connected topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

# Connected topos
* table of contents
{: toc}

## Idea

If we view a (Grothendieck) [[topos]] as a generalized [[topological space]], then a *connected topos* is a generalization of a [[connected topological space]].

More generally, a *connected geometric morphism* $p\colon E\to F$ is a "relativized" notion of this, saying that $E$ is "connected as a topos over $F$."

## Definition

### Connected geometric morphism

A [[geometric morphism]] $p\colon E\to F$ is **connected** if its [[inverse image]] part $p^*$ is [[fully faithful functor|full and faithful]].

A [[Grothendieck topos]] $E$ is **connected** if the unique geometric morphism $E \to Set = Sh(*)$ is connected.  If $E$ is the [[topos of sheaves]] on a [[topological space]] $X$ (or more generally a [[locale]]), then this is equivalent to the usual definition of connectedness for $X$ (see C1.5.7 in the [[Elephant]]).

Equivalently, a topos is connected if its [[global section]] geometric morphism exhibits [[discrete objects]].

Connected geometric morphisms are in particular [[surjective geometric morphism|surjective]].

### Connected locally connected morphisms

For geometric morphisms which are also [[locally connected geometric morphism|locally connected]], connectedness can be phrased in an especially nice form.

+-- {: .num_prop #PreservationOfTerminalObject}
###### Proposition
If $p\colon E\to F$ is locally connected, then it is connected if and only if the left adjoint $p_!$ of the inverse image functor (which exists, since $p$ is locally connected) preserves the terminal object.
=--
+-- {: .proof}
###### Proof
On the one hand, if $p^*$ is fully faithful, then the counit $p_! p^* \to \Id$ is an isomorphism, so we have $p_!(*) \cong p_!(p^*(*)) \cong *$; hence $p_!$ preserves the terminal object.

On the other hand, suppose that $p_!$ preserves the terminal object.  Suppose also for simplicity that $F=Set$.  Then any set $A$ is the [[coproduct]] $\coprod_A *$ of $A$ copies of the terminal object.  But $p^*$ and $p_!$ both preserve coproducts (since they are left adjoints) and terminal objects (since $p^*$ is left exact, and by assumption for $p_!$), so we have
$$
p_!(p^*(A)) \cong p_!(p^*(\coprod_A *)) \cong \coprod_A p_!(p^*(*)) \cong \coprod_A * \cong A
$$
Thus, the counit $p_! p^* \to \Id$ is an isomorphism, so $p^*$ is fully faithful.

When $F$ is not $Set$, we just have to replace ordinary coproducts with "$F$-indexed coproducts," regarding $E$ and $F$ as $F$-[[indexed category|indexed categories]].

(This is C3.3.3 in the [[Elephant]].)
=--

Strengthenings of this condition include

* a [[strongly connected geometric morphism]], which is locally connected such that $p_!$ preserves all finite products, and

* a [[totally connected geometric morphism]], which is locally connected such that $p_!$ preserves all finite limits.

### Connected locally connected sites

+--{: .num_prop}
###### Proposition
If $C$ is a [[locally connected site]] with a [[terminal object]], then the [[topos of sheaves]] $Sh(C)$ on $C$ is (not just locally connected) but connected.
=--
+--{: .proof}
###### Proof
As explained at [[locally connected site]], when $C$ is locally connected, the left adjoint $\Pi_0\colon Sh(C) \to Set$ is simply obtained by taking [[colimits]] over $C^{op}$.  Now by the [[co-Yoneda lemma]], the [[colimit]] over any [[representable presheaf]] is a [[singleton]] (i.e. a terminal object in [[Set]]):

$$
  \lim_\to y(V) = \int^{U \in C} C(U,V) = \int^{U \in C} C(U,V) \cdot * = *
  \,.
$$

But if $C$ has a terminal object, then that terminal object represents the terminal presheaf, which is also the terminal presheaf.  Hence under these conditions, $\Pi_0$ preserves the terminal object, so $Sh(C)$ is connected.
=--

## Properties

### Orthogonality

+--{: .num_prop}
###### Proposition
Connected geometric morphisms are left [[orthogonal]] to [[etale geometric morphisms]] in the [[2-category]] [[Topos]].
=--
+--{: .proof}
###### Proof
Since the functor $Topos^{op} \to Cat$ sending a topos to itself and a geometric morphism to its [[inverse image functor]] is 2-fully-faithful (an [[equivalence of categories|equivalence]] on [[hom-categories]]), connected morphisms are representably [[fully faithful morphism|co-fully-faithful]] in $Topos$.

Therefore, for 2-categorical orthogonality it suffices to show that in any commutative (up to iso) square
$$
  \array{
    A & \xrightarrow{f} & B
    \\
    {}^{\mathllap{p}}\downarrow & & \downarrow^{\mathrlap{q}}
    \\
    C & \xrightarrow{g} & D}
$$
of geometric morphisms in which $p$ is connected and $q$ is etale, there exists a filler $h\colon C\to B$ such that $h p \cong f$ and $q h \cong g$.

However, if $X\in D$ is such that $B \cong D/X$ (such exists by definition of $q$ being etale), then for any topos $E$ equipped with a geometric morphism $k\colon E\to D$, lifts of $k$ along $q$ are equivalent to morphisms $* \to k^*(X)$ in $C$.  In particular, $f$ is determined by a map $*\to f^*(q^*(X)) \cong p^*(g^*(X))$, and since $* \cong p^*(p_*(*))$ and $p^*$ is fully faithful, this map comes from a map $*\to g^*(X)$ in $C$, which in turn determines a geometric morphism $h\colon C\to B$ which is the desired filler.
=--

+--{: .num_prop}
###### Proposition
Any locally connected geometric morphism factors as a connected and locally connected geometric morphism followed by an etale one.
=--
+--{: .proof}
###### Proof
Given $f\colon E\to S$ locally connected, we can factor it as $E \to S/f_!(*) \to S$.  The second map is etale by definition, while the first is locally connected (the left adjoint is essentially $f_!$ again) and connected since it preserves the terminal object (by construction).
=--

In particular:

* (Connected, Etale) is a [[factorization system in a 2-category|factorization system]] on the 2-category $LCTopos$ of toposes and locally connected geometric morphisms.

* The category of etale geometric morphisms over a base topos $S$, which is equivalent to $S$ itself, is a [[reflective subcategory]] of the [[slice 2-category]] $LCTopos/S$.  The [[reflector]] constructs "$\Pi_0$ of a locally connected topos."

These results all have generalizations to [[∞-connected (∞,1)-toposes]].

## Examples

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

The [[gros topos|gros]] [[sheaf topos]] $Sh(CartSp)$ on the [[nLab:site]] [[CartSp]] -- which contains the [[quasi-topos]] of [[diffeological space]]s -- is a connected topos, since the site [[CartSp]] is a locally connected site and contains a terminal object.

=--

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

Let $\Gamma : \mathcal{E} \to Set$ be a connected and [[locally connected topos]] and
$X \in \mathcal{E}$ a connected object, $\Pi_0(X) \simeq *$.
Then the [[over-topos]] $\mathcal{E}/X$ is also connected and locally connected.

=--

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

For every object $X$, we have that $\mathcal{E}/X$ sits over $\mathcal{E}$ by the [[etale geometric morphism]].

$$
  \mathcal{E}/X
   \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
  \mathcal{E}
   \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}}
  Set
 \,.
$$

This makes $\mathcal{E}/X$ be a [[locally connected topos]].

Notice that the [[terminal object]] of $\mathcal{E}/X$ is $(X \stackrel{Id}{\to} X)$. If now $X$ is connected, then

$$
  \Pi_0 X_! (X \stackrel{Id}{\to} X)
  \simeq
  \Pi_0 X \simeq *
$$

and so the extra left adjoint $\Pi_0 \circ X_!$ preserves the terminal object. By the [above proposition](PreservationOfTerminalObject) this means that $\mathcal{E}/X$ is also connected.

=--

## Related concepts

* [[essential geometric morphism]]

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

* **connected topos** / [[∞-connected (∞,1)-topos]]

* [[strongly connected topos]]

* [[totally connected topos]] / [[totally connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

[[!redirects connected topoi]]
[[!redirects connected toposes]]
[[!redirects connected locally connected topos]]
[[!redirects connected locally connected topoi]]
[[!redirects connected locally connected toposes]]
[[!redirects connected geometric morphism]]
[[!redirects connected geometric morphisms]]
[[!redirects connected locally connected geometric morphism]]
[[!redirects connected locally connected geometric morphisms]]


connecting homomorphism

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###Context###
#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Generally, a _connecting homomorphism_ is a [[morphism]] of the kind produced by the [[snake lemma]].

Specifically, when the [[double complex]] that goes into the snake lemma is regarded as part of a [[short exact sequence]] $A_\bullet \to B_\bullet \to C_\bullet$ of [[chain complexes]], then the connecting homomorphisms induce morphisms $\delta_n : H_n(C) \to H_{n-1}(A)$ on the [[homology groups]] of these chain complexes which exhibit the corresponding [[long exact sequence in homology]] of the form

$$
  \cdots
   \to
  H_n(A) \to H_n(B) \to H_n(C)
   \stackrel{\delta_n}{\to}
  H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C)
   \to
  \cdots
  \,.
$$

This long exact sequence is the image under [[chain homology]]

$$
  H_0(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}
$$

of the long [[homotopy fiber sequence]] of chain complexes induced by the short exact sequence. Hence the connecting homomorphism is the image under $H_\bullet(-)$ of a [[mapping cone]] inclusion on chain complexes.

## For long (co)homology exact sequences

In the case that $\mathcal{A} \simeq R$[[Mod]] for some [[ring]] $R$,
the construction of the connecting homomorphism for [[homology long exact sequences]] is easily described in terms of elements and checking its properties is elementary, see _[In terms of elements](#OnHomologyInTermsOfElements)_ below. By the [embedding theorems](abelian%20category#EmbeddingTheorems) the general case can be reduced to this case. But there is also a general abstract description without recourse to elements, which we discuss further below in _[General abstract construction](#OnHomologyGeneralAbstract)_ .

