bibundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A _bibundle_ is a ([[groupoid-principal bundle|groupoid]]-)[[principal bundle]] which is equipped with a compatible second ([[groupoid action|groupoid]]-)[[action]] "from the other side".

In particular, Lie groupoid bibundles serve to exhibit "generalized morphisms"/[[Morita morphisms]] between [[Lie groupoids]]. This is in generalization of how the [[differentiable stack]]/[[smooth groupoid]] represented by a [[Lie groupoid]] is the [[moduli stack]] for [[groupoid-principal bundles]].

Therefore groupoid bibundles play a role in [[geometry]] analogous to the role played by [[bimodules]] in [[algebra]]. In this role they were originally introduced in ([Haefliger 84](#Haefliger), [Hilsum-Skandalis 87](#HilsumSkandalis87), [Pradines 89](#Pradines89)) and accordingly they are also called _Hilsum-Skandalis maps_. Independently they were seen in [[topos theory]] ([Bunge 90](#Bunge), [Moerdijk 91](#Moerdijk91)).
Historically, a central motivation for their study has been that the [[groupoid convolution algebra]] construction sends smooth bibundles between [[Lie groupoids]] to ([[Hilbert bimodule|Hilbert-]])[[bimodules]] of the corresponding [[C-star-algebra|C-star]] [[convolution algebras]], such that [[Morita equivalence]] is respected ([Muhly-Renault-Williams 87](#MuhlyRenaultWilliams), [Landsman 00](#Landsman), [Mrčun 05](#Mrcun)). This is of relevance notably for [[KK-theory]] of Lie groupoids ([Hilsum-Skandalis 87](#HilsumSkandalis87)).

Bibundles also appear as _transition bundles_ of [[nonabelian bundle gerbes]].

## Properties

### Lie groupoid bibundles and Morita/stack morphisms
 {#LieGroupoidBibundlesAndMoritaMorphisms}

We discuss how Lie groupoid bibundles correspond to [[Morita morphism]] (morphisms of [[differentiable stacks]]/[[smooth stacks]]) between the Lie groupoids.

First we set up the relevant definitions and establish our notation in

* [Lie groupoids and smooth stacks](#LieGroupoidsAndSmoothStacks)

Then we discuss smooth groupoid-principal bundles and how a Lie groupoid [[moduli stack]] for the bundles principal over it in

* [Smooth groupoid-principal bundles](#SmoothGrooupoidPrincipalBundles).

Finally we consider the corresponding smooth bibundles and how they correspond to their modulating stack morphisms in

* [Smooth groupoid-principal bibundles](#SmoothGroupoidPrincipalBibundles)

#### Lie groupoids and smooth stacks
 {#LieGroupoidsAndSmoothStacks}

A _[[smooth stack]]_ or _[[smooth groupoid]]_ is a [[stack]] on the site [[SmoothMfd]] of [[smooth manifolds]] or equivalently (and often more conveniently) on its [[dense subsite]] [[CartSp]] of just [[Cartesian spaces]] $\mathbb{R}^n, n \in \mathbb{N}$ and [[smooth functions]] between them, equipped with the standard [[coverage]] of [[good open covers]].

We write

$\;\;\; $[[SmoothGrpd]] $\coloneqq Sh_{(2,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, Grpd) $

for the [[(2,1)-category]] of [[stacks]] on this site, equivalently the result of taking [[groupoid]]-valued [[presheaves]] and then  [[simplicial localization|universally]] turning local (as seen by the [[coverage]]) [[equivalences of groupoids]] into global [[equivalence in an (infinity,1)-category]].

By generalizing here [[groupoids]] to general [[Kan complexes]] and [[equivalences of groupoids]] to [[homotopy equivalences]] of Kan complexes, one obtains _smooth [[∞-stacks]]_ or _[[smooth ∞-groupoids]]_, which we write

$\;\;\;$ [[Smooth∞Grpd]] $\coloneqq Sh_{(\infty,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, KanCplx) $.

We often write $\mathbf{H} \coloneqq$ [[Smooth∞Grpd]] for short.

By the [[(∞,1)-Yoneda lemma]] there is a sequence of [[full and faithful (∞,1)-functor|faithful inclusions]]

$\;\;\;$ [[SmoothMfd]] $\hookrightarrow$ [[SmoothGrpd]] $\hookrightarrow$ [[Smooth∞Grpd]].

This induces a corresponding inclusion of [[simplicial objects]] and hence of [[groupoid objects in an (∞,1)-category|groupoid objects]]

$$
  LieGrpd \hookrightarrow Grpd_\infty(SmoothMfd)  \hookrightarrow  Grpd_\infty(Smooth\infty Grpd)
  \,.
$$

For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object we write

$$
  \mathcal{G}_0 \to \mathcal{G} \coloneqq \underset{\longrightarrow}{\lim}_{n} \mathcal{G}_n
$$

for its [[(∞,1)-colimit|(∞,1)-colimiting]] [[cocone]], hence $\mathcal{G} \in \mathbf{H}$ (without subscript decoration) denotes the [[realization]] of $\mathcal{G}_\bullet$ as a single object in $\mathbf{H}$.

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

For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a [[groupoid object in an (infinity,1)-category|groupoid object]], we write $\mathcal{G} \coloneqq {\lim}_{n} \mathcal{G}_n \in \mathbf{H}$ for its **[[realization]]** and call the canonical [[1-epimorphism]]

$$
  \mathcal{G}_0 \to \mathcal{G}
$$

the canonical **[[atlas]]** of this realization.

=--

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

For $\mathcal{G}_\bullet \in Grpd(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd)$ a [[Lie groupoid]], we have that

1. $\mathcal{G}_0 \in SMoothMfd \hookrightarrow Smooth\infty Gprd$ is its [[smooth manifold]] of [[objects]]

1. $\mathcal{G} \in $ [[SmoothGrpd]] $\hookrightarrow$ [[Smooth∞Grpd]] is the realization of the Lie groupoid as a [[differentiable stack]], hence as a [[smooth groupoid]]

1. $\mathcal{G}_0 \to \mathcal{G}$ is the canonically induced [[atlas]] in the traditional sense of [[geometric stack]]-theory.

