and hence that under the identification $T \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$ of example \ref{EuclideanSpaceTangentBundle}
this assignment takes $f$ to its [[derivative]]

   $$
     \array{
       \mathbb{R}^{n_1} \times \mathbb{R}^{n_1}
         &\overset{ d f }{\longrightarrow}&
       \mathbb{R}^{n_2} \times \mathbb{R}^{n_2}
       \\
       (x,\vec v)
         &\mapsto&
       (f(x), d f_x(\vec v))
     }
     \,,
   $$

Conversely, in the first form above the assignment $f \mapsto f \circ (-)$ manifestly
respects [[composition]] (and [[identity functions]]). Viewed from the second perspective
this respect for composition is once again the [[chain rule]] $d(g \circ f) = (d f)\circ (d g)$:

$$
  \array{
    && Y
    \\
    & {}^{\mathllap{f}}\nearrow  && \searrow^{\mathrlap{g}}
    \\
    X && \underset{ g\circ  f}{\longrightarrow} && Z
  }
  \phantom{AAA}
    \mapsto
  \phantom{AAA}
  \array{
    && T Y
    \\
    & {}^{\mathllap{d f}}\nearrow && \searrow^{\mathrlap{d g}}
    \\
    T X &&  \underset{d(g \circ f)}{\longrightarrow} && T Z
  }
  \,.
$$

In the language of [[category theory]] this says that the assignment

$$
  \array{
    CartSp &\overset{T}{\longrightarrow}& CartSp
    \\
    X &\mapsto& T X
    \\
    {}^{\mathllap{ f }}\downarrow && \downarrow^{\mathrlap{d f}}
    \\
    Y &\mapsto& T Y
  }
$$

is an [[endofunctor]] on the [[category]] [[CartSp]] whose

1. [[objects]] are the [[Euclidean spaces]] $\mathbb{R}^n$ for $n \in \mathbb{N}$;

1. [[morphisms]] are the [[differentiable functions]] between these (for any chosen differentiability class $C^k$ with $k \gt 0$).

=--

We may now globalize the tangent bundle of Euclidean space to tangent bundles of general differentiable manifolds:

+-- {: .num_defn #TangentBundle}
###### Definition
**([[tangent bundle]] of a [[differentiable manifold]])**

Let $X$ be a [[differentiable manifold]] with [[atlas]] $\left\{ \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X\right\}_{i \in I}$.

Equip the set of all [[tangent vectors]] (def. \ref{TangentVector}), i.e. the [[disjoint union]] of the sets of tangent vectors

$$
  T X
  \;\coloneqq\;
  \underset{x \in X}{\sqcup} T_x X
  \phantom{AAA}
  \text{as underlying sets}
$$

with a [[topological space|topology]] $\tau_{T X}$ by declaring a [[subset]] $U \subset T X$ to be an [[open subset]] precisely if for all [[charts]] $\mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X$ we have that its [[preimage]] under

$$
  \array{
    \mathbb{R}^{2n}
      \simeq
    \mathbb{R}^n \times \mathbb{R}^n
      & \overset{d \phi}{\longrightarrow} &
    T X
    \\
    (x, \vec v)
      &\overset{\phantom{AAA}}{\mapsto}&
    [ t \mapsto \phi(\phi^{-1}(x) + t \vec v) ]
  }
$$

is open in the [[Euclidean space]] $\mathbb{R}^{2n}$ with its [[metric topology]].

Equipped with the [[function]]

$$
  \array{
     T X &\overset{\phantom{AA}p_x \phantom{AA}}{\longrightarrow}& X
     \\
     (x,v) &\overset{\phantom{AAAA}}{\mapsto}& x
  }
$$

this is called the _tangent bundle_ of $X$.

Equivalently this means that the tangent bundle $T X$ is the [[topological vector bundle]] which
is glued (via [this example](topological+vector+bundle#TopologicalVectorBundleFromCechCocycle)) from the
[[transition functions]] $g_{i j} \coloneqq d(\phi_j^{-1} \circ \phi_i)_{\phi^{-1}(-)}$ from lemma \ref{LinearTangentSpace}:

$$
  T X
  \;\coloneqq\;
  \left(
    \underset{i \in I}{\sqcup}
    U_i \times \mathbb{R}^n
  \right)/\left( \left\{  d( \phi_j^{-1} \circ \phi_i )_{\phi^{-1}_i(-)} \right\}_{i, j \in I} \right)
  \,.
$$

(Notice that, by examples \ref{EuclideanSpaceTangentBundle}, each $U_i \times \mathbb{R}^n \simeq T U_i$ is the tangent bundle
of the chart $U_i \simeq \mathbb{R}^n$.)

The co-projection maps of this [[quotient topological space]] construction constitute an [[atlas]]

$$
  \left\{
    \mathbb{R}^{2n}
      \underoverset{\simeq}{}{\to}
    T U_i
     \subset
    T X
  \right\}_{i \in I}
  \,.
$$

=--

+-- {: .num_lemma #DifferentiableTangentBundle}
###### Lemma
**([[tangent bundle]] is [[differentiable vector bundle]])**

If $X$ is a $(p+1)$-times [[differentiable manifold]], then the total space
of the tangent bundle def. \ref{TangentBundle} is a $p$-times [[differentiable manifold]] in that

1. $(T X, \tau_{T X})$ is a [[paracompact Hausdorff space]];

1. The [[gluing functions]] of the atlas $\left\{ \mathbb{R}^{2n} \underoverset{\simeq}{d \phi_i}{\to} T U_i \subset  T X \right\}_{i \in I}$ are $p$-times continuously differentiable.

Moreover, the projection $\pi \colon T X \to X$ is a $p$-times continuously [[differentiable function]].

In summary this makes $T X \to X$ a _[[differentiable vector bundle]]_.

=--

+-- {: .proof}
###### Proof

First to see that $T X$ is Hausdorff:

Let $(x,\vec v), (x', \vec v') \in T X$ be two distinct points. We need to produce
disjoint openneighbourhoods of these points in $T X$. Since in
particular $x,x' \in X$ are distinct, and since $X$ is Hausdorff, there exist disjoint open neighbourhoods
$U_x \supset \{x\}$ and $U_{x'} \supset \{x'\}$. Their pre-images $\pi^{-1}(U_x)$ and $\pi^{-1}(U_{x'})$
are disjoint open neighbourhoods of $(x,\vec v)$ and $(x',\vect v')$, respectively.

Now to see that $T X$ is paracompact.

Let $\{U_i \subset T X\}_{i \in I}$ be an open cover. We need to find a [[locally finite cover|locally finite]]
[[refinement]]. Notice that $\pi \colon T X \to X$ is an [[open map]] (by [this example](open+map#ProjectionsAreOpenContinuousFunctions))
so that $\{\pi(U_i) \subset X\}_{i \in I}$ is an open cover of $X$.

Let now $\{\mathbb{R}^n \underoverset{\simeq}{\phi_j}{\to} V_j \subset X\}_{j \in J}$ be an [[atlas]] for $X$ and consider the
open common refinement

$$
  \left\{ \pi(U_i) \cap V_j \subset X  \right\}_{i \in I, j \in J}
  \,.
$$

Since this is still an open cover of $X$ and since $X$ is paracompact, this has a locally finite refinement

$$
  \left\{ V'_{j'} \subset X\right\}_{j' \in J'}
$$

Notice that for each $j' \in J'$ the product topological space $V'_{j'} \times \mathbb{R}^n \subset \mathbb{R}^{2n}$
is paracompact (as a [[topological subspace]] of [[Euclidean space]] it is itself [[locally compact topological space|locally compact]]
and [[second countable topological space|second countable]] and since [[locally compact and second-countable spaces are paracompact]]).
Therefore the cover

$$
  \{ \pi^{-1}(V'_{j'}) \cap U_i \subset V'_{j'} \times \mathbb{R}^n \}_{(i,j') \in I \times J'}
$$

has a locally finite refinement

$$
  \{W_{k_{j'}} \subset V'_{j'} \times \mathbb{R}^n \}_{k_{j'} \in K_{j'}}
  \,.
$$

We claim now that

$$
  \{ W_{k_{j'}}  \subset T X \}_{j' \in J',  k_{j'} \in K_{j'}}
$$

is a locally finite refinement of the original cover.
That this is an open cover refining the original one is clear. We need to see that it is locally finite.

So let $(x,\vec v) \in T X$. By local finiteness of $\{ V'_{j'}  \subset X\}_{j' \in J'}$ there is an open neighbourhood
$V_x \supset \{x\}$ which intersects only finitely many of the $V'_{j'} \subset X$.
Then by local finiteness of $\{ W_{k_{j'}}  \subset V'_{j_'}\}$, for each such $j'$ the point $(x,\vec v)$
regarded in $V'_{j'} \times \mathbb{R}^n$ has an open neighbourhood $U_{j'}$ that intersects only finitely many of the $W_{k_{j'}}$. Hence the
intersection $\pi^{-1}(V_x) \cap \left( \underset{j'}{\cap}  U_{j'} \right)$ is a finite intersection of open subsets, hence still open,
and by construction it intersects still only a finite number of the $W_{k_{j'}}$.

This shows that $T X$ is paracompact.