### In terms of elements
 {#OnHomologyInTermsOfElements}

Let $R$ be a [[commutative ring]] and let $\mathcal{A} = R$[[Mod]]. Write $Ch_\bullet(\mathcal{A})$ for the [[category of chain complexes]] in $\mathcal{A}$.

Let

$$
  0 \to A_\bullet \stackrel{i}{\to} B_\bullet \stackrel{p}{\to}  C_\bullet \to 0
$$

be a [[short exact sequence]] in $Ch_\bullet(\mathcal{A})$.

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

For $n \in \mathbb{Z}$, define a [[group homomorphism]]

$$
  \delta_n : H_n(C) \to H_{n-1}(A)
  \,,
$$

called the **$n$th connecting homomorphism** of the short exact sequence,
by sending

$$
  \delta_n : [c] \mapsto [\partial^B \hat c]_A
  \,,
$$

where

1. $c \in Z_n(C)$ is a [[cycle]] representing a given [[homology group]];

1. $\hat c \in C_n(B)$ is any lift of that cycle to an element in $B_n$, which exists because $p$ is a [[surjection]] (but which no longer needs to be a cycle itself);

1. $[\partial^B \hat c]_A$ is the $A$-homology class of $\partial^B \hat c$  which is indeed in $A_{n-1} \hookrightarrow B_{n-1}$ by exactness (since $p(\partial^B \hat c) = \partial^C p(\hat c) = \partial^C c = 0$) and indeed in $Z_{n-1}(A) \hookrightarrow A_{n-1}$ since $\partial^A \partial^B \hat c = \partial^B \partial^B \hat c = 0$.

=--

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

Def. \ref{ConnectingForHomologyInComponents} is indeed well defined in that
the given map is independent of the choice of lift $\hat c$ involved and in that the group structure is respected.

=--

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

To see that the constructon is well-defined, let $\tilde c \in B_{n}$ be another lift. Then $p(\hat c - \tilde c) = 0$ and hence
$\hat c - \tilde c \in A_n \hookrightarrow B_n$.
This exhibits a homology-equivalence $[\partial^B\hat c]_A \simeq [\partial^B \tilde c]_A$ since
$
  \partial^A(\hat c - \tilde c)
  =
  \partial^B \hat c - \partial^B \tilde c$.

To see that $\delta_n$ is a group homomorphism, let $[c] = [c_1] + [c_2]$ be a sum. Then $\hat c \coloneqq \hat c_1 + \hat c_2$ is a lift and by linearity of $\partial$ we have $[\partial^B \hat c]_A = [\partial^B \hat c_1] + [\partial^B \hat c_2]$.

=--

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

Under [[chain homology]] $H_\bullet(-)$ the morphisms in the short exact sequence together with the connecting homomorphisms yield the [[homology long exact sequence]]

$$
  \cdots
   \to
  H_n(A) \to H_n(B) \to H_n(C)
   \stackrel{\delta_n}{\to}
  H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C)
   \to
  \cdots
  \,.
$$

=--

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

Consider first the exactness of $H_n(A) \stackrel{H_n(i)}{\to} H_n(B)
\stackrel{H_n(p)}{\to} H_n(C)$.

It is clear that if $a \in Z_n(A) \hookrightarrow Z_n(B)$ then the image of $[a] \in H_n(B)$ is $[p(a)] = 0 \in H_n(C)$.
Conversely, an element $[b] \in H_n(B)$ is in the kernel of $H_n(p)$ if there is $c \in C_{n+1}$ with $\partial^C c = p(b)$. Since $p$ is surjective let $\hat c \in B_{n+1}$ be any lift, then $[b] =  [b - \partial^B \hat c]$ but $p(b - \partial^B c) = 0$ hence by exactness $b - \partial^B \hat c \in Z_n(A) \hookrightarrow Z_n(B)$ and so $[b]$ is in the image of $H_n(A) \to H_n(B)$.

It remains to see that

1. the [[image]] of $H_n(B) \to H_n(C)$ is the [[kernel]] of $\delta_n$;

1. the [[kernel]] of $H_{n-1}(A) \to H_{n-1}(B)$ is the [[image]] of $\delta_n$.

This follows by inspection of the formula in def. \ref{ConnectingForHomologyInComponents}. We spell out the first one:

If $[c]$ is in the image of $H_n(B) \to H_n(C)$ we have a lift $\hat c$ with $\partial^B \hat c = 0$ and so $\delta_n[c] = [\partial^B \hat c]_A  = 0$. Conversely, if for a given lift $\hat c$ we have that $[\partial^B \hat c]_A = 0$ this means there is $a \in A_n$ such that $\partial^A a \coloneqq \partial^B a = \partial^B \hat c$. But then $\tilde c \coloneqq \hat c - a$ is another possible lift of $c$ for which $\partial^B \tilde c = 0$ and so $[c]$ is in the image of $H_n(B) \to H_n(C)$.

=--

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

Of course the situation for [[cochain cohomology]] is formally dual to this situation. For convenience we repeat the statement for dual chains:

Let $A^\bullet \to B^\bullet \to C^\bullet$ be a short exact sequence of cochain complexes.

For $[c]_C \in H^n(C)$ the class of a closed element $c$, by surjectivity of $B \to C$ there is an element $\hat c \in B$ mapping to it. This need not be closed anymore, but of course $d_B \hat c$ is. By the fact that $B \to C$ is a chain map we have that the image of $d_B \hat c$ in $C$ vanishes. Therefore by the exactness of the sequence the element $d_B \hat c$ may be regarded as a closed element of $A$. The cohomology class $[d_B \hat c]_A$ of this is what the connecting homomorphism

$$
  \delta^n : H^n(C) \to H^{n+1}(A)
$$

assigns to $[c]_C$:

$$
  \delta : [c]_C \mapsto [d_B\hat c]_A
  \,.
$$

This is indeed well defined, in that it is independent of the choice of $\hat c$: for $\hat c'$ another choice, we have that the difference $\hat c - \hat c'$ is in the kernel of $B \to C$ hence is in $A$. Then $d_B \hat c' = d_B \hat c + d_A(\hat c - \hat c')$. Hence $[d_B \hat c]_A = [d_B \hat c']_A$.

=--

### General abstract
 {#OnHomologyGeneralAbstract}

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

Let $0 \to A_\bullet \stackrel{f}{\to} B_\bullet \stackrel{g}{\to} C_\bullet \to 0$
be a [[short exact sequence]] of [[chain complexes]] in some [[abelian category]] $\mathcal{A}$. Then for all $n \in \mathbb{Z}$ there are natural _connecting homomorphisms_ $\partial : H_n(C) \to H_{n-1}(A)$ such that we have a [[long exact sequence]] of the form

$$
  \cdots \stackrel{g}{\to}
  H_{n+1}(C)
  \stackrel{\partial}{\to}
  H_n(A)
  \stackrel{f}{\to}
  H_n(B)
  \stackrel{g}{\to}
  H_n(C)
  \stackrel{\partial}{\to}
  H_{n-1}(A)
  \stackrel{f}{\to}
  \cdots
$$

in [[chain homology]].

=--

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

Applying the [[snake lemma]] to the [[commuting diagram]]

$$
  \array{
     && 0 && 0 && 0
     \\
     && \downarrow && \downarrow && \downarrow
     \\
     0 &\to& Z_n A &\to& Z_n B &\to & Z_n C
     \\
     && \downarrow && \downarrow && \downarrow
     \\
     0 &\to& A_n &\to& B_n &\to & C_n &\to & 0
     \\
     && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}}
     \\
     0 &\to& A_{n-1} &\to& B_{n-1} &\to & C_{n-1} &\to& 0
     \\
     && \downarrow && \downarrow && \downarrow
     \\
     && \frac{A_{n-1}}{im(d)(A_n)} &\to& \frac{B_{n-1}}{im(d)(B_n)} &\to & \frac{C_{n-1}}{im(d)(C_n)} &\to & 0
     \\
     && \downarrow && \downarrow && \downarrow
     \\
     && 0 && 0 && 0
  }
$$

shows that the rows in the commuting diagram

$$
  \array{
      && \frac{A_{n}}{im(d)(A_{n+1})} &\to& \frac{B_{n}}{im(d)(B_{n+1})} &\to & \frac{C_{n}}{im(d)(C_{n+1})} &\to & 0
      \\
     && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}}
     \\
     0 &\to& Z_{n-1} A &\stackrel{f}{\to}& Z_{n-1} B &\stackrel{g}{\to}& Z_{n-1} C
  }
$$

are [[exact sequences]]. Therefore applying the [[snake lemma]] to this, once more, yields the desired long exact sequence.

=--

## Examples

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

The [[nLab:connecting homomorphism]] of the [[nLab:long exact sequence in homology]] induced from short exact sequences of the form

$$
  A/A_{n tor} \stackrel{(-)\cdot n}{\to} A \to A/(n A)
$$

is called a _[[nLab:Bockstein homomorphism]]_.

=--

## Properties

### Relation to homotopy fiber sequences

The connecting homomorphism in a [[long exact sequence in homology]] induced from a short exact sequence $A_\bullet \stackrel{f}{\to} B_\bullet \to C_\bullet$ is equivalently the image under the [[homology group]] functor of the [[homotopy cofiber sequence]] induced by $f$. This is discussed in detail at _[[mapping cone]]_ in the section _[homology exact sequences](mapping+cone#HomologyExactSequencesAndFiberSequences)_.

## Related concepts

* [[long exact sequence in homology]]

  * [[long exact sequence in chain homology]]

  * [[long exact sequence in generalized homology]]

## References

For instance section 1.3 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

[[!redirects connecting homomorphisms]]


connection

The term **connection** can mean any of

* [[connection on a bundle]]

  * [[principal connection]]

    * [[affine connection]]

    * [[Cartan connection]]

    * [[Ehresmann connection]]

    * [[flat connection]]

  * [[meromorphic connection]]
  * [[Grothendieck connection]]
  * [[simplicial connection]]
  * [[connection in noncommutative geometry]]
* [[Hurewicz connection]]

* [[connection for a differential graded algebra]]

* [[p-connection]] for a smooth map $p$ of smooth schemes or analytic spaces

* [[connection for a coring]]

which are all very similar and often special cases of each other, or a sort of dual construction

* [[hom-connection]],

but it can also mean the non-related notions of

* [[connection on a cubical set]] and

* [[connection on a double category]],

which are quite different.