=--

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

By the [[Giraud-Rezk-Lurie axioms]] of the [[(∞,1)-topos]] $\mathbf{H}$ this morphism $\mathcal{G}_0 \to \mathcal{G}$ is a [[1-epimorphism]] and its construction establishes is an [[equivalence of (∞,1)-categories]] $Grpd_\infty(\mathbf{H}) \simeq \mathbf{H}^{\Delta^1}_{1epi}$, hence morphisms $\mathcal{G}_\bullet \to \mathcal{K}_\bullet$ in $Grpd_\infty(\mathbf{H})$ are equivalently [[diagrams]] in $\mathbf{H}$ of the form

$$
  \array{
    \mathcal{G}_0 &\to& \mathcal{K}_0
    \\
    \downarrow &\swArrow& \downarrow
    \\
    \mathcal{G} &\to& \mathcal{K}
  }
  \,.
$$

=--

This is evidently more constrained than just morphisms

$$
  \mathcal{G} \to \mathcal{K}$ in $\mathbf{H}
$$

by themselves. The latter are the _generalized morphisms_ or _[[Morita morphisms]]_ between the groupoid objects $\mathcal{G}_\bullet$, $\mathcal{K}_\bullet$.

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

Given [[groupoid objects in an (∞,1)-category|groupoid objects]] $\mathcal{G}_\bullet, \mathcal{K}_\bullet \in Grpd_\infty(\mathbf{H})$, a **[[Morita morphism]]** between them is a morphism $\mathcal{G} \to \mathcal{K}$ in $\mathbf{H}$ between their [[realizations]]. A Morita morphism that is an [[equivalence in an (infinity,1)-category|equivalence]] in $\mathbf{H}$ is called a **[[Morita equivalence]]** of groupoid objects in $\mathbf{H}$.

=--

Here we want to express these Morita morphisms $\mathcal{G} \to \mathcal{K}$ in terms of bibundle objects $\mathcal{P} \in \mathbf{H}$ on which both $\mathcal{G}_\bullet$ and $\mathcal{K}_\bullet$ [[action|act]].

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

For $X \in \mathbf{H}$ any [[object]], its [[pair groupoid]] $Pair(X)_\bullet \in Grpd_\infty(\mathbf{H})$ is

$$
  Pair(X)_n \coloneqq X^{\times^{n+1}}
  \,.
$$

The [[realization]] of this is equivalent to the point

$$
  Pair(X) \coloneqq \underset{\longrightarrow}{\lim}_n Pair(X)_n
  \simeq *
  \,.
$$

Hence all [[Morita morphisms]], def. \ref{MoritaMorphism}, to the pair groupoid are equivalent. As a groupoid object $Pair(X)_\bullet$ is non-trivial, but it is [[Morita equivalence|Morita equivalent]] to the terminal groupoid object.

=--

#### Smooth groupoid-principal bundles
 {#SmoothGrooupoidPrincipalBundles}

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

For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a [[groupoid object in an (∞,1)-category|groupoid object]], $X \in \mathbf{H}$ any object equipped with a morphism $a \colon X \to \mathcal{G}_0$ to the object of objects of $\mathcal{G}$, a **$\mathcal{G}_\bullet$-[[groupoid ∞-action]]** on $X$ with **anchor** $a$ is a [[groupoid object in an (∞,1)-category|groupoid]] $(X//\mathcal{G})_\bullet$ over $\mathcal{G}_\bullet$ of the form

$$
  \array{
    \vdots && && \vdots
    \\
    \downarrow \downarrow \downarrow \downarrow
    && &&
    \downarrow \downarrow \downarrow \downarrow
    \\
    X \underset{\mathcal{G}_0}{\times} \mathcal{G}_2
    && \to && \mathcal{G}_2
    \\
    \downarrow \downarrow \downarrow && &&
    \downarrow \downarrow \downarrow
    \\
    X \underset{\mathcal{G}_0}{\times} \mathcal{G}_1
    && \to && \mathcal{G}_1
    \\
    \downarrow \downarrow && && \downarrow \downarrow
    \\
    X  && \stackrel{a}{\to} && \mathcal{G}_0
  }
  \,,
$$

where the [[homotopy fiber products]] on the left are those of the anchor $a$ with the leftmost 0-face map $\mathcal{G}_{(\{0\} \hookrightarrow \{0, \cdots, n\})}$ and  the horizontal morphisms are the corresponding [[projections]] on the second factor.

We call $(X//\mathcal{G})_\bullet$ also the **[[action groupoid]]** of the action of $\mathcal{G}_\bullet$ on $(X,a)$ and call its [[realization]] $X \to (X//\mathcal{G})$ the [[homotopy quotient]] of the action.

=--

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

For $\mathcal{G}_\bullet = (\mathbf{B}G)_\bullet$ the [[delooping]] of a [[group object in an (∞,1)-category|group object]], def. \ref{GroupoidAction} reduces to the definition of an [[∞-action]] of the [[∞-group]] $G$.

=--

Under this relation, the discussion of [[∞-groupoid-principal ∞-bundles]] proceeds in direct analogy with that of $G$-[[principal ∞-bundles]]:

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

For $X \in $ [[Smooth∞Grpd]] any object, a morphism $f \colon X \to \mathcal{G}$ in $\mathbf{}H$ induces ("modulates") a $\mathcal{G}_\bullet$-[[groupoid action]], def. \ref{GroupoidAction}, on the [[homotopy pullback]] $f^\ast \mathcal{G}_0$

$$
  \array{
    f^* \mathcal{G}_0 &\to& \mathcal{G}_0
    \\
    \downarrow &pb_\infty& \downarrow
    \\
    X &\stackrel{f}{\to}& \mathcal{G}
    \,.
  }
$$

of the [[atlas]] of $\mathcal{G}$: the corresponding [[action groupoid]] is the [[Cech nerve]] of the [[projection]] $p \colon f^*\mathcal{G}_0 \to X$ (which as the [[(∞,1)-pullback]] of a [[1-epimorphism]] is itself a [[1-epimorphism]]):

$$
  \array{
     && \vdots && \vdots
     \\
     && \downarrow \downarrow \downarrow &&
     \downarrow \downarrow \downarrow
     \\
    && (f^\ast \mathcal{G}_0) \underset{\mathcal{G}_0}{\times} \mathcal{G}_1
    &\to&
    \mathcal{G}_1
    \\
    && \downarrow \downarrow && \downarrow \downarrow
    \\
    && f^* \mathcal{G}_0 &\stackrel{a}{\to}& \mathcal{G}_0
    \\
    && \downarrow &pb_\infty& \downarrow
    \\
    (f^\ast \mathcal{G}_0)//\mathcal{G} &\simeq& X &\stackrel{f}{\to}& \mathcal{G} &\simeq& \underset{\longrightarrow}{\lim}_n \mathcal{G}_n
    \,.
  }
$$