Finally the statement about the differentiability of the [[gluing functions]] and of the projections is immediate from the definitions

=--

+-- {: .num_prop #FunctionsBetweenDifferentiableManifoldsDifferentials}
###### Proposition
**([[differentials]] of [[differentiable functions]] between [[differentiable manifolds]])**

Let $X$ and $Y$ be [[differentiable manifolds]] and let $f \;\colon\; X \longrightarrow Y$ be a [[differentiable function]]. Then the operation of postcomposition, which takes differentiable curves in $X$ to differentiable curves in $Y$,

$$
  \array{
     Hom_{Diff}(\mathbb{R}^1, X)
       &\overset{f \circ (-)}{\longrightarrow}&
     Hom_{Diff}(\mathbb{R}^1, Y)
     \\
     \left(
       \mathbb{R}^1 \overset{\gamma}{\to} X
     \right)
     &\overset{\phantom{AAA}}{\mapsto}&
     \left(
       \mathbb{R}^1 \overset{f \circ \gamma}{\to} Y
     \right)
  }
$$

descends at each point $x \in X$ to the tangency [[equivalence relation]] (def. \ref{TangencyRelationOnSmoothCurves}, lemma \ref{TangencyIsEquivalenceRelation}) to yield a function on
sets of [[tangent vectors]] (def. \ref{TangentVector}), called the _[[differential]]_ $d f_x$ of $f$ at $x$

$$
  d f|_{x}
    \;\colon\;
  T_x X
   \longrightarrow
  T_{f(x)} Y
  \,.
$$

Moreover:

1. ([[linear function|linear dependence]] on the tangent vector) these differentials are [[linear functions]] with respect to the vector space structure on the [[tangent spaces]] from lemma \ref{LinearTangentSpace}, def. \ref{TangentSpace};

1. ([[differentiable function|differentiable dependence]] on the base point) globally they
yield a [[homomorphism]] of real [[differentiable vector bundles]]  between the [[tangent bundles]] (def. \ref{TangentBundle}, lemma \ref{DifferentiableTangentBundle}), called the global _[[differential]]_ $d f$ of $f$

   $$
     d f \;\colon\; T X \longrightarrow T Y
     \,.
   $$

1. ([[chain rule]]) The assignment $f \mapsto d f$ respects [[composition]] in that for $X$, $Y$, $Z$ three [[differentiable manifolds]] and for

   $$
     X
       \overset{\phantom{A}f\phantom{A}}{\longrightarrow}
     Y
       \overset{\phantom{A}g\phantom{A}}{\longrightarrow}
     Z
   $$

   two [[composition|composable]] [[differentiable functions]] then their [[differentials]] satisfy

   $$
     d(g \circ f) = (d g) \circ (d f)
     \,.
   $$

=--

+-- {: .proof}
###### Proof

All statements are to be tested on charts of an atlas for $X$ and for $Y$. On these charts the
statement reduces to that of example \ref{EuclideanSpaceTangentBundle}.

=--

+-- {: .num_remark}
###### Remark
**(tangent [[functor]])**

In the language of [[category theory]] the statement of prop. \ref{FunctionsBetweenDifferentiableManifoldsDifferentials} says that forming [[tangent bundles]] $T X$ of [[differentiable manifolds]] $X$ and [[differentials]] $d f$  of [[differentiable functions]] $f \colon X \to Y$ constitutes a [[functor]]

$$
  T \;\colon\; Diff \longrightarrow Vect(Diff)
$$

from the [[category]] [[Diff]] of [[differentiable manifolds]] to the category of differentiable real vector bundles.

=--

+-- {: .num_defn #VectorField}
###### Definition
**([[vector field]])**

Let $X$ be a [[differentiable manifold]] with differentiable tangent bundle $T X \to X$ (def. \ref{TangentBundle}).

A differentiable [[section]] $v \colon X \to T X$ of the tangent bundle is called a (differentiable)
_[[vector field]]_ on $X$.

=--

+-- {: .num_remark}
###### Remark
**(notation for tangent vectors in a chart)**

Under the bijection of lemma \ref{LinearTangentSpace} one often denotes the tangent vector  corresponding to the the $i$-th canonical [[linear basis|basis]] vector of $\mathbb{R}^n$ by

$$
  \frac{\partial}{\partial x^i}
  \phantom{AA} \text{or just } \phantom{AA}
  \partial_i
$$

because under the identification of tangent vectors with [[derivations]] on the algebra of [[differentiable functions]] on $X$ as [above](#AlgebraicDefinition) then it acts as the operation of taking the $i$th [[partial derivative]]. The general tangent vector corresponding to $v \in \mathbb{R}^n$ is then denoted by

$$
  \underoverset{i = 1}{n}{\sum}
  v^i
  \frac{\partial}{\partial x^i}
  \phantom{AA} \text{or just } \phantom{AA}
  \underoverset{i = 1}{n}{\sum} v^i \partial_i
  \,.
$$

Notice that this identification depends on the choice of [[chart]], which is left implicit in this notation.

Sometimes, notably in texts on [[thermodynamics]], one augments this notation to indicate the chart being used by listing the remaining coordinate functions as subscripts. For instance if two functions $f,g$ on a 2-dimensional manifold are used as coordinate functions for a local chart (i.e. so that $x^1 = f$ and $x^2 = g$ ), then one writes

$$
  (\partial/\partial f)_g
  \phantom{AA}
  (\partial/\partial g)_f
$$

for the tangent vectors $\frac{\partial}{\partial x^1}$ and $\frac{\partial}{\partial x^2}$, respectively.

=--

### Via derivations of smooth functions
 {#ViaDerivationsOfSmoothFunctions}

+-- {: .num_remark #AlgebraicDefinition}
###### Remark
**([[derivations of smooth functions are vector fields]])**

Let $X$ be a [[smooth manifold]] and write $C^\infty(X)$ for the [[associative algebra]] over the [[real numbers]]
of  [[smooth functions]] $X \longrightarrow \mathbb{R}$.

Then every smooth [[vector field]] $v \in \Gamma_X(T X)$ (def. \ref{VectorField}) induces a [[function]]

$$
  \partial_v
    \;\colon\;
  C^\infty(X) \longrightarrow C^\infty(X)
$$

by

$$
  \partial f \;\colon\; x \mapsto \frac{d}{d t} (f \circ \gamma_{v_x})_0
$$

where $\gamma_{v_x} \colon \mathbb{R}^1 \to X$ is a smooth curve which represents the [[tangent vector]] $v(x) \in T_x X$ according to def. \ref{TangentVector}.

The linearity of [[derivatives]] and the [[product rule]] of [[differentiation]]
imply that this function $\partial_v$ is a [[derivation]] on the algebra of smooth functions. Hence there is a function

$$
  \array{
    \Gamma_X(T X)
      &\longrightarrow&
    Der(C^\infty(X))
    \\
    v &\mapsto& \partial_v
  }
  \,.
$$

It turns out that this function is in fact a [[bijection]]: every derivation of the algebra of smooth functions
on a smooth manifold arises uniquely from a smooth tangent vector in this way.

For more on this see at _[[derivations of smooth functions are vector fields]]_.

=--

\begin{remark}

For manifolds of the class $C^k$, $0\lt k\lt \infty$,
the definition of a tangent vector as a [[derivation]] of the algebra
of functions ([[derivations of smooth functions are vector fields]]) remains valid if one strengthens the definition of a [[derivation]] $D$ by adjoining, on top of the algebraic condition of the [[product]] rule

$$
  D (f \cdot g) = (D f) g + f (D g)
$$

also the algebraic incarnation of the [[chain rule]]:

$$
  D(f(g_1,\ldots,g_m))
  \;=\;
  \sum_i (\partial_i f)(g_1,\ldots,g_m)D(g_i).
$$

See [Newns-Walker 56](#NewnsWalker56).

\end{remark}

### In synthetic differential geometry
 {#InSDG}

The above definitions in ordinary [[differential geometry]] suggest the slogan

+-- {: .standout}

Tangent vectors are infinitesimal curves in a space.

=--

One of the central motivations for [[synthetic differential geometry]] is the desire to provide a context in which this slogan becomes literally formally true.

+-- {: .un_defn}
###### Definition
**(tangent bundle in smooth toposes)**

Let $(\mathcal{T},(R,+,\cdot))$ be a [[smooth topos]] and write  $D = \{\epsilon \in R| \epsilon^2 = 0\}$ for the standard [[infinitesimal space|infinitesimal interval]]. For $X \in \mathcal{T}$ any object (any [[space]] in $\mathcal{T}$), the **tangent bundle** of $X$ is the morphism

$$
  p : T X \to X
$$

with

* $T X \coloneq X^D$ the [[internal hom]] of $D$ into $X$;

* $p = ev_0$ the evaluation map at the origin of $D$

  $ev_0 : (U \stackrel{v}{\to} X^D) \mapsto
  (U \times {*} \stackrel{Id \times 0}{\to} U \times D
  \stackrel{\bar v}{\to} X)$,

  where $\bar v$ is the hom-[[adjunct]] of $v$.

=--

This definition captures elegantly and usefully the notion of tangent vectors as infinitesimal curves. But it is not guaranteed that the fibers of a synthetic tangent bundle $X^D$ are fiberwise linear, i.e. are fiberwise $R$-modules the way one expects. Objects $X$ for which this is true are [[microlinear space | microlinear spaces]] in $\mathcal{T}$. See there for more details.

A [[smooth topos]] $\mathcal{T}$ is called a _well-adapted model_ for [[synthetic differential geometry]] if there is a [[full and faithful functor|full and faithful]] embedding [[Diff]] $\hookrightarrow \mathcal{T}$ of the category of [[manifold | manifolds]] into $\mathcal{T}$.

Typically, for well adapted models, under this embedding

* manifolds are [[microlinear space | microlinear spaces]]

* the synthetic definition of tangent bundle $X^D$ for $X$ a manifold does coincide with the ordinary notion of $T X$.