[[!redirects connections]]


connection for a coring

Given an $A$-[[coring]] or even a more general additive [[comonad]] with a [[grouplike element]] there are several related (but in general nonequivalent) notions of connections.

As explained in [[grouplike element]], to an $A$-coring $C$ with a grouplike element one associates a [[semi-free differential graded algebra]] $\Omega A = \Omega(A,C)$, sometimes called its (generalized) Amitsur complex. The simplest notion of a connection for the coring $C$ is a [[connection for a differential graded algebra|connection]] for the corresponding Amitsur complex.

Let now $A$ be a $k$-algebra and $(C,\Delta,\epsilon)$ be an $A$-coring with grouplike element $g$ and $(\Omega A,d)$ its [[Amitsur complex]].

A __connection__ $\nabla:M\otimes_A\Omega^\bullet\to M\otimes_A\Omega^{\bullet+1}$ on a module $M$ over a [[semifree dga]] (in the sense of the entry [[connection for a differential graded algebra]]) is determined by its value $\nabla_M|_M$ on $M\cong M\otimes_A A$. If $\rho^M:M\to M\otimes_A C$ is a right $C$-[[coaction]] then the formula

$$\nabla|_M:m\mapsto \rho^M(m)-m\otimes g$$

determines a [[flat connection]] on $M$. Conversely, any flat connection determines a right $C$-coaction by

$$\rho^M(m)=\nabla(m)+m\otimes g.$$

This amounts to a *bijection between $C$-coactions and flat connections* on $M$. Regarding that coactions correspond to descent data in the context of comonadic descent, this gives the flat connection interpretation of such descent data. A first instance is probably Grothendieck's identification of flat connections and the first order costratifications in Grothendieck's theory of differential calculus on schemes (foundations of [[crystalline cohomology]], see book by Berthelot and Ogus; cf. also [[Grothendieck connection]]).

## Connections on comodules directly

One the other hand, one can consider more generally additive comonads, and define connections on comodules over them rather directly. Or dually one can work with connection on modules over [[additive monad]]s.

Menini and Ştefan first define an intermediate notion of a quasi-connection for monads. Let $A$ be an [[additive category]] $(T,\mu,\eta)$ an additive monad in $A$ and $\nu:TM\to M$ an action on some object $M$ in $A$.
Then a __quasi-connection__ on $M$ is a map $\nabla:M\to TM$ such that

$$
\nabla \circ \nu - \mu\circ T(\nabla) = id_{TM} -  \eta\circ\nu: TM\to TM.
$$

A quasi-connection is a __connection__ if, in addition,

$$
\nu\circ\nabla = 0.
$$

For every connection in Menini--Ştefan sense, one defines its **curvature** $F_\nabla : M\to T^2 M$ by the formula

$$
F_\nabla := (id_{T^2 M} - \eta_{TM}\circ\mu_M)\circ T(\nabla)\circ\nabla.
$$

As usually, we define a **flat connection** as a connection whose curvature vanishes.

In this setting one again has a bijection between flat connections and descent data.

## Literature and links

* P. Nuss, Noncommutative descent and non-abelian cohomology,  $K$-Theory  12  (1997),  no. 1, 23--74.

* [[T. Brzeziński]], R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series __309__, Cambridge 2003.

* C. Menini, Talk at MSRI: _Connections, symmetry operators and descent data for triples_, 1999, [link](http://www.msri.org/communications/ln/msri/1999/hopfalg/menini/1/index.html)

* C. Menini, D. Ştefan, _Descent theory and Amitsur cohomology of triples_, J. Algebra __266__ (2003), no. 1, 261--304.

* [[T. Brzeziński]], _Flat connections and (co)modules_, [in:] New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35-52   [arxiv:math.QA/0608170](http://arxiv.org/abs/math.QA/0608170)

[[!redirects connection for coring]]


connection for a differential graded algebra

Given a [[semi-free differential graded algebra]] $\Omega^\bullet A$ over $k$-algebra $A$, a __connection__ in $A$-module $M$ is a $k$-linear map

$$
\nabla : M\otimes_A\Omega^\bullet A \to M\otimes_A\Omega^{\bullet+1} A
$$

such that for any homogenous element $\omega\in\Omega^k A$ and an element $\chi\in\Omega A$

$$
\nabla (\omega\chi) = \nabla(\omega)\chi + (-1)^k\omega\nabla(\chi)
$$

The curvature $F_\nabla$ of the connection $\nabla$ is the restriction of the connection squared to $M$:

$$\nabla\circ\nabla|_M : M\to M\otimes_A\Omega^2 A. $$

A connection is __flat__ (or integrable) iff its curvature vanishes.

See also [[connection in noncommutative geometry]] as some versions are close to this approach. A past query on history of the notion of connection for a dga is archived [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3107&Focus=25779#Comment_25779).

[[!redirects connection for a dga]]
[[!redirects connections for a differential graded algebra]]


connection in noncommutative geometry

In [[noncommutative geometry]] there are several versions of noncommutative bundle theory, e.g. considering vector bundles as finitely generated projective modules and the theory of [[noncommutative principal bundle]]s as [[Hopf-Galois extensions]] and their coalgebra and global analogues. Each of these formalism can be a setup in whcih one can try to develop a noncommutative analogue of the theory of [[connections on a bundle]]. The connection theory on noncommutative spaces is of course, the basis of [[gauge theories]] on noncommutative spaces. A remarkable distinction between the commutative and noncommutative case of connections on Hopf-Galois extensions is the difference between generic connection on a generic and so-called strong connections, as discovered by P. Hajac. There is an approach close to Koszul's [[connection for a differential graded algebra|for dga-s]], by defining the connections by action on noncommutative differential forms of Karoubi or by an appropriate analogue in [[cyclic homology]].

There is a rather vast literature on the subject and we should list the more important ones.

* [[Max Karoubi]], _Connexions, courbures et classes caractéristiques en K-théorie algébrique_, Current trends in algebraic topology, Part I, vol. 2, 19-27, London, Ontario 1981, [pdf](http://people.math.jussieu.fr/~karoubi/Publications/31.pdf)
* Alain Connes, _Noncommutative geometry_, Academic Press 1994, 661 p. [PDF](http://www.alainconnes.org/docs/book94bigpdf.pdf)
* [[Tomasz Brzeziński]], _Flat connections and (co)modules_, [math/0608170](http://arxiv.org/abs/math/0608170)
* [[Tomasz Brzeziński]], _A note on flat noncommutative connections_, [arxiv/1109.0858](http://arxiv.org/abs/1109.0858)
* [[Tomasz Brzeziński]], [[Shahn Majid]], _Quantum group gauge theory on quantum spaces_, Commun. Math. Phys. 157:591-638, 1993, 	[hep-th/9208007](http://arxiv.org/abs/hep-th/9208007), [doi](http://dx.doi.org/10.1007/BF02096884); Erratum 167:235, 1995,
* Piotr M. Hajac, _Strong connections on quantum principal bundles_, Comm. Math. Phys. __182__ (1996), no. 3, 579–617, [MR98e:58022](http://www.ams.org/mathscinet-getitem?mr=1461943), [euclid](http://euclid.cmp/1104288302)
* S. Majid, _Quantum and braided group Riemannian geometry_, J. Geom. Phys. __30__ (1999) 113-146, [q-alg/9709025](http://xxx.lanl.gov/abs/q-alg/9709025)
* M. Dubois-Violette, P. Michor, _Connections on central bimodules in noncommutative differential geometry_, J. Geom. Phys. __20__ (1996) 218
* [[Michel Dubois-Violette|M. Dubois-Violette]], J. Madore, T. Masson, J. Mourad, _On curvature in noncommutative geometry_, [q-alg/9512004](http://arxiv.org/abs/q-alg/9512004)

In framework of [[spectral triple]]s, see

* Branimir Ćaćić, Bram Mesland, _Gauge theory on noncommutative Riemannian principal bundles_, [arxiv/1912.04179](https://arxiv.org/abs/1912.04179)

[[!redirects connection on a noncommutative bundle]]
[[!redirects connections on a noncommutative bundle]]
[[!redirects connections in noncommutative geometry]]


connection on a 2-bundle

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _connection on a 2-bundle_ generalizes the notion of [[connection on a bundle]] from [[principal bundle]]s to [[principal 2-bundle]]s / [[gerbe]]s.

It comes with a notion of [[higher parallel transport|2-dimensional parallel transport]].

For an exposition of the concepts here see also at _[[infinity-Chern-Weil theory introduction]]_ the section _[Connections on principal 2-bundles](http://ncatlab.org/nlab/show/infinity-Chern-Weil%20theory%20introduction#ConnectionOn2Bundle)_ .

## Definition

For $G$ a [[Lie 2-group]], a _connection_ on a $G$-[[principal 2-bundle]] coming from a [[cocycle]] $g : X \to \mathbf{B}G$ is a lift of the cocycle to the  [[2-groupoid of Lie 2-algebra valued forms]] $\mathbf{B}G_{conn}$

$$
  \array{
     && \mathbf{B}G_{conn}
     \\
     & {}^{\mathllap{\nabla}}\nearrow & \downarrow
     \\
     X &\stackrel{g}{\to}& \mathbf{B}G
  }
$$

## Properties

### On trivial 2-bundles

When the underlying [[principal 2-bundle]] over a [[smooth manifold]] $X$ is topologically trivial, then the connections on it are identified with [[Lie 2-algebra valued differential forms]] on $X$.

Recall from the discussion there what such form data looks like.