=--

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

Let $f_\bullet \colon X_\bullet \to \mathcal{G}_\bullet$ be a morphism of 1-groupoid objects, say of [[Lie groupoids]]. Then, as discussed at _[[homotopy pullback]]_, the [[(∞,1)-pullback]] of the [[atlas]] $\mathcal{G}_0 \to \mathcal{G}$ along the [[realization]] $f$ is computed as the 1-categorical [[pullback]]

$$
  \array{
   &\to& (\mathcal{G}_0 \underset{\mathcal{G}}{\times} \mathcal{G}^{\Delta^1})_\bullet
   \\
   \downarrow &pb& \downarrow
   \\
   \mathcal{X}_\bullet &\to& \mathcal{G}_\bullet
  }
$$

in $Sh(CartSp)^{\Delta^{op}}$. Schematically the groupoid on the right has morphisms $ \gamma_0 \to \gamma_1$ which are [[commuting diagrams]] in $\mathcal{G}$ of the form
$$
  \array{
    && g
    \\
    & {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}}
    \\
    g_0 && \to && g_1
  }
  \,.
$$

Therefore the pullback is the sheaf of groupoids which is schematically of the form

$$
  f^\ast \mathcal{G}_0
  \;\simeq\;
  \left\{
    \array{
      && g
      \\
      & {}^{\mathllap{\gamma_0}}\swarrow &&   \searrow^{\mathrlap{\gamma_1}} & && \in \mathcal{G}
      \\
      f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1)
      \\
      x_0 && \stackrel{\xi}{\to}&& x_1 && \in \mathcal{X}
    }
  \right\}
  \,.
$$

In this presentation now

$$
  \mathcal{G}_1
  \;\simeq\;
  \left\{
    \array{
      && g_0
      \\
      && \downarrow^{\gamma}
      \\
      && g_1
      \\
      & {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}}
      \\
      g_{1 0} && \to && g_{11}
    }
  \right\}
  \,.
$$

and the target map $\mathcal{G}_1 \to \mathcal{G}_0$ is given by forgetting the top vertical morphism in this diagram, while the source map is given by composing (!) the top vertical morphism with the two diagonal morphism.

Pullback of these two maps induces the left and right vertical map in

$$
  \array{
    f^\ast \mathcal{G}_0 \underset{\mathcal{G}_0}{\times}
    \mathcal{G}_1
    &\to& \mathcal{G}_1
    \\
    \downarrow\downarrow && \downarrow \downarrow
    \\
    f^\ast \mathcal{G}_0 &\stackrel{a}{\to}& \mathcal{G}_0
  }
  \,.
$$

from

$$
  f^\ast \mathcal{G}_0 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1
  \;\simeq\;
  \left\{
    \array{
      && g_0
      \\
      && \downarrow^{\gamma}
      \\
      && g_1 && && \in \mathcal{G}
      \\
      & {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} &
      \\
      f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1)
      \\
      x_0 && \stackrel{\xi}{\to} && x_1 & & \in \mathcal{X}
    }
  \right\}
  \,.
$$

The left one just forgets the top vertical morphism, the right one composes it with the diagonal morphisms. This composion is the $\mathcal{G}_\bullet$-action on $f^\ast \mathcal{G}_0$.

=--

#### Smooth groupoid-principal bibundles
 {#SmoothGroupoidPrincipalBibundles}

Finally then for $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ two Lie groupoids and $f \;\colon\; \mathcal{X} \to \mathcal{G}$ a morphism in [[Smooth∞Grpd]] between the corresponding [[differentiable stacks]], we obtain first the $\mathcal{G}$-[[groupoid principal bundle]] $f^* \mathcal{G}_0 \stackrel{p}{\to} \mathcal{X}$ and then by further homotopy pullback also the left $\mathcal{X}$-[[groupoid principal bundle]] $p^* \mathcal{X}_0$:

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

For $\mathcal{X}_\bullet, \mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ two [[groupoid object in an (infinity,1)-category|groupoid objects]] and $f \colon \mathcal{X} \to \mathcal{G}$ a [[Morita morphism]] between them, def. \ref{MoritaMorphism}, we say that the corresponding **$\mathcal{X}_\bullet-\mathcal{G}_\bullet$-bibundle** $\mathcal{P}(f)$ is the $\mathcal{G}_\bullet$-[[groupoid-principal bundle]] $f^\ast \mathcal{G}_0$ pulled back to the canonical [[atlas]] of $\mathcal{X}$ and equipped with the induced $\mathcal{X}_\bullet$-[[groupoid action]]:

$$
  \array{
    \mathcal{P}(f)& \coloneqq& p^* \mathcal{X}_0 &\to&
    f^* \mathcal{G}_0&\to& \mathcal{G}_0
    \\
    && \downarrow &\pb_\infty& \downarrow^{\mathrlap{p}} &pb_\infty& \downarrow
    \\
    && \mathcal{X}_0
    &\to&
    \mathcal{X}
    &\stackrel{f}{\to}&
    \mathcal{G}
  }
  \,.
$$

=--

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

Here the $\mathcal{G}_\bullet$-action on $\mathcal{P}(f)$ is principal over $\mathcal{X}_0$, in that the quotient map is

$$
  \mathcal{P}(f) \to \mathcal{P}(f)//\mathcal{G} \simeq \mathcal{X}_0
  \,,
$$

since $\mathcal{P}(f)$ is the pullback of a $\mathcal{G}_\bullet$-principal bundle (modulated by the bottom composite map in the above diagram).

On the other hand the $\mathcal{X}_\bullet$-action on $\mathcal{P}(f)$ is not principal over $\mathcal{G}_0$ -- unless $f$ is an [[equivalence in an (infinity,1)-category]] (hence a ([[Morita equivalence]]) from $\mathcal{X}_\bullet$ to $\mathcal{G}_\bullet$.) It is instead always principal over $f^\ast \mathcal{G}_0$.