Let $\mathbb{L} = (C^\infty Ring^{fin})^{op}$ be the category of [[generalized smooth algebra|smooth loci]]. For $M$ a [[manifold]], the exponential $M^D$ does exist in $\mathbb{L}$ and is isomorphic to the ordinary tangent bundle $T X$ of $X$. (For instance  [[Models for Smooth Infinitesimal Analysis|MSIA, chapter II, prop 1.12]].

There are well-adapted [[smooth topos | smooth topoi]] $\mathcal{Z}$ and $\mathcal{B}$ presented as [[category of sheaves|categories of sheaves]] on $\mathbb{L}$: the first for the [[Grothendieck topology]] where covers are _finite_ open covers, the second where covers are finite open covers and projections ([[Models for Smooth Infinitesimal Analysis|MSIA, chapter VI]]). Both topologies are [[subcanonical coverage|subcanonical]], hence the [[Yoneda embedding]] $Y : \mathbb{L} \to Sh(\mathbb{L})$ does preserve the above property.

Hence in these models for $X \in Diff$ a [[manifold]], $T X \in Diff$ its ordinary tangent bundle and $s : Diff \to Sh(\mathbb{L})$ the full and faithful embedding, we have isomorphisms

$$
  (s(X))^D  \simeq s(T X)
$$

which respect the bundle maps.

### In supergeometry

The tangent bundle of a manifold $X$ may be interpreted as a [[supermanifold]] in which $X$ has degree $0$ and the tangent vectors have degree $1$.  See [[shifted tangent bundle]].

### For other generalized smooth spaces

There are useful categories of [[generalized smooth space | generalized smooth spaces]] which are neither categories of manifolds nor [[smooth topos | smooth topoi]] modeling [[synthetic differential geometry]]. But most of them admit useful notions of tangent bundles, too, sometimes more than one.

See [[Frölicher space]] and [[diffeological space]] for the definitions in their context.

## Related concepts

* [[tangent]]

* [[synthetic tangent bundle]], [[kinematic tangent bundle]], [[operational tangent bundle]]

* [[cotangent bundle]]

* [[normal bundle]]

* [[G-structure]]

* [[stable tangent bundle]]

* [[tangent cone]]

* [[Lie algebra]]

[[!include infinitesimal and local - table]]

* in [[homotopy theory]]/[[Goodwillie calculus]]: [[tangent (∞,1)-category]]

## References

An early account of  [[derivations of smooth functions are vector fields|tangent vectors as derivations]], including the $C^k$-case for $0\lt k\lt \infty$ is in

* {#NewnsWalker56} [[W. F. Newns]], [[A. G. Walker]],  _Tangent Planes To a Differentiable Manifold_.  Journal of the London Mathematical Society s1-31:4 (1956), 400–407 ([doi:10.1112/jlms/s1-31.4.400](https://doi.org/10.1112/jlms/s1-31.4.400))

A textbook account of tangent bundles in the context of [[synthetic differential geometry]]:

* [[Ieke Moerdijk]], [[Gonzalo Reyes|Gonzalo E. Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_ Springer (1991)

Further discussion of axiomatizations in this context is in

* {#CockettCruttwell} [[J.R.B. Cockett]], [[Geoff Cruttwell]], _Differential structure, tangent structure, and SDG_ (2012) ([pdf](https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/sman3.pdf))

Discussion for [[diffeological spaces]] is in

* Carlos Torre, _A tangent bundle for diffeological spaces_ ([arXiv:math/9801046](http://arxiv.org/abs/math/9801046))

[[!redirects tangent vector]]
[[!redirects tangent space]]
[[!redirects tangent vector space]]
[[!redirects tangent vector field]]
[[!redirects tangent bundles]]
[[!redirects tangent vectors]]
[[!redirects tangent spaces]]
[[!redirects tangent vector spaces]]
[[!redirects tangent vector fields]]


tangent bundle category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **tangent bundle category** is a [[category]] equipped with a "[[tangent bundle]]" [[endofunctor]] satisfying some natural axioms.

Usually these are called simply **tangent categories**, but on the nLab the page [[tangent category]] is about "the tangent category of a given category" constructed by abelianization.  In other words, tangent bundle categories are about *abstraction* of the tangent bundle construction, while tangent categories are a *categorification* thereof (in some vague sense).

## Definition

Tangent bundle categories have three equivalent definitions: the first is due to [[Jiří Rosický]] and was rediscovered/refined by [[Robin Cockett|Cockett]] and [[Geoff Cruttwell|Cruttwell]]. In his thesis, Poon Leung found an equivalent definition of tangent categories using a category structure that acts as an abstract Weil prolongation, and later [[Richard Garner]] gave a definition of a tangent category as a sort of enriched category.

### Classical Definition

Tangent bundle categories were originally introduced by Rosicky to model the behaviour of the tangent bundle on the category of smooth manifolds, and of [[microlinear spaces]] in a [[smooth topos]]. For smooth manifold $M$ and a point $m \in M$, on may consider a coordinate patch $U \subset \mathbb{R}^n$ around $m$. Looking at the tangent space of $U$, we see $T(U) \cong U \times \mathbb{R}^n$, and similarly
$T^2(U) \cong (U \times \mathbb{R}^n) \times (\mathbf{R^n} \times \mathbb{R}^n)$. Besides the vector bundle structures on $p, p_T, T(p)$, there are two important morphisms:

* The vertical lift: $\ell(m,v) = (m,0,0,v)$.

* The canonical flip $c(m,u,v,w) = (m,v,u,w)$.

These morphisms are important in the axiomatization of differential structure given in cartesian differential categories.

#### Additive bundles

Tangent categories were originally defined by Rosicky using abelian group bundles, however Cockett and Cruttwell's definition uses commutative monoid bundles in order to capture examples of tangent structure that arise in theoretical computer science.

An additive bundle $q: E \to B$ in a category $\mathbb{X}$ is a commutative monoid in $\mathbb{X}/B$. A bundle morphism $(f,g): q \to q'$ is _additive_ if it preserves the fibered commutative monoid structure.

#### Tangent structure

A tangent structure $\mathbb{T}$ on a category $\mathbb{X}$ is a tuple:

$$
  \left(
    T: \mathbb{X} \to \mathbb{X}, p: T \Rightarrow \mathsf{id},
    0: \mathsf{id} \Rightarrow T, +: T \times_p T \Rightarrow T,
    \ell: T \Rightarrow T^2, c: T^2 \Rightarrow T^2
  \right)
$$

Denote pullback powers of $p$ as $T_n(M)$.

* $T$ preserves all pullback powers of $p$.

* $(p, +,0)$ is a natural additive bundle.

* The flip $c$ is a natural involution $cc = 1_{T^2M}$,
  and the following bundle morphism is additive:

\begin{center}

\begin{tikzcd}
   T^2(M) \ar[r, "c"] \ar[d, "p_T"] & T^2(M) \ar[d, "{T(p)}"]
   \\
   T(M) \ar[r, equals] & T(M)
 \end{tikzcd}

\end{center}

* The vertical lift $\ell$ gives an additive bundle morphism $(\ell, 0): p \Rightarrow T(p)$.

* We also require the following coherences between the vertical lift and the canonical flip:

\begin{center}

\begin{tikzcd}
      T \ar[r, "\ell"] \ar[d, "\ell"'] & T^2 \ar[d, "{T(\ell)}"]
      \\
      T^2 \ar[r, "\ell_T"] & T^3
\end{tikzcd}

\end{center}

\begin{center}

\begin{tikzcd}
      T^3 \ar[d, "c_T"] \ar[r, "{T(c)}"] & T^3 \ar[r, "c_T"] & T^3
      \ar[d, "T(c)"] \\
      T^3 \ar[r, "{T(c)}"] & T^3 \ar[r, "c_T"] & T^3
\end{tikzcd}

\end{center}

\begin{center}

\begin{tikzcd}
      T^2 \ar[r, "\ell_T"] \ar[d, "c"']
      & T^3 \ar[r, "{T(c)}"]
      & T^3 \ar[d, "c_T"]
     \\
      T^2 \ar[rr, "{T(\ell)}"] && T^3
\end{tikzcd}

\end{center}

\begin{center}

\begin{tikzcd}
       T \ar[r, "\ell"] \ar[rd, "\ell"]
       & T^2 \ar[d, "c"]
       \\ & T^2
\end{tikzcd}

\end{center}

In the monoidal category $[\mathbb{X}, \mathbb{X}]$, the first diagram corresponds to $\ell: T \Rightarrow T^2$ being a cosemigroup. The second diagram corresponds to $c: T^2 \Rightarrow T^2$ acting as a symmetry, and the third and fourth diagrams state that $\ell$ is a symmetric cosemigroup.

* Universality of the vertical lift: Define the map $\mu := T(+) \circ \langle \ell \circ \pi_0, 0_T \circ \pi_1 \rangle$, we require the following diagram be a pullback:

\begin{center}

\begin{tikzcd}
      T_2(M) \ar[r, "\mu"] \ar[d, "p \circ \pi_0 = p \circ \pi_1"]
      & T^2(M) \ar[d, "{T(p)}"] \\
      M \ar[r, "0"] & T(M)
\end{tikzcd}

\end{center}

### Tangent Structure as abstract Weil Prolongation

There is a vast literature on the notion of a "Weil functor". A particularly important theorem is that every product preserving endofunctor on the category of smooth manifolds is given by a prolongation operation with a Weil algebra. To simplify this section, we will only consider the case of tangent categories with negatives - see Leung's thesis to see the generalization to commutative rigs.