Let $\mathfrak{g}$ be some [[Lie 2-algebra]]. For instance for discussion of connections on $G$-[[gerbe]]s ($G$ a [[Lie group]]) this would be the [[automorphism 2-group|derivation Lie 2-algebra]] of the [[Lie algebra]] of $G$.

Let $\mathfrak{g}_0$ and $\mathfrak{g}_1$ be the two vector spaces involved and let

$$
  \{t^a\} \,, \;\;\; \{b^i\}
$$

be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general [[Chevalley-Eilenberg algebra]]

$$
  CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}
$$

with these generators.

We thus have

$$
  d_{CE(\mathfrak{g})} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c - r^a{}_i b^i
$$

$$
  d_{CE(\mathfrak{g})} b^i = -\alpha^i_{a j} t^a \wedge b^j
    -
    r^i{}_{a b c} t^a \wedge t^b \wedge t^c
  \,,
$$

for collections of structure constants $\{C^a{}_{b c}\}$ (the bracket on $\mathfrak{g}_0$) and $\{r^i_a\}$ (the differential $\mathfrak{g}_1 \to \mathfrak{g}_0$) and $\{\alpha^i{}_{a j}\}$ (the [[action]] of $\mathfrak{g}_0$ on $\mathfrak{g}_1$) and $\{r_{a b c}\}$ (the "Jacobiator" for the bracket on $\mathfrak{g}_0$).

These constants are subject to constraints (the weak [[Jacobi identity]] and its higher [[coherence law]]s) which are precisely equivalent to the condition

$$
  (d_{CE(\mathfrak{g})})^2 = 0
  \,.
$$

Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism

$$
  \Omega^\bullet(U) \leftarrow W(\mathfrak{g})
   :
  (A,B)
$$

from the [[Weil algebra]] $W(\mathfrak{g})$.

This is given by a 1-form

$$
  A \in \Omega^1(U, \mathfrak{g}_0)
$$

and a 2-form

$$
  B \in \Omega^2(U, \mathfrak{g}_1)
  \,.
$$

The [[curvature]] of this is $(\beta, H)$, where the 2-form component ("fake curvature") is

$$
  \beta^a = d_{dR} A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c
  + r^{a}_{i} B^i
$$

and whose 3-form component is

$$
  H^i
     =
   d_{dR} B^i
   +
   \alpha^i{}_{a j} A^a \wedge B^j
   +
   r^i{}_{a b c} A^a \wedge A^b \wedge A^c
  \,.
$$

### Differential Čech cocycle data

We spell out the data of a connection on a 2-bundle over a [[smooth manifold]] $X$ with respect to a given [[open cover]] $\{U_i \to X\}$, following
([FSS](#FSS), [SchreiberCohesive](#SchreiberCohesive))

(...)

## Examples

* A [[differential string structure]] (untwisted) is a 2-connection with coefficients in the [[string 2-group]] / [[string Lie 2-algebra]].

* The [[worldvolume]] theory of the [[fivebrane]] is expected to be a [[6d (2,0)-supersymmetric QFT]] containing a [[self-dual higher gauge theory]] whose fields are 2-connections (see _[Self-dual 2-connections](http://ncatlab.org/nlab/show/NS5-brane#SelfDual2Connections)_ there).

* A connection on a [[twisted vector bundle]] is naturally a 2-connection.

## Related concepts

* [[connection on a bundle]]

* **connection on a 2-bundle** / [[connection on a gerbe]] / [[connection on a bundle gerbe]]

  * [[2-groupoid of Lie 2-algebra valued forms]]

* [[connection on a 3-bundle]] / [[connection on a bundle 2-gerbe]]

* [[connection on an ∞-bundle]]

* [[parallel transport]], [[higher parallel transport]]

* [[holonomy]]

## References

### General

Connections on 2-bundles with vanishing 2-form curvature and arbitrary 3-form curvature are defined in terms of their [[higher parallel transport]] are discussed in

* [[Urs Schreiber]],  [[Konrad Waldorf]],

  _Smooth Functors and Differential Forms_, Homology, Homotopy Appl., 13(1), 143-203 (2011) ([arXiv:0802.0663](http://arxiv.org/abs/0802.0663))

  _Connections on non-abelian gerbes and their holonomy_, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. ([TAC](http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html), [arXiv:0808.1923](http://arxiv.org/abs/0808.1923), web)

expanding on

* [[John Baez]], [[Urs Schreiber]], _Higher gauge theory_ ([arXiv:math/0511710](http://arxiv.org/abs/math/0511710)) ([web](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos+--+references/#BaezSchreiber))

Much further discussion and illustration and relation to [[tensor networks]] is in

* {#Parzygnat18} [[Arthur Parzygnat]], _Two-dimensional algebra in lattice gauge theory_, Journal of Mathematical Physics 60, 043506 (2019) ([arXiv:1802.01139](https://arxiv.org/abs/1802.01139), [doi:10.1063/1.5078532](https://doi.org/10.1063/1.5078532))

Examples of 2-connections with vanishing 2-form curvature obtained from [[geometric quantization]] are discusssed in

* Olivier Brahic, _On the infinitesimal Gauge Symmetries of closed forms_ ([arXiv](http://arxiv.org/abs/1010.2189))

The [[cocycle]] data for 2-connections with coeffcients in [[automorphism 2-group]]s but without restrictions on the 2-form curvature  have been proposed in

* [[Lawrence Breen]], [[William Messing]], _Differential Geometry of Gerbes _ Advances in Mathematics, Volume 198, Issue 2, 20 (2005) ([arXiv:math/0106083](http://arxiv.org/abs/math/0106083))

* [[Lawrence Breen]], _Differential Geometry of Gerbes and Differential Forms_ ([arXiv:0802.1833](http://arxiv.org/abs/0802.1833))

and

* Paolo Aschieri, Luigi Cantini, [[Branislav Jurco]], _Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory_ , Communications in Mathematical Physics Volume 254, Number 2 (2005) 367-400,([arXiv:hep-th/0312154](http://arxiv.org/abs/hep-th/0312154)).

A discussion of fully general local 2-connections is in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_\infty$-connections_ ([arXiv:0801.3480](http://arxiv.org/abs/0801.3480))

and the globalization is in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]]_

For a discussion of all this in a more comprehensive context see section xy of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_
 {#SchreiberCohesive}

See also [[connection on an infinity-bundle]] for the general theory.

### Applications to physics

Nonabelian 2-connections appear for instance as [[orientifold]] [[B-fields]] in [[type II string theory]], as [[differential string structure]] in [[heterotic string theory]], and as fields in non-abelian [[7-dimensional Chern-Simons theory]]. See at these pages for references.

An appearance in 4-dimensional [[Yang-Mills theory]] and [[4d TQFT]] is reported in

* [[Sergei Gukov]], [[Anton Kapustin]], _[[Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories]]_ ([arXiv:1307.4793](http://arxiv.org/abs/1307.4793))
 {#GukovKapustin13}

[[!redirects connections on a 2-bundle]]
[[!redirects connections on 2-bundles]]

[[!redirects 2-connection]]
[[!redirects 2-connections]]

[[!redirects principal 2-connection]]
[[!redirects principal 2-connections]]

[[!redirects connection on a principal 2-bundle]]

[[!redirects principal 2-bundle with connection]]
[[!redirects principal 2-bundles with connection]]


connection on a 3-bundle

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

#Contents#
* table of contents
{:toc}

## Definition

For $G$ a [[Lie 3-group]], a _connection_ on a $G$-[[principal 3-bundle]] coming from a [[cocycle]] $g : X \to \mathbf{B}G$ is a lift of the cocycle to the  [[3-groupoid of Lie 3-algebra valued forms]] $\mathbf{B}G_{conn}$

$$
  \array{
     && \mathbf{B}G_{conn}
     \\
     & {}^{\mathllap{\nabla}}\nearrow & \downarrow
     \\
     X &\stackrel{g}{\to}& \mathbf{B}G
  }
$$

## Examples

* [[11-dimensional supergravity]] can be understood as the theory of 3-connections with coefficients in the [[super L-infinity algebra]] called the [[supergravity Lie 3-algebra]]. See [[D'Auria-Fre formulation of supergravity]].

## Related concepts

* [[connection on a bundle]]

* [[connection on a 2-bundle]] / [[connection on a gerbe]] / [[connection on a bundle gerbe]]

* **connection on a 3-bundle** / [[connection on a bundle 2-gerbe]]

  * [[chiral anomaly]], [[baryogenesis]]

  * [[supergravity C-field]]

* [[connection on an ∞-bundle]]

## References

The [[higher parallel transport]] of local 3-connections is considered in

* [[Joao Faria Martins]], [[Roger Picken]], _The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module_ ([arXiv:abs/0907.2566](http://arxiv.org/abs/0907.2566))

Discussion in terms of [[Gray-categories]] is in

* Wei Wang, _On 3-gauge transformations, 3-curvature and Gray-categories_ ([arXiv:1311.3796](https://arxiv.org/abs/1311.3796))

Examples of 3-connections obtained from fibrations of [[Courant algebroid]]s are discussed in

* Olivier Brahic, _On the infinitesimal Gauge Symmetries of closed forms_ ([arXiv](http://arxiv.org/abs/1010.2189))

A discussion of fully general local 3-connections is in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_\infty$-connections_ ([arXiv:0801.3480](http://arxiv.org/abs/0801.3480))

and the globalization is in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]]_

For a discussion of all this in a more comprehensive context see section xy of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_
 {#SchreiberCohesive}

See also [[connection on an infinity-bundle]] for the general theory.

[[!redirects connections on a 3-bundle]]

[[!redirects connections on 3-bundles]]

[[!redirects 3-connection]]
[[!redirects 3-connections]]


connection on a bundle

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

#Contents#
* table of contents
{:toc}

## Idea

A **connection** on a [[bundle]]  $P \to X$ --  a [[principal bundle]] or an [[associated bundle]] like a [[vector bundle]] -- is a rule that identifies [[fiber]]s of the bundle along paths in the base space $X$.