=--

Thus we arrive at an equivalent, however more basic definition of Lie groupoid bibundle:
+-- {: .num_defn }
###### Definition

Given Lie groupoids $G:=G_1\Rightarrow G_0$ and $H:=H_1\Rightarrow H_0$, a $G$-$H$-**bibundle** is a principal $H$-bundle $E \xrightarrow{\pi_G} G_0$ over $G_0$ with anchor $E\xrightarrow{\pi_H} H_0$ together with a left $G$-action (see [here](http://ncatlab.org/nlab/show/groupoid+principal+bundle#lie_groupoid_principal_bundles) ) with anchor $\pi_G$, such that the two actions commute. If the $G$-action also gives rise to a principal bundle over $H_0$, then $E$ induces a [[Morita equivalence]] between $G$ and $H$ and it is sometimes called a **Morita bibundle** in this case.
=--

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

Given a manifold $M$, and two open covers $\{U_i\}$ and $\{V_\alpha\}$, we may form two Cech groupoids (see [here](http://ncatlab.org/nlab/show/Lie+groupoid#examples_for_lie_groupoids) ) $\sqcup U_{ij} \Rightarrow \sqcup U_i$ and $\sqcup V_{\alpha \beta} \Rightarrow \sqcup V_\alpha$. Then $\sqcup_{i, \alpha} U_i \times_{M} V_\alpha
$ (which is a common refinement of $\{U_i\}$ and $\{V_\alpha\}$) is a Morita bibundle. The actions are

$$(x_i, x_j)\cdot (x_j, x_\alpha)=(x_i, x_\alpha), \quad (x_i, x_\alpha)\cdot (x_\alpha, x_\beta)=(x_i, x_\beta). $$

Obviously these actions are free. Moreover, it is also not hard to see that $\sqcup U_i\times_M V_\alpha/\sqcup V_{\alpha \beta} = \sqcup U_i$ and $\sqcup U_i\times_M V_\alpha/\sqcup U_{ij} = \sqcup V_\alpha$. When a free action has representible quotient, it must automatically be proper.

=--

##### Composition

Given a bibundle functor $E: G\to H$ and a bibundle functor $F: H\to K$ between Lie groupoids, the composition $E\circ F: G\to K$ is the quotient manifold $E\times_{H_0} F/H_1$ equipped with the remaining $G$ and $K$ action. Here $H$ acts on $E\times_{H_0} F$ from right by $(x, y)\cdot h_1=(x\cdot h_1^{-1}, y \cdot h_1)$. It is free and proper because the right action of $H$  on $E$ is so. Then $G$ action and $K$ action descend to the quotient $E\times_{H_0} F/H_1$. Moreover, those who free and proper is (are), remains so.

Thus bibundle functors compose to a bibundle functor, and Morita bibundles compose to a Morita bibundle.

Then we see that  there is a $(2,1)$-category $BUN$ with objects Lie groupoids, 1-morphisms bibundle functors, and 2-morphisms isomorphisms of bibundles. It is $(2,1)$-category because 2-morphisms are obviously invertible. This $(2,1)$-category is equivalent to the one obtained by generalised morphism or by anafunctors.

##### Bundlisation #####

Given a strict morphism $G\xrightarrow{f} H$, then we may form a bibundle $E:= G_0\times_{f_0, H_0, t} H_1$ with right $H$ action induced by $H$-multiplication and with left $G$ action induced by $G$-action on $G_0$. Bundlisation preserves composition.

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

If both [[atlases]] are [[0-truncated]] objects ([[smooth spaces]]) $\mathcal{X}_0, \mathcal{G}_0 \in Sh(CartSp) \simeq \tau_1 \mathbf{H} \hookrightarrow \mathbf{H}$, then by the [[pasting law]] for [[homotopy pullbacks]] we have that $\mathcal{P}(f)$ is [[n-truncated|(n-1)-truncated]] if $\mathcal{G}$ is [[n-truncated]].

In particular therefore the total space of a [[smooth groupoid|smooth 1-groupoid]] bibundle is 0-truncated hence is a [[smooth space]].

=--

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

In order to discuss Lie-groupoid bibundles we continue the discussion in  example \ref{LieGroupoidPrincipalBundle} of Lie-groupoid principal bundles. Proceeding for the second homotopy pullback diagram as discussed there for the first one, one finds that the total space $\mathcal{P}$ of the bibundle is presented by the sheaf of groupoids whose schematic depiction is

$$
  f^\ast \mathcal{G}_0
  \;\simeq\;
  \left\{
    \array{
      && g
      \\
      & {}^{\mathllap{\gamma_0}}\swarrow &&   \searrow^{\mathrlap{\gamma_1}} & && \in \mathcal{G}
      \\
      f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1)
      \\
      x_0 && \stackrel{\xi}{\to}&& x_1
      \\
      & \searrow && \swarrow &  && \in \mathcal{X}
      \\
      && x
    }
  \right\}
  \,.
$$

Here the vertically-running morphisms are the [[objects]] and two such are related by a [[morphism]] if they fit into a commuting diagram complete by horizontal morphisms as indicated. Since $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ both are groupoids, these morphisms are unique if they exist, and hence, as predicted by remark \ref{TruncationOfBibundle}, $\mathcal{P}(f)$ is 0-truncated, hence is a [[smooth space]]. Moreover, since the [[isomorphism]] [[equivalence relation]] here is free, the [[quotient]] [[smooth space]] is actually a [[smooth manifold]] (since $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ are [[Lie groupoids]]).

This then recovers the definition of bibundles for Lie groupoids as often found in the literature.

The right $\mathcal{G}_\bullet$-action is by precomposition of these diagram with morphisms in $\mathcal{G}$, while the left $\mathcal{X}$-action is by postcomposition with morphisms in $\mathcal{X}$.

=--

Conversely, given a $\mathcal{X}_\bullet$-$\mathcal{G}_\bullet$-Lie groupoid bibundle which is principal on the left

$$
  \array{
    \mathcal{P} &\to& \mathcal{B} &\to& \mathcal{G}_0
    \\
    \downarrow && && \downarrow
    \\
    \mathcal{X}_0 &\to& \mathcal{X} && \mathcal{G}
  }
$$

we recover the Morita morphism $f$ that it coresponds to by the [[Giraud-Rezk-Lurie axioms]]: first $p$ is the induced map between the [[homotopy colimits]] of the Cech nerves of the two left horizontal maps

$$
  \array{
    \mathcal{P} &\to& \mathcal{B} &\to& \mathcal{G}_0
    \\
    \downarrow && \downarrow^{\mathrlap{p}} && \downarrow
    \\
    \mathcal{X}_0 &\to& \mathcal{X} && \mathcal{G}
  }
$$

and then $f$ is similarly the map between the homotopy colimits of the Cech nerves of the two right vertical maps.

(...)