#### Weil Algebras

Consider a commutative ring $R$. For this section an $R$-algebra is a commutative, unital, associative $R$-algebra. A _Weil Algebra_ over $R$ is an augmented $R$-algebra $V$ so that:

1. The underlying $R$-module of $V$ is $R^n$.
1. The _kernel of augmentation_ of $\pi: V \to R$ is nilpotent: there exists
  a natural number $k$ so that for every $x \in \mathsf{ker}(\pi)$, $x^k = 0$.

The category of Weil algebras is the full subcategory of $R-\mathsf{alg}/R$ whose objects are Weil algebras.

+-- {: .num_prop #cat_of_R_weil }
###### Proposition

1. $R$-Weil algebras have products.
1. $R$-Weil algebras have coproducts.
1. $R$ is a zero object in the category of $R$-Weil algebras.

=--

It is often useful to consider a presentation of $R$-Weil algebras.

+-- {: .num_prop #presentation_of_R_weil }
###### Proposition

* (1) Weil algebras may be presented as $R[X_i]/I$, where $I$ is an ideal of $R[X_i]$.

* (2) The product of Weil algebras $R[X_i]/I, R[Y_j]/J$ may be presented as  $R[X_i,Y_j]/(I \cup J \cup XY)$, where $XY = \{ X_i Y_j \}$, the coproduct as $R[X_i,Y_j]/(I \cup J)$.

* (3) The following diagram is a [[pullback]]:

\begin{center}

\begin{tikzcd}
        R[X,Y]/(X^2,Y^2, XY)
        \ar[r, "X \mapsto X Y", "Y \mapsto Y"']
        \ar[d, "X{,}Y \mapsto 0"]
        & R[X,Y]/(X^2,Y^2) \ar[d, "Y \mapsto 0{,} X \mapsto X"] \\
        R \ar[r, "0"] & R[X]/X^2
      \end{tikzcd}

\end{center}

* (4) For any Weil algebra $U$, the endofunctor $U \oplus (-)$ preserves products and the above pullback.

=--

We finally restrict our attention to the category $\mathsf{Weil}_1$. Let $W$ denote the $R$-Weil algebra $R[X]/X^2$, then we may consider the full subcategory whose objects are the closure of $\{ W^n | n \in \mathbb{N}\}$ under coproduct.

#### Abstract Weil Prolongation

Consider the category $\mathsf{Weil}_1$ over the the integers as a symmetric monoidal category with $(\mathsf{Weil}_1,\oplus, R)$. If a category $\mathbb{X}$ has a tangent structure, then it has an [[actegory]] structure

$$
  \propto: \mathbb{X} \times \mathsf{Weil}_1 \to \mathbb{X}
$$

so that for any object $M$ in $\mathbb{X}$, $\propto(M,-)$ preserves connected limits
of $\mathsf{Weil}_1$.

1. The tangent bundle functor is the action by $R[X]/X^2$.
  \[
   ( M \propto R[X]/X^2) \propto R[Y]/Y^2 = M \propto R[X,Y]/(X^2,Y^2)
  \]
1. The addition map is given by:
  \[
    M \propto (X,Y \mapsto X): M \propto R[X,Y]/X^2,Y^2,XY \to M \propto R[X]/X^2
  \]
  similarly $0$ is given by:
  \[
    M \propto 0: M \propto R \to M \propto R[X]/X^2
  \]
1. The canonical flip is given by:
  \[
    M \propto (X,Y \mapsto Y,X): M\propto R[X,Y](X^2,Y^2) \to M\propto R[X,Y](X^2,Y^2)
  \]
1. The vertical lift is given by:
  \[
    M \propto (X \mapsto XY): M \propto R[X]/X^2 \to M\propto R[X,Y](X^2,Y^2)
  \]

### Tangent Structure as a Weighted Limit

Using a theorem due to [[Richard Wood]], a
## References

The definition is due to

* [[Jiri Rosicky]], _Abstract tangent functors_, Diagrammes 12 (1984)

For developments of his ideas, see

* [[Robin Cockett]] and [[Geoff Cruttwell]], _Differential structure, tangent structure, and SDG_, ([pdf](https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/sman3.pdf))

* [[Robin Cockett]] and [[Geoff Cruttwell]], _Differential bundles and fibrations for tangent categories_, ([arXiv:1606.08379](https://arxiv.org/abs/1606.08379))

* [[Robin Cockett]] and [[Geoff Cruttwell]], _Connections in tangent categories_, ([arXiv:1610.08774](https://arxiv.org/abs/1610.08774))

* [[Geoff Cruttwell]], [[Rory Lucyshyn-Wright]], _A simplicial foundation for differential and sector forms in tangent categories_, ([arXiv:1606.09080](https://arxiv.org/abs/1606.09080))

* Poon Leung, _Classifying tangent structures using Weil algebras_, Theory and Applications of Categories, 32(9):286–337, 2017, ([tac](http://www.tac.mta.ca/tac/volumes/32/9/32-09abs.html))

* [[Geoff Cruttwell]] and [[Jean-Simon Lemay]], _Tangent categories as a bridge between differential geometry and algebraic geometry_, 2023. ([arXiv:2301.05542](https://arxiv.org/abs/2301.05542))

Representation of this tangent structure as exponentiation by a tangent vector is given in

* [[Richard Garner]], _An embedding theorem for tangent categories_, ([arXiv:1704.08386](https://arxiv.org/abs/1704.08386))

The enriching category from the above paper was discussed earlier in

* [[Eduardo Dubuc]]. _Sur les modeles de la géométrie différentielle synthétique._ Cahiers de topologie et géométrie différentielle catégoriques 20, no. 3 (1979): 231-279. ([pdf](http://www.numdam.org/article/CTGDC_1979__20_3_231_0.pdf))

* Wolfgang Bertram _Weil spaces and Weil-Lie groups._ arXiv preprint arXiv:1402.2619 (2014),([arXiv:1402.2619](https://arxiv.org/abs/1402.2619))

Weil Prolongation is discussed in the following papers:

* [[Anders Kock]] (1986). _Convenient vector spaces embed into the Cahiers topos._ Cahiers de topologie et géométrie différentielle catégoriques, 27(1), 3-17 ([pdf](http://www.numdam.org/article/CTGDC_1986__27_1_3_0.pdf))

* Wolfgang Bertram and Arnaud Souvay. _A general construction of Weil functors._ arXiv preprint arXiv:1111.2463 (2011).([arXiv:1111.2463](https://arxiv.org/pdf/1111.2463.pdf))

An extension of the concept of a tangent bundle category to $(\infty, 1)$-categories is in:

* {#BauerBurkeChing21} [[Kristine Bauer]], [[Matthew Burke]], [[Michael Ching]], _Tangent $\infty$-categories and Goodwillie calculus_ ([arXiv:2101.07819](https://arxiv.org/abs/2101.07819)).

Two further examples of tangent structures on an $(\infty,1)$-category, on [[(∞,1)-Topos]] and its [[opposite (infinity,1)-category|opposite]], are given in:

* [[Michael Ching]], _Dual tangent structures for infinity-toposes_, ([arXiv:2101.08805](https://arxiv.org/abs/2101.08805))

[[!redirects tangent bundle categories]]


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]]*.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **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 _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|abelian]] [[group objects]] and restricting the morphisms such as to respect the abelian group object structure.

## Definition

Let $C$ be a [[category]] with [[pullback]]s. Then the **tangent category** $T_C$ of $C$ is the category whose

* [[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 $f \colon A \to B$ any morphism in $C$, the [[pullback]] $f^* \mathcal{B} \coloneqq A \times_B \mathcal{B}$ in $C$ is naturally equipped with the structure of an [[abelian group|abelian]] [[group object]] [[internalization|internal to]] the [[slice category]] $C/A$;

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

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

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

* 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) $.

## Properties

### General

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

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

The fiber over $Id_A$ of this functor is the category of abelian group objects in the [[overcategory]] $C/A$:

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

There is also another functor $q : T_C \to C$, inherited from the [[domain cofibration]]

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

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 $C$).

### Of a site

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

> check

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 $T_C$

$$
  (C \stackrel{i}{\to} T_C \stackrel{p}{\to} C) = Id_C
  \,.
$$

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]]

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

## Examples

### Modules as tangents to rings {#OrdinaryModules}

+-- {: .num_prop}
###### Proposition

For $C = $ [[CRing]] we have $T_C \simeq $ [[Mod]].

=--

(More generally for $C =$ [[Ring]] then $T_C$ is the category of [[bimodules]], see at _[Beck module -- Over associative algebras](Beck+module#OverAssociativeAlgebras)_).

+-- {: .proof}
###### Proof

Consider the functor
$$
 F: Mod \to T_{CRing}
$$

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 $R$,

$$
F_R:  Mod_R \to Ab(CRing/R)
$$

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

=--

+-- {: .num_prop}
###### Proposition

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]].

=--

+-- {: .proof}
###### 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
$$f_1 (x y) = f_0 (x) f_1 (y) + f_0 (y) f_1 (x)$$
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 $Mod$.

=--

### Modules over smooth algebras {#ModulesOverSmoothAlgebras}

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

$$
  U : SmoothAlg \to Ring
$$

to the underlying ordinary [[ring]]s.

+-- {: .num_theorem}
###### Theorem
{#TangentOfSmoothAlgTheorem}

There is an [[equivalence of categories]]

$$
  T_{SmoothAlg} \stackrel{\simeq}{\to}
  SmoothAlg \times_{Ring} T_{Ring}
  \,,
$$

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.

+-- {: .num_remark}
###### Remark

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.

=--

+-- {: .num_cor}
###### Corollary

For any category $S$ we have that

$$
  (T_{SmoothAlg})^S \simeq
  SmoothAlg^S \times_{Ring^S} (T_{Ring})^S
  \,.
$$

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.