There are several different but equivalent formalizations of this idea:

* as a  __[[parallel transport]]__ functor,

* as a rule for a __[[covariant derivative]]__,

* as a __distribution (field) of horizontal subspaces__  -- an [[Ehresmann connection]] -- and via a __connection $1$-form__ which annihilates the distribution of horizontal subspaces. The connection in that sense induces a smooth version of [[Hurewicz connection]].

The usual textbook convention is to say just connection for the distribution of horizontal subspaces, and the objects of the other three approaches one calls more specifically covariant derivative, connection $1$-form and [[parallel transport]].

In the remainder of this Idea-section we discuss a bit more how to understand connections in terms of parallel transport.

Given a smooth [[bundle]] $P \to X$, for instance a $G$-[[principal bundle]] or a [[vector bundle]], a _connection_ on $P$ is a prescription to associate with each path

$$
  \gamma : x \to y
$$

in $X$  (which is a [[morphism]] in the [[path groupoid]] $\mathbf{P}_1(X)$) a morphism $tra(\gamma)$ between the fibers of $P$ over these points

$$
  \array{
     P_x &\stackrel{tra(\gamma)}{\to}& P_y
     \\
     x &\stackrel{\gamma}{\to}& y
  }
$$

such that

* this assignment respects the structure on the fibers $P_x$ (for instance is $G$-equivariant in the case that $P$ is a $G$-bundle or that is linear in the case that $P$ is a vector bundle);

* this assignment is smooth in a suitable sense;

* this assignment is [[functor]]ial in that for all pairs $x \stackrel{\gamma}{\to} y$, $y \stackrel{\gamma'}{\to} z$ of composable paths in $X$ we have
  $$
    \array{
       P_x &\stackrel{tra(\gamma)}{\to}& P_y &\stackrel{tra(\gamma')}{\to}& P_z
       \\
       x &\stackrel{\gamma}{\to}& y &\stackrel{\gamma'}{\to}& z
    }
    \;\;\;
    =
    \;\;\;
    \array{
       P_x &\stackrel{tra(\gamma' \circ \gamma)}{\to}& P_z
       \\
       x &\stackrel{\gamma'\circ \gamma}{\to}& z
    }
  $$

In other words, a connection on $P$ is a [[functor]]

$$
  tra : \mathbf{P}_1(X) \to At''(P)
$$

from the [[path groupoid]] of $X$ to the [[Atiyah Lie groupoid]] of $P$ that is smooth in a suitable sense and _splits the Atiyah sequence_ in that $\mathbf{P}_1(X) \stackrel{tra}{\to} At''(X) \to \mathbf{P}_1(X)$ (see the notation at [[Atiyah Lie groupoid]]).

**Terminology**

The functor $tra$ is called the **[[parallel transport]]** of the connection. This terminology comes from looking at the orbits of all points in $P$ under $tra$ (i.e. from looking at the [[category of elements]] of $tra$): these trace out paths in $P$ sitting over paths in $X$ and one thinks of the image of a point $p \in P_x$ under $tra(\gamma)$ as the result of propagating $p$ parallel to these curves in $P$.

**Flat connections**

It may happen that the assignment $tra : \gamma \mapsto tra(\gamma)$ only depends on the [[homotopy]] class of the path $\gamma$ relative to its endpoints $x, y$. In other words: that $tra$ factors through the functor $P_1(X) \to \Pi_1(X)$ from the [[path groupoid]] to the [[fundamental groupoid]] of $X$. In that case the connection is called a **[[flat connection]]**.

### More concrete picture

By [[Lie theory|Lie differentiation]] the functor $tra$, i.e. by looking at what it does to very short pieces of paths, one obtains from it a splitting of the [[Atiyah Lie groupoid|Atiyah Lie algebroid sequence]], which is a morphism

$$
  \nabla : T X \to at(P)
$$

of vector bundles. Locally on $X$ -- meaning: when everything is pulled back to a [[cover]] $Y \to X$ of $X$ -- this is a $Lie(G)$-[[differential form#twisted|valued]] 1-form $A \in \Omega^1(Y, Lie(G))$ with certain special properties.

In particular, since every $G$-[[principal bundle]] canonically trivializes when pulled back to its own total space $P$, a connection in this case gives rise to a 1-form $A \in \Omega^1(P)$ satisfying two conditions. This data is called an **[[Ehresmann connection]]**.

If instead $P = E$ is a vector bundle, then the two conditions satisfies by $A$ imply that it defines a linear map

$$
  \nabla : \Gamma(E) \to \Omega^1(X,E):=\Gamma(X,T_X^*\otimes E)
$$

from the space $\Gamma(E)$ of section of $E$ that satisfies the properties of a **covariant derivative**.

If again the connection is flat, then this is manifestly the datum of a [[Lie infinity-algebroid representation]] of the [[Lie algebroid|tangent Lie algebroid]] $T X$ of $X$ on $E$: it defines the action [[Lie algebroid]] which is the [[Lie theory|Lie version]] of the [[Lie groupoid]] classified by the parallel transport functor.

...

### More abstract picture

Recall from the discussion at $G$-[[principal bundle]] that the $G$-bundle $P \to X$ is encoded in a a suitable morphism

$$
  X \to \mathbf{B}G
$$

(namely a morphism in the right [[(infinity,1)-category]] which may be represented by an [[anafunctor]]).

It turns out that similarly suitable morphisms

$$
  \mathbf{P}_1(X) \to \mathbf{B}G
$$

encode in one step the $G$-bundle together with its [[parallel transport]] functor.

This is described in great detail in the reference by Schreiber--Waldorf below.

(...am running out of time... )

## Definition {#Definition}

Let $G$ be a [[Lie group]]. We recall briefly the following discussion of $G$-[[principal bundle]]s. For an in-depth discussion see [[Smooth∞Grpd]].

Write

$$
  \mathbf{B}G : U \mapsto ( Hom_{Diff}(U,G) \stackrel{\to}{\to} *)
$$

for the [[functor]] that sends a [[Cartesian space]] $U$ to the [[delooping]] [[groupoid]] of the group of $G$-valued [[smooth function]]s on $U$: the groupoid with a single object and the [[group]] $Hom_{Diff}(U,G)$ of maps as its set of morphisms.

This is a groupoid-valued [[sheaf]] on the [[site]] [[CartSp]]${}_{smooth}$ and in fact is a [[(2,1)-sheaf]]/[[stack]].

For $X$ a [[paracompact space|paracompact]] [[smooth manifold]], we may also regard it as a [[(2,1)-sheaf]] on [[CartSp]] in an evident way.

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

The groupoid $G Bund(X)$ of $G$-[[principal bundle]]s on $X$ is equivalent to the hom-groupoid

$$
  \mathbf{H}(X,\mathbf{B}G) \simeq G Bund(X)
$$

taken in the [[(2,1)-topos]] of [[(2,1)-sheaves]] on [[CartSp]]${}_{smooth}$.

=--

A detailed discussion of this is at [[Smooth∞Grpd]] in the section on Lie groups.

Now write $\mathfrak{g}$ for the [[Lie algebra]] of $G$. Then consider the functor

$$
  \mathbf{B} G_{conn}
  : U \mapsto
  [\mathbf{P}_1(U),\mathbf{B}G]
  =
  \left\{
    A \stackrel{g}{\to} (g^{-1} A g + g^{-1} d g)
    |
    A \in \Omega^1(U,\mathfrak{g})\,, g \in C^\infty(U,G)
  \right\}
$$

that sends a [[Cartesian space]] $U$ to the **[[groupoid of Lie-algebra valued 1-forms]]** over $U$.

There is an evident morphism of [[(2,1)-sheaves]]

$$
  \mathbf{B}G_{conn} \to \mathbf{B}G
$$

that forgets the 1-forms on each object $U$.

+-- {: .num_defn}
###### Definition
**(connection)**

A **connection** on a smooth $G$-[[principal bundle]] $g : X \to \mathbf{B}G$ is a lift $\nabla$ to $\mathbf{B}G_{conn}$

$$
  \array{
    && \mathbf{B}G_{conn}
    \\
    & {}^{\mathllap{\nabla}}\nearrow & \downarrow
    \\
    X &\stackrel{g}{\to}& \mathbf{B}G
  }
  \,.
$$

The **groupoid of $G$-principal bundles with connection** on $X$ is

$$
  G Bund_\nabla(X) := Hom(X,\mathbf{B}G_{conn})
  \,.
$$

=--

Explicitly, a morphism $g : X \to \mathbf{B}G$ is a nonabelian [[Cech cohomology]] [[cocycle]] on $X$ with values in $G$:

1. a choice of [[good open cover]] $\{U_i \to X\}$ of $X$;

1. a collection of [[smooth function]]s $(g_{i j} \in C^\infty(U_i \cap U_j), G)$

such that on $U_i \cap U_j \cap U_k$ the equation

* $g_{i j} g_{j k} = g_{i k}$

holds.

A lift $\nabla : X \to \mathbf{B}G_{conn}$ of this is in addition

1. a choice of [[Lie-algebra valued 1-form]]s
   $(A_i \in \Omega^1(U_i, \mathfrak{g}))$

such that on $U_i \cap U_j$ the equation

* $A_j = g^{-1} A_i g + g^{-1} d g$

holds, where on the right we have the pullback $g^* \theta$ of the [[Maurer-Cartan form]] on $G$ (see there).

## Properties
 {#Properties}

### Existence of connections
 {#ExistenceOfConnections}

+-- {: .num_defn}
###### Definition
**(existence of connections)**

Every $G$-principal bundle admits a connection. In other words, the forgetful functor

$$
  Hom(X, \bar \mathbf{B}G_{conn}) \to Hom(X,\mathbf{B}G)
$$

is an [[essentially surjective functor]].

=--

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

Choose a [[partition of unity]] $(\rho_i \in C^\infty(X,\mathbb{R}))$ subordinate to the [[good open cover]] $\{U_i \to X\}$ with respect to which a given cocycle $g : X \to \mathbf{B}G$ is expressed:

* $(x \;not\; in\; U_i) \Rightarrow \rho_i(x) = 0$;

* $\sum_i \rho_i = 1$.