### Relation to groupoid convolution bimodules
 {#RelationToGroupoidConvolutionBimodules}

There should be a [[2-functor]] from [[Lie groupoids]] to [[C-star-algebras]] and [[Hilbert C-star-bimodules]] between them given by forming [[groupoid convolution algebras]] and naturally exhibited by Lie groupoid bibundles: the [[groupoid convolution algebra]] of the total space of the bibindle becomes a [[bimodule]] over the two other groupoid convolution algebras.

Some aspects of this are in the literature, e.g. ([Mrčun 99](#Mrcun99)) for [[étale Lie groupoids]] and ([Landsman 00](#Landsman00)) for general [[Lie groupoids]]. The follwing is taken from the latter article.

+-- {: .num_defn #BundleOfVertical}
###### Definition/Notation

For $p \colon E \to X$ a [[smooth function]] between smooth manifolds, we write $T^p E \hookrightarrow T E$ for the bundle of [[vertical vector fields]], the sub-bundle of the [[tangent bundle]] of $E$ on those vectors in the [[kernel]] of the [[differentiation]] maps $d p|_{e} \colon T_e E \to T_{\tau(e)} X$.

We write ${\vert \Lambda\vert^{1/2}}(T^\tau E)$ for the bundle of [[half-densities]] on vertical vector fields.

=--

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

Let $\mathcal{G}_\bullet$ be a [[Lie groupoid]] and let ($E \stackrel{\tau}{\to} \mathcal{G}_0, \rho)$ be a $\mathbb{G}_\bullet$-[[groupoid-principal bundle]] $E \to E//\mathcal{G}$ (with anchor $\tau$ and action map $\rho$).

Then the bundle of [[vertical vector fields]] $T^\tau E$ equipped with the anchor map $T^\tau E \stackrel{d \tau}{\to} T \mathcal{G}_0 \to \mathcal{G}_0$ inherits a canonical $\mathcal{G}_\bullet$-action itself.

The [[quotient]] map

$$
  {\vert \Lambda\vert^{1/2}}(T^\tau E)/\mathcal{G} \to E/\mathcal{G}
$$

exists and is naturall a vector bundle again.

=--

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

In the situation of remark \ref{ActionOnVerticalVectors}, write

* $C^\infty_{c/G}(E, {\vert\Lambda\vert}^{1/2}(T^\tau) E)^G$

  for the space of [[smooth function|smooth]] [[sections]] of the [[half-density]]-bundle of $T^\tau E$ which are $\mathcal{G}$-equivariant and which have [[compact support]] up to $\mathcal{G}$-action;

* $C^\infty_c(E/\mathcal{G}, {\vert \Lambda\vert}^{1/2}(T^\tau E))$

  for the space of smooth sections with compact support of the quotient bundle.

=--

The following constructions work by repeatedly applying the following identification:

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

In the situation of def. \ref{TwoVersionsOfHalfDensities}, there is a [[natural isomorphism]]

$$
  C^\infty_{c/G}(E, {\vert\Lambda\vert}^{1/2}(T^\tau) E)^G
  \simeq
  C^\infty_c(E/\mathcal{G}, {\vert \Lambda\vert}^{1/2}(T^\tau E)  )
  \,.
$$

=--

The central definition here is now:

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

For $(E_1, \tau_1)$, $(E_2, \tau_2)$ two principal $\mathcal{G}_\bullet$ manifolds, set

$$
  (E_1, E_2)_{\mathcal{G}}
  \coloneqq
  C^\infty_c( E_1 \underset{\mathcal{G}_0}{\times}) E_2, {\vert\Lambda\vert^{1/2}(T^\tau E_1)} \otimes  {\vert\Lambda\vert^{1/2}(T^\tau E_2)}
$$

=--

And the central fact is:

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

Given 3 $\mathcal{G}_\bullet$-manifolds $(E_i, \tau_i)$, $i \in \{1,2,3\}$, there is a  [[smooth function]]

$$
  \star
  \;\colon\;
  (E_1, E_2)_{\mathcal{G}}
  \times
  (E_2, E_3)_{\mathcal{G}}
  \to
  (E_1, E_3)_{\mathcal{G}}
$$

given on sections $\sigma_1, \sigma_2$ and points $(e_1, e_3)$ by

$$
  \sigma_1 \star \sigma_2
  \colon
  (e_1, e_3)
  \mapsto
  \int_{\tau_2^{-1}(\tau_1 e_1)}
  \sigma_1(e_1, -)  \otimes \sigma_2(-,e_3)
  \,,
$$

where the [[integration]] is against the [[measure]] that appears by tensoring two (of the four) [[half-densities]] in the integrand.

This operation is an associative and invoutive partial composition operation and hence defines a [[star-category]] whose [[objects]] are $\mathcal{G}_\bullet$-principal manifolds and whose spaces of morphisms are as in def. \ref{BisectionsOnFiberProductOfTwoGActionSpaces}.

=--

In particular one has the following identifications.

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

For $\mathcal{G}_1 \to \mathcal{G}_0$ regarded as a $\mathcal{G}_\bullet$-principal action space, there is a [[natural isomorphism]]

$$
  (\mathcal{G}_1, \mathcal{G}_1)_{\mathcal{G}}
  \simeq
  C^\infty_c(\mathcal{G}_1, {\vert\Lambda\vert}^{1/2}(T^s \mathcal{G}_1) \otimes {\vert\Lambda\vert}^{1/2}(T^t \mathcal{G}_1))
$$

and the algebra structure on this by prop. \ref{TheBimdoduleConvolutionProduct} is isomorphic to the [[groupoid convolution algebra]] of smooth sections over $\mathcal{G}_\bullet$.

=--

More generally:

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

For $E \stackrel{\tau}{\to} \mathcal{G}_0$ any $\mathcal{G}$-principal manifold, we have a [[natural isomorphism]]

$$
  (\mathcal{G}_1, E)_{\mathcal{G}}
  \simeq
  C^\infty_c(E_1, {\vert\Lambda\vert}^{1/2}(T^G E) \otimes {\vert\Lambda\vert}^{1/2}(T\tau E))
  \,.
$$

=--

> We consider [[completion]] of all this to the [[C-star-algebra]] context (...)

Now we can put the pieces together and sends groupoid-bindunles to $C^\ast$-bimodules over the two [[groupoid convolution algebras]].