=--

For proving the [above theorem](#TangentOfSmoothAlgTheorem) the main step is the following lemma.

+-- {: .num_lemma}
###### Lemma

For a fixed smooth algebra $R$, the [[forgetful functor]]

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

is an [[equivalence of categories]].

=--

+-- {: .proof}
###### 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 ([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 _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

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

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

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

We now use that any [[smooth algebra]] $A$ 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

$$
  A(f)(r+e)
  =
  A(f)(r) +
  \sum_{l = 1}^k w_l \cdot A\left(\frac{\partial f}{\partial x_l}\right)(r)
  +
  \sum_{i,j}
   e_i \cdot e_j A(h_{i,j})(r,e)
  \,.
$$

Now if $R \in SmoothAlg$  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 [above theorem](#OrdinaryModules) 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

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

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

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

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 $N$:

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

since $\epsilon_i \cdot \epsilon_j = 0$ 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]].

=--

Finally we come to the proof of the [full theorem above](#TangentOfSmoothAlgTheorem)

+-- {: .proof}
###### 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 Relative free T-algebra adjunctions), checking bijection on [[hom-set]]s

$$
  T_{SmoothAlg}(A\to R_1,B \to R_2) \to SmoothAlg \times_{Ring} T_{Ring} ((U(A), R_1), (U(B), R_2))
$$

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

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

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

=--

## Related concepts

* [[tangent (∞,1)-category]].

## References

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

* [[Daniel Quillen]], ...

A discusson of $T_{SmoothAlg}$ is in

* [[Herman Stel]], _[[schreiber:master thesis Stel|∞-Stacks and their Function Algebras]]_
{#Stel}

[[!redirects tangent categories]]


tangent cohesive (∞,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[tangent (∞,1)-category]] $T\mathbf{H}$
to a [[cohesive (∞,1)-topos]] is itself cohesive: the
_tangent cohesive (∞,1)-topos_.

This $T \mathbf{H}$ is the $\infty$-topos of [[parameterized spectra]] in $\mathbf{H}$, hence is the context for cohesive [[stable homotopy theory]].

## Properties

### Stable extension of cohesion
 {#StableExtensionOfCohesion}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]].

By the discussion at
_[tangent ∞-category -- Examples -- Of an ∞-topos](tangent+%28infinity%2C1%29-category#TangentTopos)_ the tangent $\infty$-topos $T \mathbf{H}$ constitutes an [[extension]] of $\mathbf{H}$ by its [[stabilization]] $Stab(\mathbf{H})$:

$$
  \array{
    && Stab(\mathbf{H})
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
    &
    Stab(\infty Grpd) \simeq Spectra
    \\
    && \simeq && \simeq
    \\
    && T_\ast \mathbf{H} && T_\ast \infty Grpd
    \\
    && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}}
    \\
    \mathbf{H}
    &\stackrel{\overset{d}{\longrightarrow}}{\underset{\Omega^\infty \circ tot}{\leftarrow}}&
    T \mathbf{H}
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  &
  T \infty Grpd
  \\
  &&
  {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}}
  &&
  {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}}
  \\
  &&
    \mathbf{H}
    &
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  &
  \infty Grpd
  }
  \,.
$$

Here

* $\Omega^\infty \circ tot \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}$ assigns the _total space_ of a spectrum bundle;

  its [[left adjoint]] is the [tangent complex functor](tangent+%28infinity%2C1%29-category#CotangentComplex);

* $base \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}$ assigns the _base space_ of a spectrum bundle;

  its [[left adjoint]] produces the 0-bundle;

  together these exhibit $T \mathbf{H}$ as an [[infinitesimal cohesive (infinity,1)-topos]] over $\mathbf{H}$.

### Stable homotopy types

In a tangent cohesive $\infty$-topos $T \mathbf{H}$ all the [[homotopy types]] in $T_\ast \mathbf{H} \hookrightarrow T\mathbf{H}$ are [[stable homotopy types]].

### Cohomology -- Twisted bivariant generalized geometric cohomology theory
 {#Cohomology}

Where the [[(∞,1)-categorical hom-space]] in a general [[(∞,1)-topos]] constitute a notion of [[cohomology]], those of a [[tangent (∞,1)-topos]] specifically constitute [[twisted generalized cohomology]], in fact [[twisted bivariant cohomology]].

For consider a [[spectrum object]] $E \in T_\ast \mathbf{H}$ and write $GL_1(E) \in Grp(\mathbf{H})$ for its [[∞-group of units]]. Then the [[∞-action]] of this on $E$ is (by the discussion there) exhibited by an object

$$
  \left(
   \array{
     E//GL_1(E)
     \\
     \downarrow
     \\
     \mathbf{B}GL_1(E)
   }
  \right)
  \;\;\;
   \in
  \;\;\;
  T_{\mathbf{B}GL_1(E)}\mathbf{H} \hookrightarrow T\mathbf{H}
  \,.
$$

More generally, for $Pic(E) \in \mathbf{H}$ the [[Picard ∞-groupoid]] of $E$ there is the universal [[(∞,1)-line bundle]]

$$
  (\widehat{Pic(E)} \to Pic(E)) \in T \mathbf{H}
  \,.
$$

Now for any object $X \in \mathbf{H}$ we have the trivial [[sphere spectrum]] [[spectrum bundle]] over $X$

$$
  X \times \mathbb{S}
  \simeq
  \left(
   \array{
     X \times \mathbb{S}
     \\
     \downarrow
     \\
     X
   }
  \right)
  \;\;\;
   \in
  \;\;\;
  T_{X}\mathbf{H} \hookrightarrow T\mathbf{H}
  \,.
$$

then morphisms in $T \mathbf{H}$ from the latter to the former

$$
    \left(
   \array{
     X \times \mathbb{S}
     \\
     \downarrow
     \\
     X
   }
  \right)
  \longrightarrow
  \left(
   \array{
     E//GL_1(E)
     \\
     \downarrow
     \\
     \mathbf{B}GL_1(E)
   }
  \right)
$$

are equivalently [[homotopy]] [[commuting diagrams]] of the form

$$
  \array{
    X \times \mathbb{S} &\stackrel{\sigma}{\longrightarrow}& E//GL_1(E)
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\chi}{\longrightarrow}& \mathbf{B}GL_1(E)
  }
$$

and hence

1. a choice of [[twisted cohomology|twist of E-cohomology]] $\chi \;\colon \; X  \longrightarrow \mathbf{B}GL_1(E)$, modulating a $GL_1(E)$-[[principal ∞-bundle]];

1. an element in the $\chi$-twisted $E$-cohomology of $X$, $\sigma \in E^{\bullet + \chi}(X,E)$, hence a [[section]] of the [[associated ∞-bundle|associated]] [[(∞,1)-line bundle]].

If we consider the [[internal hom]] instead of the external [[(∞,1)-categorical hom space]] then things work even more nicely and we can use just $X$ instead of $X \times \mathbb{S}$:

+-- {: .num_prop #MappingSpectrumOutOfUnstableTypeAsCartesianInternalHom}
###### Proposition

For $X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}$ a [[geometric homotopy type]] and $E \in Stab(\mathbf{H}) \simeq T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ a [[spectrum object]], then the [[internal hom]]/[[mapping stack]]

$$
  [X,E]_{T \mathbf{H}} \in T \mathbf{H}
$$

(with respect to the Cartesian [[closed monoidal (∞,1)-category]] structure on the [[(∞,1)-topos]] is equivalently the [[mapping spectrum]]

$$
  [\Sigma^\infty X, E]_{Stab(\mathbf{H})} \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}
  \,,
$$

in that

$$
  [X,E]_{T \mathbf{H}} \simeq [\Sigma^\infty X,E]_{Stab(\mathbf{H})}
  \,.
$$

=--

+-- {: .proof}
###### Proof

Notice that as an object of $T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}$, the object $X$ is the constant [[(∞,1)-presheaf]] on $seq$. By the formula for the [[internal hom]] in an [[(∞,1)-category of (∞,1)-presheaves]] we have

$$
  [X,E]_\bullet \simeq \mathbf{H}^{seq}(X \times \bullet, E)
  \,.
$$

But since $X$ is constant the object $X \times \bullet$ is for each object of $seq$ the [[representable functor|presheaf represented]] by that object. Therefore by the [[(∞,1)-Yoneda lemma]] it follows that

$$
  [X,E]_\bullet \simeq [X,E_\bullet]
  \,.
$$

This is manifestly the same formula as for the [[mapping spectrum]] out of $\Sigma^\infty X$.

=--

Similar kind of arguments give the following more general statement.

+-- {: .num_prop}
###### Proposition

For $X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}$ a [[geometric homotopy type]], for $E \in E_\infty(\mathbf{H})$ an [[E-∞ ring]]  with $(\widehat{Pic(E)} \to Pic(E)) \hookrightarrow T \mathbf{H}$ its universal [[(∞,1)-line bundle]] over its [[Picard ∞-groupoid]], then the [[internal hom]]/[[mapping stack]]

$$
  [X,\widehat{Pic(E)}]_{T \mathbf{H}} \in T \mathbf{H}
$$

is the object whose

* base homotopy type is the [[E-∞ ring]] $[X, Pic(E)]$ of $E$-[[twisted cohomology|twist]] on $X$;

* whose [[spectrum bundle]] is the collection of $\chi$-[[twisted cohomology|twisted E-cohomology spectra]] for all twists $\chi$.

=--

In full generality we may formulate the [[internal hom]] [[mapping space]] in $T \mathbf{H}$ in [[homotopy type theory]] notation as follows.