Then set

$$
  A_i := \sum_{i_0} \rho_{i_0}|_{U_{i_0}} (g_{i_0 i}|^{-1}_{U_{i_0}}) d_{dR} (g_{i_0 i}|_{U_{i_0}})
  \,.
$$

By slight abuse of notation we shall write this and similar expressions simply as

$$
  A_i := \sum_{i_0} \rho_{i_0}(g_{i_0 i}^{-1} d_{dR} g_{i_0 i})
  \,.
$$

Using the that $(g_{i j})$ satisfies its cocycle condition, one checks that this satisfies the cocycle condition for the 1-forms:

$$
  \begin{aligned}
     A_j - g_{i j}^{-1} A_i g_{i j}  &=
     \sum_{i_0} \rho_{i_0} ( g_{i_0 j}^{-1} d g_{i_0 j} -
       ( g_{i_0 i} g_{i j}) ^{-1} (d g_{i_0 i}) g_{i j} )
     \\
     & =
     \sum_{i_0} \rho_{i_0} ( g_{i j}^{-1} d g_{i j} )
     \\
     & =
     g_{i j}^{-1} d g_{i j}
  \end{aligned}
  \,.
$$

=--

## Special cases

### Connections on the tangent bundle

Connections on [[tangent bundle]]s are also called [[affine connection]]s, or [[Levi-Civita connection]]s.

They play a central role for instance on [[Riemannian manifold]]s and [[pseudo-Riemannian metric|pseudo-Riemannian manifold]]s. From the end of the 19th century and the beginning of the 20th centure originates a language to talk about these in terms of [[Christoffel symbol]]s.

### Connections in physics

In [[physics]] connections on bundles model [[gauge field]]s.

* The [[electromagnetic field]] is a connection on a [[circle group]]-principal bundle;

* A [[Yang-Mills field]] more generally is a connection on a [[unitary group]]-principal bundle.

* The field of [[gravity]] is encoded in a connection on the [[orthogonal group]]-principal bundle to which the [[tangent bundle]] is [[associated bundle|associated]].

For more on this see [[higher category theory and physics]].

## Generalizations

### Superconnections

Generalizing the parallel transport definition from ordinary manifolds to [[supermanifold]]s yields the notion of [[superconnection]].

### Simons-Sullivan structured bundles

When the notion of connection on a principal bundle is slightly coarsened, i.e. when more connections are regarded as being ismorphic than usual, one arrives at a structure called a [[Simons-Sullivan structured bundle]]. This has the special property that for $G = U$  the [[unitary group]], the corresponding [[Grothendieck group]] of such bundles is a model for [[differential K-theory]].

### Connections on a principal $\infty$-bundle

See [[connection on a principal ∞-bundle]].

## Related concepts

* **connection on a bundle**, [[universal connection]]

  * [[principal connection]]

    * [[affine connection]], [[Levi-Civita connection]] [[Cartan connection]]

    * [[Chen connection]]

    * [[covariant derivative]]

    * [[equivariant connection]]

    * [[logarithmic connection]]

  * [[Chern connection]]

  * [[moduli space of connections]]

  * [[fiber bundles in physics]]

* [[connection on a 2-bundle]] / [[connection on a gerbe]] / [[connection on a bundle gerbe]]

* [[connection on a 3-bundle]] / [[connection on a bundle 2-gerbe]]

* [[connection on an ∞-bundle]]

* [[parallel transport]], [[higher parallel transport]]

* [[holonomy]], [[higher holonomy]]

  * [[holonomy group]]

  * [[special holonomy]]

  * [[nonabelian Stokes theorem]]

* [[fiber bundles in physics]]

[[!include higher Atiyah groupoid - table]]

[[!include gauge field - table]]

## References

The notion of [[Ehresmann connections]] originates with:

* [[Charles Ehresmann]], *Les connexions infinitésimales dans un espace fibré différentiable*,

   Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris (1951) 29–55

  [[Séminaire Bourbaki]], **1** 24 (1952) 153–168 [[numdam:SB_1948-1951__1__153_0](http://www.numdam.org/item/?id=SB_1948-1951__1__153_0), [pdf](http://www.numdam.org/article/SB_1948-1951__1__153_0.pdf)]

The understanding of [[gauge potentials]] in [[Yang-Mills theory]] as [[connection on a bundle|connections]] on [[fiber bundles]] originates with (see also at *[[fiber bundles in physics]]*):

* {#WuYang75} [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]

Textbook account:

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 9 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* [[Gerd Rudolph]], [[Matthias Schmidt]], Chapter 1 of: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

The formulation of connections in terms of their smooth [[parallel transport]] [[functors]] is in

* {#SchreiberWaldorf07} [[Urs Schreiber]], [[Konrad Waldorf]], _Parallel Transport and Functors_,  J. Homotopy Relat. Struct. 4, 187-244 (2009) ([arXiv:0705.0452](http://arxiv.org/abs/0705.0452))

based on a series of classical observations. I farily comprehensive commented list of related references is here:

* _differential cohomology in a cohesive topos -- References -- Connections -- In terms of parallel transport_

Basic facts about connections, such as the existence proof in terms of Cech cocycles, are collected in the brief lecture note

* [[Theodore Voronov]], _Differential Geometry, §3 Connection on a vector bundle_ ([pdf](http://www.maths.manchester.ac.uk/~tv/Teaching/Differential%20Geometry/2008-2009/lecture3.pdf))

Discussion of connection form (mostly on trivial bundles) in [[synthetic differential geometry]] includes the following

* {#Kock06} [[Anders Kock]], _Connections and path connections in groupoids_, 2006 ([web]( http://math.au.dk/publs?publid=619))

* {#Kock07} [[Anders Kock]], _Group valued differential forms revisited_, Feb 2007 ([web]( http://math.au.dk/publs?publid=636))

[[!redirects connections on a bundle]]
[[!redirects connections on bundles]]

[[!redirects connection on a principal bundle]]
[[!redirects connections on a principal bundle]]
[[!redirects connections on principal bundles]]

[[!redirects bundle with connection]]
[[!redirects bundle with connections]]
[[!redirects bundles with connection]]
[[!redirects bundles with connections]]

[[!redirects connection on a circle-bundle]]
[[!redirects connections on a circle-bundle]]
[[!redirects connections on circle-bundles]]

[[!redirects connection on a circle bundle]]
[[!redirects connections on a circle bundle]]
[[!redirects connections on circle bundles]]


connection on a bundle 2-gerbe

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#### Differential cohomology
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=--
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=--

* [[connection on a bundle]]

* [[connection on a 2-bundle]] / [[connection on a gerbe]] / [[connection on a bundle gerbe]]

* [[connection on a 3-bundle]] / **connection on a bundle 2-gerbe**

* [[connection on an ∞-bundle]]

***

## Idea

A connection of a [[bundle 2-gerbe]] is essentially a [[Cech cohomology|Cech]]-realization of a degree 4-cocycle in [[Deligne cohomology]].

## References

* [[Danny Stevenson]], _Bundle 2-gerbes_ ([arXiv:math/0106018](http://arxiv.org/abs/math/0106018))

[[!redirects connection on a bundle 2-gerbe]]


connection on a bundle gerbe

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#### Differential cohomology
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=--
=--
=--

* [[connection on a bundle]]

* [[connection on a 2-bundle]] / [[connection on a gerbe]] / **connection on a bundle gerbe**

* [[connection on a 3-bundle]] / [[connection on a bundle 2-gerbe]]

* [[connection on an ∞-bundle]]

***

#Contents#
* table of contents
{:toc}

## Idea

A _connection on a [[bundle gerbe]]_ is a slight variant of a [[Cech cohomology|Cech]]-realization of a degree 3 [[Deligne cohomology]] cocycle.

> old content, needs to be polished

Like a [[connection on a bundle|connection]] on a locally trivialized bundle is encoded in a  Lie algebra-valued connection $1$-form on $Y$, the connection on a bundle gerbe
gives rise to a Lie-algebra valued $2$-form on $Y$ (this shift in degree
is directly related to the step from second to third integral cohomology). This $2$-form
is sometimes addressed as the _curving $2$-form_ of a bundle gerbe.

But there is more data necessary to describe a connection on a bundle gerbe.
The details of the definition -- which is evident for line bundle gerbes but more
involved for principal bundle gerbes -- can be naturally derived from a functorial
concept of parallel surface transport, just like connection $1$-forms on bundles
can be derived from parallel line transport.

### Definitions

#### For line bundle gerbes

A connection (also known as "connection and curving") on a line bundle gerbe
$$
  B \stackrel{p}{\to} Y^{[2]} \stackrel{\to}{\to} Y \stackrel{\pi}{\to} X
$$
is

* a 2-form on $Y$
    $$
      B \in \Omega^2(Y)
    $$

* a connection $\nabla$ on the line bundle $B \to Y^{[2]}$

* such that
   $$
      \pi_1^*B \; -\;  p_2^*B \;=\; F_\nabla
   $$

* together with an extension of the bundle gerbe product $\mu$ to an isomorphism
    $$
        \mu_\nabla \;:\; p_{12}^* (B,\nabla) \;\; \otimes p_{23}^* (B,\nabla) \;\to\; p_{13}^* (B,\nabla)
    $$
    of line bundles with connection.

Notice that this condition ensures that $d B$ is a $3$-form on $Y$ which agrees on
double intersections
$$
  p_1^* d B \;\; = \;\; p_2^* d B
  \,.
$$
This means that $d B$ actually descends to a 3-form on $X$.

The __curvature__ associated with the connection on a line bundle gerbe
is the unique 3-form on $X$
$$
  H \in \Omega^3(X)
$$
such that
$$
  \pi^* H = d B
  \,.
$$

The deRham class $[H]$ of this 3-form is the image in real cohomology of the class in integral coholomology classifying the bundle gerbe.