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

Given two [[Lie groupoids]] $\mathcal{G}_\bullet$ and $\mathcal{K}_\bullet$ and given a [[Morita equivalence]] groupoid [[bibundle]] $E$ between them, we have

$$
  N \coloneqq (\mathcal{G}_1, E)_{\mathcal{G}} \simeq (E, \mathcal{K})_{\mathcal{K}}
$$

and this identification makes $N$ into a $C^\ast(\mathcal{G}_\bullet)-C^\ast(\mathcal{K}_\bullet)$-pre-[[Hilbert bimodule]] as follows:

1. The identification $N \simeq (E, \mathcal{K}_1)_{\mathcal{K}}$ defines the right $C^\ast(\mathcal{K}_\bullet)$-action by example \ref{GroupoidConvolutionAlgebraFromBibundleCase}; and similarly the identification $N \simeq (\mathcal{G}_1, E)_{\mathcal{G}}$ defines a left $C^\ast(\mathcal{G}_\bullet)$-action.

1. The $C^\ast(\mathcal{K})$-valued [[inner product]] on $N$ is that induced by the composite

   $$
     (E,\mathcal{K}_1)_{\mathcal{K}}^\ast
     \times
     (E,\mathcal{K}_1)_{\mathcal{K}}
     \stackrel{\simeq}{\to}
     (\mathcal{K}_1)_{\mathcal{K}, E}
     \times
     (E,\mathcal{K}_1)_{\mathcal{K}}
     \to
     (\mathcal{K}_1, \mathcal{K}_1)_{\mathcal{K}}
     \hookrightarrow
     C^\ast(\mathcal{K}_\bullet)
     \,.
   $$

=--

## References
 {#References}

### General

Groupoid bibundles were first considered for [[foliation groupoids]] in

* {#HilsumSkandalis87} [[Michel Hilsum]], [[Georges Skandalis]], _Morphismes K-orientés d'espaces de feuilles et functorialité en théorie de Kasparov_. Ann. Scient. Ec. Norm. Sup. 20 (1987), 325–390. ([numdam:ASENS_1987_4_20_3_325_0](http://www.numdam.org/item?id=ASENS_1987_4_20_3_325_0))

The generalization to arbitrary [[topological groupoids]] was considered in

* {#Haefliger84} [[André Haefliger]], _Groupoïdes d'holonomie et classifiants_, Astérisque 116 (1984), 70–97.

* {#Pradines89} [[Jean Pradines]], _Morphisms between spaces of leaves viewed as fractions_. Cahiers Top. Géom. Diff. Cat. XXX-3 (1989), 229–246 ([numdam:CTGDC_1989__30_3_229_0](http://www.numdam.org/item/?id=CTGDC_1989__30_3_229_0))

and independently in [[topos theory]] in

* {#Bunge90} [[Marta Bunge]], _An application of descent to a classification theorem. Math. Proc. Cambridge Phil. Soc. 107 (1990), 59–79.

* {#Moerdijk91} [[Ieke Moerdijk]], _Classifying toposes and foliations_. Ann. Inst. Fourier, Grenoble 41, 1 (1991), 189–209.

Groupoid bibundles are used in the context of [[groupoid convolution algebras]] as geometric analogs of [[bimodules]] in

* {#MuhlyRenaultWilliams} [[Paul Muhly]], [[Jean Renault]], and D. Williams, _Equivalence and isomorphism for groupoid $C^\ast$-algebras_, J. Operator Th. 17 (1987), 3–22.

* [[Klaas Landsman]], _The Muhly-Renault-Williams theorem for Lie groupoids and its classical counterpart_, Lett. Math. Phys. 54 (2000), no. 1, 43–59. ([arXiv:math-ph/0008005](http://arxiv.org/abs/math-ph/0008005))
 {#Landsman}
 {#Landsman00}

* {#Mrcun05} [[Janez Mrcun]], _Stability and invariants of Hilsum-Skandalis maps_ ([arXiv:math/0506484](http://arxiv.org/abs/math/0506484))

A review of [[Lie groupoid]]-bibundles and maps of [[differentiable stacks]] is in section 2 of

* {#Blohmann} [[Christian Blohmann]], _Stacky Lie groups_, Int. Mat. Res. Not. (2008) Vol. 2008: article ID rnn082 ([arXiv:math/0702399](http://arxiv.org/abs/math/0702399))

Discussion of [[Lie group cohomology]] and the [[string 2-group]] [[infinity-group extension]] in terms of Lie groupoid bibundles is in

* [[Chris Schommer-Pries]] _Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group_ [arXiv:0911.2483](http://arxiv.org/abs/0911.2483/)
 {#Sp}

Talk notes on bibundles include

* [[Michael Murray]], _Bispaces and bibundles_ ([pdf slides](http://www.mpim-bonn.mpg.de/digitalAssets/7050_Murray.pdf))

* [[Jeffrey Morton]], ([pdf slides](http://www.math.uwo.ca/~jmorton9/seminar/gpdrep.pdf))

See also

* [[Michael Murray]], [[David Roberts]], [[Danny Stevenson]], _On the existence of bibundles_ LMS (2012)([arXiv:1102.4388](http://arxiv.org/abs/1102.4388))

### Convolution to $C^\ast$-bimodules

For groupoid bibundles between [[étale Lie groupoids]] the assignment of the [[groupoid convolution algebra]]-[[bimodule]] to them is shown to be [[functor|functorial]] in

* {#Mrcun99} [[Janez Mrcun]], _Functoriality of the bimodule associated to a Hilsum-Skandalis map_. K-Theory 18 (1999) 235–253.

For more references along these lines see for the moment at [groupoid convolution algebra -- Extension to bibundles and bimodules](http://ncatlab.org/nlab/show/category+algebra#ExtensionToBibundlesAndBimodules)

[[!redirects bibundle]]
[[!redirects bibundles]]

[[!redirects groupoid bibundle]]
[[!redirects groupoid bibundles]]

[[!redirects Hilsum-Skandalis morphism]]
[[!redirects Hilsum-Skandalis morphisms]]

[[!redirects Hilsum-Skandalis map]]
[[!redirects Hilsum-Skandalis maps]]

[[!redirects ∞-groupoid bibundle]]
[[!redirects ∞-groupoid bibundles]]
[[!redirects infinity-groupoid bibundle]]
[[!redirects infinity-groupoid bibundles]]


bicartesian category

A [[category]] is __bicartesian__ if it is both cartesian and cocartesian, that is if both it and its [[opposite category|opposite]] may be made into [[cartesian monoidal categories]].
Note this is different than a [[semiadditive category]], i.e. a category with *coinciding* products and coproducts. Thus every semiadditive category is bicartesian, but not every bicartesian category is semiadditive.