+-- {: .num_prop #InternalHomGeneral}
###### Proposition

For

$$
  (a \colon A) \;\vdash\; E_a \colon Spectra(\mathbf{H})
$$

and

$$
  (b \colon B) \;\vdash\; F_b \colon Spectra(\mathbf{H})
$$

two [[spectrum bundle]] [[dependent type|dependent types]] over base [[homotopy types]], $A,B \colon \mathbf{H}$, respectively, then the [[function type]] $(E \to F) \colon T\mathbf{H}$ between them (regarded as [[homotopy types]] in $T \mathbf{H}$) is

$$
  \chi \colon (A \to B); \sigma \colon \underset{a \colon A}{\prod} SpMap(E_a,F_{\chi(a)})
  \;\vdash\;
  \underset{a \colon A}{\prod} F_{\chi(a)}
  \,.
$$

=--
+-- {: .proof}
###### Proof
Let $(x:X)\vdash M_x : Spectra$ be another [[spectrum bundle]].  The [[cartesian product]] $M\times E$ in $T \mathbf{H}$ is then $(x:X),(a:A) \vdash M_x \oplus E_a$, with $\oplus$ also the [[coproduct]] (hence the [[direct sum]]), since [[spectra]] are [[stable (infinity,1)-category|stable]] and hence [[additive (infinity,1)-category|additive]].  We compute the [[mapping space]] $T\mathbf{H}(M\times E,F)$ as follows:
$$
\begin{aligned}
  \sum_{(\phi:X\times A \to B)} \prod_{((x,a):X\times A)} SpMap(M_x\oplus E_a,F_{\phi(x,a)})
  &=& \sum_{(\phi:X \to A \to B)} \prod_{(x:X)} \prod_{(a:A)} SpMap(E_a,F_{\phi(x,a)}) \times SpMap(M_x,F_{\phi(x,a)})\\
  &=& \prod_{(x:X)} \sum_{(\chi:A \to B)} \left(\prod_{(a:A)} SpMap(E_a,F_{\chi(a)})\right) \times \left( \prod_{(a:A)} SpMap(M_x,F_{\chi(a)}) \right)\\
  &=& \prod_{(x:X)} \sum_{\psi : \sum_{(\chi:A \to B)} \prod_{(a:A)} SpMap(E_a,F_{\chi(a)})} SpMap\left(M_x, \prod_{(a:A)} F_{pr_1(\psi)(a)}\right)\\
  &=& \sum_{\rho : X \to \sum_{(\chi:A \to B)} \prod_{(a:A)} SpMap(E_a,F_{\chi(a)})} \prod_{(x:A)} SpMap\left(M_x, \prod_{(a:A)} F_{pr_1(\rho(x))(a)}\right)
\end{aligned}
$$
In the first line, we [[currying|curry]] $\phi$, apply the [[induction principle]] for [[dependent type|dependent]] maps out of $X\times A$, and also apply the [[universal property]] of the [[coproduct]] $M_x \oplus E_a$.  In the second line, we apply the [[universal property]] for mapping into [[∞-types]] (the "[[type-theoretic axiom of choice]]") and also that for dependent functions into a [[Cartesian product|product]].  In the third line we apply the associativity of [[∞-types]], and also the [[universal property]] for mapping into the [[dependent product]] $\prod$ of [[spectra]].  Finally, in the fourth line, we apply the [[type-theoretic axiom of choice]] again in the other direction.  The resulting [[type]] is the [[mapping space]] from $M$ to the claimed [[function type]] $(E\to F)$ defined above.  (See also [this discussion](http://nforum.mathforge.org/discussion/5321/parameterized-cohesive-spectra/?Focus=42394#Comment_42394).)
=--

+-- {: .num_example}
###### Example

We have the following special cases of prop. \ref{InternalHomGeneral}.

1. If $E_a = 0$ for all $a \colon A$, and if $B = \ast$, then the function type is

   $$
     \vdash \; \underset{a \colon A}{\prod} F
   $$

   which reproduces the [[mapping spectrum]] $SpMap(\Sigma^\infty A, F)$ from prop. \ref{MappingSpectrumOutOfUnstableTypeAsCartesianInternalHom}.

1. If $A = B = \ast$ then the mapping type is

   $$
     \sigma \colon SpMap(E,F) \;\vdash \; F \colon Spectra
   $$

1. If $E_a = 0$ for all $a \colon A$ and $F_b = 0$ for all $b \colon B$ then the mapping type is

   $$
     \chi \colon (A \to B)\;\vdash \; 0 \colon Spectra
     \,.
   $$

=--

### Cohesive and differential refinement
 {#CohesiveAndDifferentialRefinement}

Let $T\mathbf{H}$ be a tangent cohesive $(\infty,1)$-topos and write $T_\ast \mathbf{H}$ for the [[stable (∞,1)-category]] of [[spectrum objects]] inside it. We discuss how every [[stable homotopy type]] here canonically sits in the middle of a _[[differential cohomology diagram]]_.

+-- {: .num_prop}
###### Proposition

For every $A \in T_\ast \mathbf{H}$ the [[natural transformation|naturality square]]

$$
  \array{
    A &\stackrel{}{\longrightarrow}& A/\flat A
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A)
  }
$$

(of the [[shape modality]] applied to the [[homotopy cofiber]] of the [[counit of a comonad|counit]] of the [[flat modality]]) is an [[(∞,1)-pullback]] square.

=--

This was observed in ([Bunke-Nikolaus-Völkl 13](#BunkeNikolausVoelkl13)). It is an incarnation of a [[fracture theorem]].

+-- {: .proof}
###### Proof

By [[cohesive (infinity,1)-topos|cohesion]] and [[stable (infinity,1)-category|stability]] we have the diagram

$$
  \array{
    \flat A &\longrightarrow & A &\stackrel{}{\longrightarrow}& A/\flat A
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow && \downarrow
    \\
    \Pi(\flat A) &\longrightarrow& \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A)
  }
$$

where both rows are [[homotopy fiber sequences]]. By [[cohesive (infinity,1)-topos|cohesion]] the left vertical map is an [[equivalence in an (infinity,1)-category|equivalence]]. The claim now follows with the [homotopy fiber characterization](homotopy+pullback#HomotopyFiberCharacterization) of [[homotopy pullbacks]].

=--

+-- {: .num_remark}
###### Remark

This means that in stable cohesion every cohesive stable homotopy type is in controled sense a cohesive extension/refinement of its [[geometric realization of cohesive infinity-groupoids|geometric realization]] [[discrete infinity-groupoid|geometrically discrete]] ("bare") stable [[homotopy type]] by the non-[[discrete object|discrete]] part of its cohesive structure;

In particular, $A/\flat A$ may be identified with differential cycle data. Indeed, by stability and cohesion it is the [flat de Rham coefficient object](cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology)

$$
  A/\flat A = \flat_{dR}\Sigma A
$$

of the [[suspension]] of $A$, and the map to this quotient is thus the [[Maurer-Cartan form]] $\theta_A$. So

$$
  \array{
    A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A)
  }
$$

exhibits $A$ as a [[differential cohomology]]-coefficient of the [[generalized cohomology theory]] $\Pi(A)$.

It follows by the discussion at [[schreiber:differential cohomology in a cohesive topos]] that the further differential refinement $\widehat{A}$ of $A$ should be given by a further [[homotopy pullback]]

$$
  \array{
    \widehat{A} &\longrightarrow& \Omega^1(-,Lie(A))
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A)
  }
  \,.
$$

But of course by the generality of the above  proposition, such an $\widehat{A}$ sits itself again in its fracture-like pullback diagram.

=--

Dually:

+-- {: .num_prop}
###### Proposition

For every $A \in T_\ast \mathbf{H}$ the [[natural transformation|naturality square]]

$$
  \array{
    \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Sigma^{-1} A)
    \\
    \downarrow && \downarrow
    \\
    \flat A &\stackrel{}{\longrightarrow}& A
  }
$$

(of the [[flat modality]] applied to the [[homotopy fiber]] of the [[unit of a monad|unit]] of the [[shape modality]]) is an [[(∞,1)-pullback]] square.

=--

+-- {: .proof}
###### Proof

As before but dually, the diagram extends to a morphism of
[[homotopy cofiber]] diagrams of the form

$$
  \array{
    \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Omega A)
    \\
    \downarrow && \downarrow
    \\
    \flat A &\stackrel{}{\longrightarrow}& A
    \\
    \downarrow && \downarrow
    \\
    \flat \Pi(A) &\stackrel{\simeq}{\longrightarrow}& \Pi(A)
  }
  \,,
$$

and by [[cohesion]] the bottom horizontal morphism is an [[equivalence in an (infinity,1)-category|equivalence]].

=--

Combining these two statements yields the following ([Bunke-Nikolaus-Völkl 13](#BunkeNikolausVoelkl13)).

+-- {: .num_cor #TheDifferentialDiagram}
###### Corollary

For $\mathbf{H}$ a [[cohesive (∞,1)-topos]]
every [[stable homotopy type]] $A \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}$ sits inside a [[diagram]] of the form

$$
  \array{
    &&  \Pi_{dR} \Omega A && \longrightarrow && \flat_{dR}\Sigma A
    \\
    & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow
    \\
    \flat \Pi_{dR} \Omega A  && && A && && \Pi \flat_{dR}\Sigma A
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow_{\mathrlap{\Pi \theta_A}}
    \\
    && \flat A && \longrightarrow && \Pi A
  }
  \,,
$$

where the two squares are [[homotopy pullback]] squares and the two diagonals are the [[fiber sequences]] of the [[Maurer-Cartan form]] $\theta_A$ and its dual.