#### For principal bundle gerbes

A connection on a $G$-principal bundle gerbe is

* a $\mathrm{Lie}(G)$-valued 2-form on $Y$
    $$
       B \in \Omega^2(Y,\mathrm{Lie}(G))
    $$

* together with a $\mathrm{Lie}(\mathrm{Aut}(G))$-valued 1-form on $Y$
    $$
       A \in \Omega^1(Y,\mathrm{Lie}(\mathrm{Aut}(G)))
    $$

* and a certain twisted notion of connection on the $G$-bundle $B$

* satisfying a couple of conditions that reduce to those described above in the case $G = U(1)$.

For the case that $F_{A} + \mathrm{ad} B = 0$, these conditions are nothing but
a tetrahedron law on a 2-functor from 2-paths in $Y$ to the category
$\Sigma(G\mathrm{BiTor})$. This is discussed in
[math.DG/0511710](http://arxiv.org/abs/math.DG/0511710).

For the more general case a choice for these conditions that harmonizes with the
conditions found for (proper) gerbes with connection by Breen & Messing in
[math.AG/0106083](http://arxiv.org/abs/math.AG/0106083) has
been given by Aschieri, Cantini & Jurčo in
[hep-th/0312154](http://arxiv.org/abs/hep-th/0312154).

### Surface transport

From a line bundle gerbe with connection one obtains a notion of [[parallel transport]] along surfaces in a way that generalizes the procedure for locally trivialized fiber bundles with connection.

Recall that in the case of fiber bundles, the holonomy associated to a based loop $\gamma$ is obtained by

* choosing a triangulation of the loop (i.e., a decomposition into intervals) such that each vertex sits in a double intersection $U_{ij}$ and such that each edge sits in a patch $U_i$

* choosing for each edge a lift into $Y = \sqcup_i U_i$

* choosing for each vertex a lift into $Y^{[2]} = \sqcup_{ij} U_i\cap U_j$

* assigning to each edge lifted to $U_i$ the transport computed from the connection
    1-form $a_i$

* assigning to each vertex  lifted to $U_i \cap U_j$ the value of the transition function
    $g_{ij}$ at that point

* multiplying these data in the order given by $\gamma$ .

For bundle gerbes this generalizes to a procedure that assigns a triangulation to a closed surface, that lifts faces, edges, and vertices to single, double and triple intersections,
respectively, and which assigns the exponentiated integrals of the 2-form over faces, of the connection 1-form over edges, and assigns the isomorphism $\mu_{ijk}$ to vertices.

For the abelian case (line bundle gerbes) this procedure has been first described in

* K. Gawedzki & N. Reis, _WZW branes and Gerbes_
([arXiv](http://arxiv.org/abs/hep-th/0205233))

based on

* O. Alvarez, _Topological quantization and cohomology_
Commun. Math. Phys. 100 (1985), 279-309.

Further discussion can be found in

* A. Carey, S. Johnson & M. Murray, _Holonomy on D-branes_, ([arXiv](http://arxiv.org/abs/hep-th/0204199))

Gawedzki and Reis showed this way that the Wess-Zumino term in the WZW-model is nothing but the surface holonomy of a (line bundle) gerbe.

In terms of string physics this means that the string (the $2$-particle) couples to the Kalb-Ramond field -- which hence has to be interpreted as the connection ("and curving") of a gerbe -- in a way that categorifies the coupling of the electromagnetically charged ($1$-)particle to a line bundle.

The necessity to interpret the Kalb--Ramond field as a connection on a gerbe was originally discussed in

* D. Freed and E. Witten
_Anomalies in string theory with D-branes_, Asian J. Math. 3 (1999), 819-851 ([arXiv](http://arxiv.org/abs/hep-th/9907189))

Underlying the Gawedzki--Reis formula is a general mechanism of transition of transport $2$-functors, described in

* [[Urs Schreiber]],  [[Konrad Waldorf]], _Connections on non-abelian gerbes and their holonomy_, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. ([TAC](http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html), [arXiv:0808.1923](http://arxiv.org/abs/0808.1923), web)

and similarly in

* Joao Faria Martins, Roger Picken, _A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces_ ([arXiv](http://arxiv.org/abs/0808.3964))

 This applies to more general situations than ordinary line bundle gerbes with connection.

The generalization ("[[Jandl gerbes]]") to unoriented surfaces (hence to type I strings)

* K. Waldorf, C. Schweigert & U. S., _Unoriented WZW Models and Holonomy of Bundle Gerbes_ ([arXiv](http://arxiv.org/abs/hep-th/0512283))

## Related concepts

* [[B-field]]

* [[Deligne cohomology]]

## Further references

(...)

In relation to [[ordinary differential cohomology]]:

* [[Byungdo Park]], *Differential cohomology and gerbes: An introduction to higher differential geometry*, lecture notes (2023) [[pdf](https://byungdo.github.io/seminars/IMSRS.pdf), [[Park-DifferentialCohomology.pdf:file]]]

Discussion of [[transgression]] and [[higher holonomy]] in [[Deligne cohomology]] (for [[bundle gerbes with connection]]) over [[orbifolds]]:

* [[Ernesto Lupercio]], [[Bernardo Uribe]], *Holonomy for Gerbes over Orbifolds*, J. Geom.Phys. **56** (2006) 1534-1560 [[arXiv:math/0307114](https://arxiv.org/abs/math/0307114), [doi:10.1016/j.geomphys.2005.08.006](https://doi.org/10.1016/j.geomphys.2005.08.006)]

[[!redirects connections on a bundle gerbes]]
[[!redirects connections on bundle gerbes]]

[[!redirects bundle gerbe with connection]]
[[!redirects bundle gerbes with connection]]


connection on a comodule

See [[connection for a coring]]


connection on a cubical set

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

## Idea

In a [[cubical set]], you are guaranteed for every $n$-cell (which can be drawn as a 1-cell)

$$a\stackrel{f}{\to}b$$

that there is the identity $(n+1)$-cell (which can be drawn as a 2-cell) of the form

$$\array{
a & \stackrel{f}{\to} & b \\
\darr^{Id} & \Downarrow^{Id} & \darr^{Id} \\
a & \stackrel{f}{\to} & b
}$$

A cubical set is said to have _connections_ if in addition it has for every $n$-cell $a\stackrel{f}{\to}b$ also $(n+1)$-cells of the form

$$\array{
a & \stackrel{f}{\to} & b \\
\darr^{f} & \Downarrow & \darr^{Id} \\
b & \stackrel{Id}{\to} & b
}$$

And so forth. You should think of this as saying that the "thin" cell

$$a\underoverset{f}{f}{\rightrightarrows}b$$

is regarded as a degenerate cube by the cubical set in all the possible ways.

So it's a very natural condition,  particularly if you think of all these cubical cells as cubical paths in some space.

## Symmetric and asymmetric connections

In the literature, two different definitions of connections
on a cubical set are considered.
The first definition treats edges of cubes as directed
from 0 to 1, whereas the second definition assumes them
to be completely symmetric.
Hence, we talk about symmetric or asymmetric connections.

We emphasize that ordering of edges has nothing to do
with the ordering of coordinates,
the latter is always present
and is analogous to the ordering of vertices in simplices.

In complete analogy to [[simplicial sets]],
cubical sets with symmetric or asymmetric connections
can be defined as presheaves on certain categories,
which happen to be [[strict test categories]].

## Definition of symmetric connections using generators and relations

If $K= \{K_n| n \geq 0\}$ is a [[cubical set]], then a **connection structure on $K$** consists of functions $\Gamma^+_i, \Gamma^- _i: K_n \to K_{n+1}$, $i=1, \ldots \, , n; n \geq 1$, satisfying the relations for $\alpha, \beta=\pm$:

1. $\Gamma^\alpha_i\Gamma^\beta_j= \Gamma^\beta_{j+1} \Gamma^\alpha _i$ if $i \lt j$;

1.  $\Gamma^\alpha_i\Gamma^\alpha_i= \Gamma^\alpha_{i+1} \Gamma^\alpha _i$;

1. $\partial^\alpha_j \Gamma^\alpha_j= \partial ^\alpha_{j+1} \Gamma ^\alpha_j = id$;

1. $\partial^\alpha_j \Gamma^{-\alpha}_j= \partial ^\alpha_{j+1} \Gamma ^{-\alpha}_j = \varepsilon _j \partial^\alpha_j$;

1. $ \partial^\alpha_i \Gamma ^\beta_j = \begin{cases}\Gamma^\beta_{j-1} \partial ^\alpha _i & \text{if }\; i \lt j \\    \Gamma^\beta_j \partial ^\alpha _{i-1} & \text{if }\;  i \gt j+1; \end{cases} $

1. $ \Gamma^\alpha_j \varepsilon_j = \varepsilon^2_j = \varepsilon_{j+1}\varepsilon _j$;

1. $\Gamma^\alpha_i \varepsilon _j = \begin{cases}
\varepsilon_{j+1} \Gamma^\alpha _i & \text {if }\;  i \lt j \\
\varepsilon _j \Gamma^\alpha_{i-1} & \text{if }\;  i \gt j ;
\end{cases}$

The connections are to be thought of as "extra degeneracies". A degenerate cube of type $\varepsilon_j x$ has opposite faces equal and all other faces degenerate.

A cube of
type  $ \Gamma_i^\alpha  x$  has a pair  of adjacent faces equal and
all other faces of type $\Gamma_j^\alpha  y$  or $\varepsilon_j y$ . So this makes the cubical theory nearer to the simplicial.
Cubical complexes with this, and other, structures  have  also
been considered by Evrard.

The first appearance of this notion in dimension $2$ was in the paper by Brown and Spencer listed below, and used to obtain an equivalence between crossed modules and edge symmetric double groupoids with connection.

Such connections on cubical sets were introduced in 1981 by Brown and Higgins in order to obtain the equivalence of their "cubical [[∞-groupoids]]" with [[crossed complexes]]. They are also essential to allow the notion of "commutative $n$-shell" in such a structure.