A bicartesian category which is also [[cartesian closed category|cartesian closed]] is a __[[bicartesian closed category]]__. Bicartesian closed categories are usually *not* [[cocartesian closed category|cocartesian closed]].

## See also

* [[bicartesian preordered object]]

[[!redirects bicartesian]]
[[!redirects bicartesian categories]]


bicartesian closed category

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

A __bicartesian closed category__ is a [[cartesian closed category]] with finite [[coproducts]]. In the case where this is furthermore a [[preorder]] or [[poset]], it is called a [[Heyting prealgebra]] or [[Heyting algebra]], respectively. They provide the semantics and proof theory of [[intuitionistic logic|intuitionistic]] [[propositional logic]].

Note that a bicartesian closed category is [[bicartesian category|bicartesian]] (that is, it is both [[cartesian monoidal category|cartesian]] and [[cocartesian monoidal category|cocartesian]]), and furthermore it is cartesian closed, but it is usually *not* [[cocartesian closed category|cocartesian closed]] (as the only such category is the trivial terminal category), nor co-(cartesian closed) (i.e., the dual of a cartesian closed category; aka, cocartesian coclosed). Thus the terminology could be confusing, but since the only categories which are both cartesian closed and co-(cartesian closed) are preorders, there is not much danger.

Also note that a bicartesian closed category is automatically a [[distributive category]]. This follows since the functors $X\mapsto A\times X$ have right adjoints (by closedness), so they preserve colimits.

A bicartesian closed category is one kind of [[2-rig]].

## See also

* [[bicartesian closed preordered object]]


bicartesian closed preordered object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The notion of a *bicartesian closed preordered object* or *Heyting prealgebra object* is the generalization of that of *[[bicartesian closed preordered set]]* or *[[Heyting prealgebra]]* as one passes from the ambient [[category of sets]] [[internalization|into]] more general ambient  [[categories]] with suitable properties.

## Definition

In a [[finitely complete category]] $C$, a **bicartesian closed preordered object** or **Heyting prealgebra object** is a [[bicartesian preordered object]]
$$(X, R, s, t, \rho, \tau_p \wedge, \top, \tau, \lambda_l, \lambda_r, \vee, \bot, \beta, \kappa_l, \kappa_r)$$
with a morphism $(-)\Rightarrow(-):X \times X \to X$ and functions
$$\epsilon_l:((* \to X) \times (* \to X)) \to (* \to R)$$
$$\epsilon_r:((* \to X) \times (* \to X)) \to (* \to R)$$
such that for all global elements $a:* \to X$ and $b:* \to X$,

* $s \circ \epsilon_l(a, b) = (a \Rightarrow b) \wedge a$
* $t \circ \epsilon_l(a, b) = b$
* $s \circ \epsilon_r(a, b) = a$
* $t \circ \epsilon_r(a, b) = b \Rightarrow (a \wedge b)$

## See also

* [[Heyting prealgebra]]

* [[bicartesian preordered object]]

* [[bicartesian closed category]]

* [[Boolean prealgebra object]]

* [[Heyting algebra object]]

[[!redirects bicartesian closed preordered objects]]
[[!redirects Heyting prealgebra object]]
[[!redirects Heyting prealgebra objects]]


bicartesian preordered object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

In a [[finitely complete category]] $C$, a **bicartesian preordered object** or **prelattice object** is a [[preordered object]] $X$ that is both a [[cartesian monoidal preordered object]] and a [[cocartesian monoidal preordered object]].

## See also

* [[cartesian monoidal preordered object]]

* [[cocartesian monoidal preordered object]]

* [[bicartesian closed preordered object]]

* [[lattice object]]

* [[bicartesian category]]

[[!redirects bicartesian preordered objects]]
[[!redirects prelattice object]]
[[!redirects prelattice objects]]


bicategorical trace

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of [[trace]] in a [[monoidal category]] to a [[bicategory]] context, based on equipping the bicategory with a "shadow" or categorified trace (including the [[trace of a category]]).

## Examples

The following traditional notions of "traces" turn out to be special cases of bicategorical traces

* All symmetric monoidal traces, such as the [[Lefschetz number]].

* The [[Reidemeister trace]].

## Related concepts

* [[trace]]

* [[higher trace]]

* [[span trace]]

## References

* [[Kate Ponto]], [[Michael Shulman]], _Shadows and traces in bicategories_ ([arXiv:0910.1306](https://arxiv.org/abs/0910.1306))

A survey is in

* [[Mike Shulman]], _Traces in monoidal categories and bicategories_ (2008) ([pdf slides](http://home.sandiego.edu/~shulman/papers/ccrtraces.pdf))

Relating the [[bicategorical trace]] to the [[Dennis trace]] on [[algebraic K-theory]]:

* {#CLMPZ20} [[Jonathan Campbell]], [[John Lind]], [[Cary Malkiewich]], [[Kate Ponto]], [[Inna Zakharevich]], _K-theory of endomorphisms, the TR-trace, and zeta functions_ [[arXiv:2005.04334](https://arxiv.org/abs/2005.04334)]

[[!redirects 2-bicategorical traces]]

[[!redirects 2-categorical trace]]
[[!redirects 2-categorical traces]]


bicategory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

# Bicategories
* table of contents
{:toc}

## Idea

A **bicategory** is a particular [[algebraic definition of higher category|algebraic]] notion of _weak [[2-category]]_ (in fact, the earliest to be formulated, and still the one in most common use).  The idea is that a bicategory is a category _[[weak enrichment|weakly enriched]]_ over [[Cat]]: the [[hom-objects]] of a bicategory are [[hom-category|hom-categories]], but the associativity and unity laws of [[enriched category|enriched categories]] hold only up to coherent isomorphism.

For information on [[morphisms]] of bicategories, see [[pseudofunctor]].