=--

+-- {: .num_remark}
###### Remark

The bottom horizontal morphisms in the diagram in prop. \ref{TheDifferentialDiagram} are the canonical [points-to-pieces transform](cohesive%20topos#CanonicalComparison).

=--

+-- {: .num_remark}
###### Remark

This kind of diagram under forming $\pi_0$ has been traditionally known from [[ordinary differential cohomology]] and from [[differential K-theory]], and had been used in proposals to axiomatize [[differential cohomology]], see for instance ([Bunke 12, prop. 4.57](http://arxiv.org/abs/1208.3961)) and see at _[[differential cohomology diagram]]_. Here we see that this holds fully generally for every stable cohesive homotopy type. If one still regards this diagram as characteristic of "differential" refinement it hence exhibits every cohesive stable type as a [[coefficients]] of _some_ [[differential cohomology]] theory. This is a strong version of the synthetic notion
"[[schreiber:differential cohomology in a cohesive topos]]" . For more on this see also at _[[smooth spectrum]]_.

=--

## Related concepts

* [[sheaf of spectra]]

* [[smooth spectrum]]

* [[twisted generalized cohomology]]

* [[twisted bivariant cohomology]]

* [[twisted differential cohomology]]

* [[twisted smooth cohomology in string theory]]

## References

The idea of forming $T_\ast \mathbf{H}$ as a home for nontrivial [[stable homotopy types]] was originally suggested by [[Georg Biedermann]] and [[André Joyal]], see section 35 of

* {#Joyal08} [[André Joyal]], _Notes on Logoi_, 2008 ([pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf))

and see the further references at _[[tangent (infinity,1)-topos]]_.

Discussion of [[differential cohomology]] in $T_\ast Smooth \infty Grpd \simeq Stab(Smooth \infty Grpd)$ is in

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

For details see at _[[differential cohomology hexagon]]_.

The above discussion of geometric twisted generalized cohomology as cohomology in the tangent cohesive $\infty$-topos was presented in

* {#SchreiberTwists2013} [[Urs Schreiber]], talk at _[Twists, generalised cohomology and applications](http://wwwmath.uni-muenster.de/sfb/sfb878/2013/twists.html)_, October 2013 ([talk notes](twisted+smooth+cohomology+in+string+theory#LocalPrequantumFieldTheory))

Discussion in a comprehensive context of cohesion is in section 4.2.3 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

[[!redirects cohesive tangent (∞,1)-topos]]

[[!redirects stable cohesive homotopy theory]]
[[!redirects stable cohesive homotopy type theory]]

[[!redirects cohesive stable homotopy theory]]
[[!redirects cohesive stable homotopy type theory]]

[[!redirects tangent cohesion]]

[[!redirects stable cohesion]]

[[!redirects tangent cohesive ∞-topos]]
[[!redirects tangent cohesive ∞-toposes]]
[[!redirects tangent cohesive infinity-topos]]
[[!redirects tangent cohesive infinity-toposes]]


tangent complex

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _tangent complex_ of an algebro-geometric object is meant to behave as the (sheaf of) sections of the [[tangent bundle]].

{#TangentComplexAsInfinitesimalDiskBundleInIntroduction} At least in the generality of  [[derived stack|derived]] [[algebraic stacks]] $X$, the tangent complex is equivalently (up to a shift in degree) the module of sections of the [[infinitesimal disk bundle]] of $X$ (the [[formal completion]] of the [[diagonal]] $\Delta_X \colon X \to X \times X$) ([Hennion 13, theorem 1](#Hennion13)).

For $A$ an ordinary [[associative algebra]], its _tangent complex_ is its [[module]] of [[derivations]].

For $A$ a [[dg-algebra]], its _tangent complex_ is the essentially the value of the [[derived functor]] of the derivations-assigning functor on $A$. This is closely related to the [[automorphism ∞-Lie algebra]] of $A$.

## Related concepts

* [[deformation theory]]

* **tangent complex**, [[André-Quillen cohomology]], [[Hochschild cohomology]]

* [[cotangent complex]], [[André-Quillen homology]], [[Hochschild homology]]

## References

The concept goes back to

* M. Schlessinger, [[Jim Stasheff]], _The Lie algebra structure of tangent cohomology and deformation theory_, J. Pure Appl. Algebra, 38(1985), 313–322.

The tangent complex of an [[algebra over an operad]] in [[chain complex]]es is discussed in section 8 of

* {#Hinich} [[Vladimir Hinich]],  _Homological algebra of homotopy algebras_ Communications in algebra, 25(10). 3291-3323 (1997)([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), _Erratum_ ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453)))

See also

* [[Jonathan Block]], A. Lazarev, _André-Quillen cohomology and rational homotopy of function spaces_ , Advances in mathematics, 193 (2005) ([pdf](http://www.math.upenn.edu/~blockj/papers/BlockLazarevRat.pdf))

* {#Hennion13}  [[Benjamin Hennion]], _Tangent Lie algebra of derived Artin stacks_,  Journal für die reine und angewandte Mathematik (Crelles Journal), December 2015 ([arXiv:1312.3167](http://arxiv.org/abs/1312.3167), [DOI:10.1515/crelle-2015-0065](http://dx.doi.org/10.1515/crelle-2015-0065))

* Sam Raskin, p. 2,3 of _The cotangent stack_ (pdf)

[[!redirects tangent complexes]]


tangent cone

#Contents#
* table of contents
{:toc}

## Idea

The [[tangent bundle]] to an [[algebraic variety]] $X$ is abstractly defined by the methods of [[synthetic differential geometry]] as the space of maps

$$
  D \longrightarrow X
  \,,
$$

where $D$ is the [[spectrum of a commutative ring|spectrum]] of the [[ring of dual numbers]].

If $X$ is sufficiently regular, then this is the naive "Zariski [[tangent space]]". More generally the correct construction is given by the _tangent cone_ construction.

## Definition

+-- {: .num_defn}
###### Definition

For $X \hookrightarrow \mathbb{A}^n$ an [[affine algebraic variety]] defined by an [[ideal]] $I \hookrightarrow R[x_1, \cdots, x_n]$, hence $X \simeq Spec(R[x_1, \cdots, x_n]/I)$, then  its _tangent cone_ is

$$
  C X \coloneqq Spec(R[x_1, \cdots, x_n]/I_\ast)
$$

where $I_\ast$ is the [[ideal]] obtained from $I$ by truncating each [[polynomial]] $f \in I \hookrightarrow R[x_1, \cdots, x_n]$ to its homogeneous part of lowest monomial degree.

=--

## Properties

### Relation to étale morphisms

A [[homomorphism]] $f \colon X \longrightarrow Y$ of [[algebraic varieties]] over an [[algebraically closed field]] is an [[étale morphism]] if for all points $x \in X$ it induces an [[isomorphism]] of tangent cones

$$
  C_x X \longrightarrow C_{f(x)} Y
  \,.
$$

e.g. ([Costa, def. 1.1.8](#Costa)).

+-- {: .num_remark}
###### Remark

This is the same kind of characterization as for [[local diffeomorphisms]] (see there) in terms of [[tangent spaces]].

More generally, both cases are special cases of the definition of [[formally étale morphisms]] in terms of an [[infinitesimal shape modality]] $\Pi_{inf}$ as those morphisms $f \colon X \to Y$ such that the [[unit of a monad|unit]] [[natural transformation|naturality square]]

$$
  \array{
    X &\longrightarrow& \Pi_{inf}(X)
    \\
    \downarrow && \downarrow
    \\
    Y &\longrightarrow& \Pi_{inf}(Y)
  }
$$

is a [[pullback]].

=--

## References

Reviews include

* [[James Milne]], p. 18 of _[[Lectures on Étale Cohomology]]_

* Edgar Costa, around def. 1.1.6  of _Étale cohomology_ ([pdf](https://dspace.ist.utl.pt/bitstream/2295/686086/1/tese.pdf))
  {#Costa}

See also

* [[Igor Shafarevich|Igor R. Shafarevich]], _Basic algebraic geometry_

* Wikipedia [Tangent cone](http://en.wikipedia.org/wiki/Tangent_cone)

* [[eom]]: [tangent cone](http://www.encyclopediaofmath.org/index.php/Tangent_cone)

[[!redirects tangent cones]]


tangent function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Trigonometry
+-- {: .hide}
[[!include trigonometry -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[trigonometry]], the _tangent function_ "$tan$" is one of the basic [[trigonometric functions]].

\begin{imagefromfile}
    "file_name": "Mathnet-TangentFunction.jpg",
    "float": "left",
    "width": 220,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 10,
        "right": 15,
        "left": 0
    },
    "caption": "From [Mathnet](#Mathnet)"
\end{imagefromfile}

The tangent function may be understood as computing the [[length]] of the segment (shown in blue) of a [[tangent]]  (whence the name) to a unit [[circle]]  inside the [[Euclidean plane]], between its point of tangency and its intersection with any radial line, as a [[function]] of the [[angle]] ${\color{red}\theta}$ between that radial line and the one [[orthogonal]] to the [[tangent]].

Since, by definition of the [[sine function]] "$sin$" and the [[cosine function]] "$cos$", that radial line has [[distance]] $sin(\theta)$ from the perpendicular line where it crosses the unit circle at $cos(\theta)$ along that perpendicular line, the length of that tangential line segment [[equality|equals]] the [[ratio]] $sin(\theta)/cos(\theta)$, and this is how the tangent function is often defined.