## Definition as presheaves on strict test categories

\begin{definition}
The category of cubes with symmetric connections is defined as follows.
We start with the [[monoidal category]]
whose objects are finite [[ordered sets]], denoted $(S,\le)$,
morphisms $(S,\le)\to(T,\le)$ are maps of sets $\{0,1\}^S\to\{0,1\}^T$,
and monoidal products are given by disjoint unions on objects
and cartesian products on morphisms.
The category of cubes with symmetric connection
is the monoidal subcategory of this monoidal category
generated by the following four types of morphisms:
all morphisms $(S,\le)\to(\emptyset,\le)$ and $(\emptyset,\le)\to(S,\le)$,
and morphisms $(S,\le)\to(1,\le)$ that compute the maximum or minimum
of an $S$-tuple of zeros and ones.
For cubes with asymmetric connection we take the maps
that compute the maximum, but not the minimum.
\end{definition}

\begin{definition}
The category of cubical sets with symmetric or asymmetric connections
is the category of [[presheaves of sets]]
on the category of cubes with symmetric or asymmetric connections.
\end{definition}

It would seem that...

Alternatively, let $\mathcal{P}_{\subset}$ be the category whose objects are [[finite set]]s with morphisms from $m$ to $n$ the functions $\mathcal{P}m \to \mathcal{P}n$ preserving $\subseteq$.  There is a faithful essentially-surjective functor $J : \box \to \mathcal{P}_{\subset}$ such that restriction of presheaves along $J$ is exactly "forget connection structure".

There is also a faithful representation of $\mathcal{P}_{\subset}$ in polyhedra-with-boundary and piecewise-linear maps, valued on objects $n\mapsto [0,1]^n$ with the coordinate-ordered simplicial subdivision of $[0,1]^n$.  This induces a connection structure on the singular cubical set of a topological space.

## Properties

### As a model for homotopy theory

The ordinary [[cube category]] is a _[[test category]]_. This means that bare [[cubical sets]] carry the structure of a [[category with weak equivalences]] whose [[homotopy category]] is that of [[∞-groupoids]].

But the category of cubes _with connection_  is even a _[[strict test category]]_ ([Maltsiniotis, 2008](#Maltsiniotis)). This means that under [[geometric realization]] (see the discussion at [[homotopy hypothesis]]) the [[cartesian product]] of cubical sets with connection is sent to the correct product [[homotopy type]].

The lack of this property for cubical sets without connections was one of the original reasons for abandoning Kan's initial cubical approach to combinatorial [[homotopy theory]] in favour of the simplicial approach; the implications of this new result have yet to be thought through. Another reason was that cubical groups were in general not Kan complexes; however cubical groups with connection are Kan complexes. See the paper by Tonks listed below.

## Examples

The prime example of  a  cubical   set  with  connections  is the
singular cubical complex  $KX$  of a [[topological space]] $X$.   Here for $n
\ge 0$ $K_n$ is  the  set  of singular $n$-cubes in $X$ (i.e.
continuous maps $I^n   \to   X$)  and the connection $
\Gamma_i^\alpha :K_{n } \to   K_{n+1}$  is induced by the map
$\gamma_i^\alpha   : I^{n+1} \to
 I^{n}$    defined by

   $$   \gamma _i^\alpha (t_1 ,t_2 ,\ldots \, ,t_{n+1} ) =
             (t_1 ,t_2 ,\ldots\, ,t_{i-1},A(t_i ,t_{i+1}),t_{i+2},\ldots \, ,t_{n+1} )
             $$

 where $A(s,t)=\max(s,t), \min(s,t)$ as $\alpha=-,+$ respectively.

The first hint of such a general structure came in the paper by Brown and Spencer given below. The term "[[connection]]" was used there because of a relation of a generalisation of this idea to [[connection on a bundle|path-connections]] in differential geometry. A principal $G$-bundle $E$ over $B$ gives rise to the Ehresmann  groupoid $Equ(E)$ of $G$-maps between the fibres, and the Moore paths $\Lambda$ on this form a double category $D$ with $Equ(E)$ and $\Lambda(B)$ as edge categories. A connection $\Gamma$ is then a functor from $\Lambda(B)$ to one of the category structures on $D$ which gives a smooth lifting of paths to transport of the fibres. This is the origin of the term [[transport law]] for the relation of connections to composition.

**Discussion**

There is a discussion of cubical vs simplicial singular homology and for other aspects at ([this mathoverflow] (http://mathoverflow.net/questions/3656/cubical-vs-simplicial-singular-homology)).

A complete cubical approach to algebraic topology at the border between homotopy and homology is given in the book discussed in these pages at [[Nonabelian Algebraic Topology]] see ([here] (https://ncatlab.org/nlab/show/Nonabelian+Algebraic+Topology)). As an example, the theory shows how the [[Relative Hurewicz Theorem]] follows from a Higher Homotopy Seifert-van Kampen Theorem, withut using singular homology.

## Model structure

The category of cubical sets with connections is a [[strict test category]]
and therefore admits a [[cartesian model structure]]
that is Quillen equivalent to the [[Kan–Quillen model structure]] on [[simplicial sets]].
This was proved by [Maltsiniotis](#Maltsiniotis).

See also the article [[model structure on cubical sets]].

## Joyal-type model structures

In complete analogy to [[simplicial sets]], there is also an analogue of the [[Joyal model structure]] on [[cubical sets]], with or without connection.  See the article [[model structures for cubical quasicategories]].

## Related concepts

* [[cubical set]]

* [[model structure on cubical sets]]

* [[model structures for cubical quasicategories]]

## References

* [[Ronnie Brown]] and C.B. Spencer,  "Double groupoids and crossed modules'', Cah.  Top. Géom. Diff. 17 (1976) 343--362.

* Evrard, M., "Homotopie des complexes simpliciaux et cubiques", Preprint(1976).

* Brown, R.  and  Higgins, P.J., "On the algebra of cubes", _J. Pure Appl.  Algebra_ 21 (1981) 233--260.

* F. Al-Agl, R. Brown and R. Steiner, "Multiple categories: the equivalence between a globular and cubical approach", _Advances in Mathematics_, 170 (2002), 71--118.

* M. Grandis and L. Mauri, "Cubical sets and their site",  _Theory Applic. Categories_, 11 (2003) 185--201.

*  P.J. Higgins,
"Thin elements and commutative shells in cubical  $\omega$-categories", _Theory Appl. Categ._ 14 (2005) 60--74.

The statement that cubical groups with connections are Kan complexes is due to

* A. Tonks,  "Cubical groups which are Kan", _J. Pure Appl. Algebra_, 81 (1992) 83--87.

Cubical sets with connection are used in the following paper:

* I. Patchkoria "Cubical approach to derived functors" Homology Homotopy Appl.
Volume 14, Number 1 (2012), 133-158.

and in the following in preference to simplicial methods to take resolutions in an analytic motivic setting:

* J. Ayoub - "L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle, I"  [pdf](http://user.math.uzh.ch/ayoub/PDF-Files/GaloisMotivic-1.pdf),

* A Vezzani - "A motivic version of the theorem of Fontaine and Wintenberger"  arXiv:1405.4548,

The statement that cubes with max-connections form a [[strict test category]] is due to

* [[Georges Maltsiniotis]], _La catégorie cubique avec connections est une catégorie test stricte_, HHA 11:2 (2009), 1--16. ([web](http://www.intlpress.com/HHA/v11/n2/a15/))
 {#Maltsiniotis}

based on

* [[Denis-Charles Cisinski]], _Les préfaisceaux comme modeèle des types d'homotopie_, Astérisque, 308 (2006)

The case of cubes with both max-connections and min-connections is analogous and was treated explicitly in Corollary 3 and Theorem 3 of

* Ulrik Buchholtz, Edward Morehouse, _Varieties of Cubical Sets_, [arXiv](https://arxiv.org/abs/1701.08189).

[[!redirects connection on cubical sets]]
[[!redirects connections on a cubical set]]
[[!redirects connections on cubical sets]]
[[!redirects cubical set with connections]]
[[!redirects cubical sets with connections]]


connection on a double category

Let $D=(D_1, D_0)$ be a [[double category]].  A **connection** on $D$ is given by a pair of functions $\Gamma, \Gamma' \colon \mathrm{arr} D_0 \to \mathrm{arr} D_1$, that assign to each vertical morphism $f \colon Y \to X$ cells of the form
$$
\begin{matrix}
  Y & \overset{f^*X}{\to} & X \\
  \mathllap{f} \downarrow & \mathllap{\Gamma f} \Downarrow &   \downarrow \mathrlap{1} \\
  X & \underset{\iota_X}{\to} & X
\end{matrix}
\qquad \qquad
\begin{matrix}
  Y & \overset{\iota_Y}{\to} & Y \\
  \mathllap{1} \downarrow & \Downarrow \mathrlap{\Gamma' f} & \downarrow \mathrlap{f} \\
  Y & \underset{f^*X}{\to} & X
\end{matrix}
$$
(where $\iota_X$ is the horizontal identity on $X$) such that $\Gamma$ and $\Gamma'$ behave suitably with respect to composition and identities in $D_0$:

...

(that is, they are identity-on-objects functors) and such that the vertical composite $\Gamma'(f) \cdot \Gamma(f) = \iota_f$ and the horizontal composite $\Gamma(f) \circ \Gamma'(f) = 1_{f^*X}$, (that is, $f$ and $f^*X$ are [[companions]]).


connection on a gerbe

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###Context###
#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#### Differential cohomology
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[[!include differential cohomology - contents]]
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* [[connection on a bundle]]

* [[connection on a 2-bundle]] / **connection on a gerbe** / [[connection on a bundle gerbe]]

* [[connection on a 3-bundle]] / [[connection on a bundle 2-gerbe]]

* [[connection on an ∞-bundle]]

***

#Contents#
* table of contents
{:toc}

## Idea

A [[connection on a 2-bundle]] that is presented in the form of a [[gerbe]].

## References

* [[Larry Breen]], [[William Messing]], _Differential geometry of gerbes_ ([arXiv:math/0106083](http://arxiv.org/abs/math/0106083))

* [[Larry Breen]], _Differential Geometry of Gerbes and Differential Forms_ ([arxiv:0802.1833](http://arxiv.org/abs/0802.1833))

[[!redirects connections on a gerbe]]