\linebreak

## Definition

The tangent function is the [[ratio]] of the [[sine]] and the [[cosine]], where this is defined:

$$
  \theta
    \,\in\,
  \mathbb{R}
    \setminus
  \big\{
    (k+1/2)\pi
    \,\big\vert\,
    k \in \mathbb{Z}
  \big\}
  \;\;\;\;\;\;\;\;\;\;\;\;\;
    \vdash
  \;\;\;\;\;\;\;\;\;\;\;\;\;
  \tan(\theta)
  \,\coloneqq\,
  \frac{\sin(\theta)}{\cos(\theta)}
$$

## Properties

* The tangent function $f(x) = \tan x$ is of course odd, and obeys a functional equation ('addition formula')
$$f(x + y) = \frac{f(x) + f(y)}{1 - f(x)f(y)}.$$

* The [[hyperbolic functions|hyperbolic]] analog $\tanh x$, related to the tangent by
$$\tan(i x) = i\tanh x,$$
arises in [[special relativity theory]] in connection with [[rapidity]].

* The [[MacLaurin series]] for the tangent function arises in enumerative combinatorics, especially when bundled with the [[secant function]]. Here the sum
$$\sec x + \tan x = \sum_{n \geq 0} \frac{E_n x^n}{n!}$$
is the exponential generating function of a sequence of natural numbers $E_k$ that count "zigzag permutations" on the set $\{1, \ldots, k\}$, i.e., permutations $a_1 a_2 \ldots a_k$ where $a_1 \gt a_2 \lt a_3 \gt \ldots$. The *tangent numbers*, which are the coefficients $E_n$ when $n$ is odd, are related to [[Bernoulli numbers]] through the formula
$$(-1)^{k-1} B_{2k} = \frac{2k \cdot E_{2k-1}}{2^{2k}(2^{2k} - 1)}.$$
They are related to values of the [[zeta function]] by the formula
$$2(2^{2k}-1)\zeta(2k) = \frac{E_{2k-1}\pi^{2k}}{(2k-1)!}.$$
Both of these facts may be verified by putting together corresponding facts for the [[cotangent function]] (see there) with the double angle formula
$$\tan x = \cot x - 2\cot(2x).$$

## Related concepts

* [[tangent]], [[tangent vector]]

* [[cotangent function]]

* [[arctan]]

* [[trigonometric identity]]

## References

* Wikipedia, _[Trigonometric functions -- tan](https://en.wikipedia.org/wiki/Trigonometric_functions#tan)_

* {#Mathnet} Mathnet, *[tangent](https://www.math.net/tangent)*

[[!redirects tangent functions]]

[[!redirects tan]]


tangent Lie algebra

Given an [[algebraic group]] $G$ in characteristic zero, or a finite dimensional [[Lie group]], one associates to it a Lie algebra, its __tangent Lie algebra__ which is the Lie subalgebra $\mathcal{X}^{linv}(G)$ of the Lie algebra $\mathcal{X}^(G)$ left [[invariant vector field]]s on $G$ with respect to the usual Lie bracket of vector fields.

The value of a left-invariant vector field $X$ at the unit element $e$ is a tangent vector $X_e$ at $e$. It appears that the specialization/evaluation at the unit element map $\mathcal{X}^{linv}(G)\to T_e G$ is an isomorphism of vector spaces, which is often considered as an identification. However, one needs to look into vector fields in order to find the bracket, hence defining the tangent Lie algebra as the tangent vector space at $e$ misses the bracket (which come from consideration of infintesimals of second order). One can instead work with right invariant vector fields $\mathcal{X}^{rinv}(G)$ and obtain an isomorphic Lie algebra; the isomorphism is of course, by  comparing the specialization at $e$.

Within $\mathcal{X}^(G)$, all right invariant
vector fields commute with all left invariant vector fields.

The correspondence $G\mapsto\mathcal{X}^{linv}(G)$ is functorial.

A generalization of a tangent Lie algebra of a Lie group is the [[tangent Lie algebroid]].

The first idea of the tangent Lie algebra was explained in a letter of [[Sophus Lie]] to his friend Meyer in 1874
(See the historical appendix to Bourbaki, Lie groups and Lie algebras vol. 3).


tangent Lie algebroid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Definition ##

Let $X$ by a differentiable [[manifold]]. The _tangent Lie algebroid_ $T X$ of $X$ is the [[Lie algebroid]] that corresponds -- in the sense of  [[Lie theory]] -- to

* the [[codiscrete groupoid]] $X \times X$ of $X$.

* the [[fundamental groupoid]] $\Pi_1(X)$ of $X$.

When the tangent Lie algebroid is regarded as a [[Lie ∞-algebroid]] it corresponds to

* the [[path ∞-groupoid]] $\Pi_\infty(X)$.

The space of objects of $T X$ is $X$ itself and its elements in degree 1 are the vectors on $X$, i.e. the elements in the [[tangent bundle]] of $X$. These are to be thouhght of as the [[infinitesimal space|infinitesimal]] paths in $X$.

So to some extent the tangent Lie algebroid _is_ the [[tangent bundle]] $T X$ of $X$. More precisely, when using the definition of a [[Lie algebroid]] $E$ over $X$ as a diagram

$$
  \array{
    E &&\stackrel{\rho}{\to}&& T X
    \\
    & \searrow && \swarrow
    \\
    && X
  }
$$

where $\rho$ is the map called the _anchor map_, the tangent Lie algebroid is that whose anchor map is the identity map

$$
  \array{
    E = T X &&\stackrel{\rho = Id}{\to}&& T X
    \\
    & \searrow && \swarrow
    \\
    && X
  }
  \,.
$$

Therefore the tangent Lie algebroid of $X$ is usually denoted $T X$, just as the [[tangent bundle]] itself.

The [[Chevalley-Eilenberg algebra]] of $T X$ is correspondingly fundamental: it is the [[deRham dg-algebra]] of [[differential form]]s on $X$:

$$
  CE(T X) = (\Omega^\bullet(X), d_{dR})
  \,.
$$

So regarded as an [[NQ-supermanifold]] the tangent Lie algebroid is the [[shifted tangent bundle]] $\Pi T X$ equipped with its canonical odd vector field.

## Relation to flat $\infty$-Lie algebroid valued differential forms ##

For $\mathfrak{a}$ any other [[Lie algebroid]] or [[Lie infinity-algebroid]], a morphism of [[Lie infinity-algebroid]]s

$$
  \omega : T X \to \mathfrak{a}
$$

is _flat $\mathfrak{a}$-valued [[differential form]] datum_ .

For instance if $\mathfrak{a} = \mathfrak{g}$ is an ordinary [[Lie algebra]] then a morphism $T X \to \mathfrak{g}$ is a flat Lie algebra valued differential form on $X$, i.e. an element $A \in \Omega^1(X) \otimes \mathfrak{g}$ such that $d A + [-,-](A \wedge A) = 0$.

If $\mathfrak{g} = \mathfrak{u}(1) = \mathbb{R}$ is the abelian 1-dimensional Lie algebra, then a morphism $T X \to \mathfrak{g}$ is just a closed [[differential form|differential 1-form]] on $X$.

More on $\infty$-Lie algebroid-valued differential forms is (eventually) at [[∞-Lie algebroid valued differential forms]].

## Related concepts

The higher-order version of tangent Lie algebroids are [[jet bundle]] [[D-scheme]]s.

## References ##

One of the earliest reference seems to be

* [[Ted Courant]], _Tangent Lie algebroids_.  Journal of Physics A: Mathematical and General 27:13 (1994), 4527-4536.  [doi](http://dx.doi.org/10.1088/0305-4470/27/13/026).

[[!redirects tangent Lie algebroids]]


tangent map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

Given a [[differentiable map]] $f:M\to N$ of [[differentiable manifold | differentiable manifolds]], for each $p \in M$ we have the differential $d f_p : T_p M \to T_{f(p)} N$ of $f$ at $p$. These maps define a unique _tangent map_ $T f : T M \to T N$ of [[tangent bundles]] (as [[vector bundles]]) such that the [[diagram]]

$$\array{
T_p M &\stackrel{d_p f}\to & T_{f(p)}N\\
\downarrow &&\downarrow
\\
T M &\stackrel{T f}\to & T N
}$$

[[commuting diagram|commutes]] for each $p$. The differential $d f_p$ of $f$ at $p$ (or sometimes the composition $T_p M\stackrel{d_p f}\rightarrow T_{f(p)}N\hookrightarrow  T N$) is then denoted $T_p f$,
making $T$ into a [[functor]] (see *[[differentiation]]*).

category: geometry, differential geometry
[[!redirects tangent maps]]


tangent sheaf

The tangent sheaf $\mathcal{T}_X$ of a [[space]] $X$ is the [[sheaf]] of [[section]]s of the [[tangent bundle]] $T X\to X$.

category: sheaf theory


tangent spaces of mapping spaces

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

As shown in [[manifold structure of mapping spaces]], the space of [[smooth map]]s from a [[sequentially compact space|sequentially compact]] [[Frölicher space]] (or [[diffeological space|diffeological]] or Chen space) in to a [[smooth manifold]] is again a smooth manifold.  Its [[tangent space]] is straightforward to identify.  A tangent vector is an [[infinitesimal object|infinitesimal]] deviation of a smooth map; that is, it defines a direction in which to deform that map.  As a smooth map is determined by its values at points, when deforming a smooth map it is enough to explain how to deform each point.  Thus a tangent vector at $\alpha \colon S \to M$ defines, for each $p \in S$, a tangent vector at $\alpha(p) \in M$.  Thus we obtain a map $S \to T M$.  It is not unbelievable that this map is again smooth, whence we have:
\[
T C^\infty(S,M) \to C^\infty(S, T M).
\]
The claim of this page is that this is a [[diffeomorphism]], and further a [[vector bundle]] [[isomorphism]], where each is considered a vector bundle over $C^\infty(S, M)$.