## Definition

A **bicategory** $B$ consists of

*  A [[collection]] of __[[objects]]__ $x,y,z,\dots$, also called __$0$-cells__;
*  For each pair of $0$-cells $x,y$, a [[category]] $B(x,y)$, whose objects are called __[[morphisms]]__ or __$1$-cells__ and whose morphisms are called __[[2-morphisms]]__ or __$2$-cells__;
*  For each $0$-cell $x$, a distinguished $1$-cell $1_x\in B(x,x)$ called the __[[identity morphism]]__ or __identity $1$-cell__ at $x$;
*  For each triple of $0$-cells $x,y,z$, a functor ${\circ}\colon B(y,z)\times B(x,y) \to B(x,z)$ called __[[horizontal composition]]__;
*  For each pair of $0$-cells $x,y$, [[natural isomorphisms]] called __[[unitors]]__: $\left(
\begin{array}{rcl}
f&\mapsto&f \circ 1_x\\
\theta&\mapsto&\theta \circ 1_{1_x}
\end{array}
\right)  \cong id_{B(x,y)} \cong \left(
\begin{array}{rcl}
f&\mapsto&1_y\circ f\\
\theta&\mapsto&1_{1_y} \circ \theta
\end{array}
\right):B(x,y)\rightarrow B(x,y)$
*  For each quadruple of $0$-cells $w,x,y,z$, a natural isomorphism called the __[[associator]]__ between the two functors from $B(y,z) \times B(x,y) \times B(w,x)$ to $B(w,z)$ built out of ${\circ}$

such that

*  The [[pentagon identity]] is satisfied by the [[associators]];
*  And the triangle identity is satisfied by the [[unitors]].

If there is exactly one $0$-cell, say $*$, then the definition is exactly the same as a monoidal structure on the category $B(*,*)$.  This is one of the motivating examples behind the [[delooping hypothesis]] and the general notion of [[k-tuply monoidal n-category]].

### Details
{#detailedDefn}

Here we spell out the above definition in full detail.  Compare to the [detailed definition of strict $2$-category](/nlab/show/strict+2-category#detailedDefn), which is written in the same style but is simpler.

A bicategory $B$ consists of

*  a collection $Ob B$ or $Ob_B$ of _objects_ or _$0$-cells_,
*  for each object $a$ and object $b$, a collection $B(a,b)$ or $Hom_B(a,b)$ of _morphisms_ or _$1$-cells_ $a \to b$, and
*  for each object $a$, object $b$, morphism $f\colon a \to b$, and morphism $g\colon a \to b$, a collection $B(f,g)$ or $2Hom_B(f,g)$ of _$2$-morphisms_ or _$2$-cells_ $f \Rightarrow g$ or $f \Rightarrow g\colon a \to b$,

equipped with

*  for each object $a$, an _identity_ $1_a\colon a \to a$ or $\id_a\colon a \to a$,
*  for each $a,b,c$, $f\colon a \to b$, and $g\colon b \to c$, a _composite_ $f ; g\colon a \to c$ or $g \circ f\colon a \to c$,
*  for each $f\colon a \to b$, an _identity_ or _$2$-identity_ $1_f\colon f \Rightarrow f$ or $\Id_f\colon f \to f$,
*  for each $f,g,h\colon a \to b$, $\eta\colon f \Rightarrow g$, and $\theta\colon g \Rightarrow h$, a _vertical composite_ $\theta \bullet \eta\colon f \Rightarrow h$,
*  for each $a,b,c$, $f,g\colon a \to b$, $h\colon b \to c$, and $\eta\colon f \Rightarrow g$, a _left whiskering_ $h \triangleleft \eta \colon h \circ f \Rightarrow h \circ g$,
*  for each $a,b,c$, $f\colon a \to b$, $g,h\colon b \to c$, and $\eta\colon g \Rightarrow h$, a _right whiskering_ $\eta \triangleright f\colon g \circ f \Rightarrow h \circ f$,
*  for each $f\colon a \to b$, a _left unitor_ $\lambda_f\colon \id_b \circ f \Rightarrow f$, and an _inverse left unitor_ $\bar{\lambda}_f\colon f \Rightarrow \id_b \circ f$,
*  for each $f\colon a \to b$, a _right unitor_ $\rho_f\colon f \circ \id_a \Rightarrow f$ and an _inverse right unitor_ $\bar{\rho}_f\colon f \Rightarrow f \circ \id_a$, and
*  for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, an _associator_ $\alpha_{h,g,f}\colon (h \circ g) \circ f \Rightarrow h \circ (g \circ f)$ and an _inverse associator_ $\bar{\alpha}_{h,g,f}\colon h \circ (g \circ f) \Rightarrow (h \circ g) \circ f$,

such that

*  for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\eta \bullet \Id_f$ and $\Id_g \bullet \eta$ both equal $\eta$,
*  for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b$, the vertical composites $\iota \bullet (\theta \bullet \eta)$ and $(\iota \bullet \theta) \bullet \eta$ are equal,
*  for each $a \overset{f}\to b \overset{g}\to c$, the whiskerings $\Id_g \triangleright f$ and $g \triangleleft \Id_f$ both equal $\Id_{g \circ f }$,
*  for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b$ and $i\colon b \to c$, the vertical composite $(i \triangleleft \theta) \bullet (i \triangleleft \eta)$ equals the whiskering $i \triangleleft (\theta \bullet \eta)$,
*  for each $f\colon a \to b$ and $g \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c$, the vertical composite $(\theta \triangleright f) \bullet (\eta \triangleright f)$ equals the whiskering $(\theta \bullet \eta) \triangleright f$,
*  for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\lambda_g \bullet (\id_b \triangleleft \eta)$ and $\eta \bullet \lambda_f$ are equal,
*  for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\rho_g \bullet (\eta \triangleright \id_a)$ and $\eta \bullet \rho_f$ are equal,
*  for each $a \overset{f}\to b \overset{g}\to c$ and $\eta\colon h \Rightarrow i\colon c \to d$, the vertical composites $\bar{\alpha}_{i,g,f} \bullet (\eta \triangleright (g \circ f))$ and $((\eta \triangleright g) \triangleright f) \bullet \bar{\alpha}_{h,g,f}$ are equal,
*  for each $f\colon a \to b$, $\eta\colon g \Rightarrow h\colon b \to c$, and $i\colon c \to d$, the vertical composites $\bar{\alpha}_{i,h,f} \bullet (i \triangleleft (\eta \triangleright f))$ and $((i \triangleleft \eta) \triangleright f) \bullet \bar{\alpha}_{i,g,f}$ are equal,
*  for each $\eta\colon f \Rightarrow g\colon a \to b$ and $b \overset{h}\to c \overset{i}\to d$, the vertical composites $\bar{\alpha}_{i,h,g} \bullet (i \triangleleft (h \triangleleft \eta))$ and $((i \circ h) \triangleleft \eta) \bullet \bar{\alpha}_{i,h,f}$ are equal,
*  for each $\eta\colon f \Rightarrow g\colon a \to b$ and $\theta\colon h \Rightarrow i\colon b \to c$, the vertical composites $(i \triangleleft \eta) \bullet (\theta \triangleright f)$ and $(\theta \triangleright g) \bullet (h \triangleleft \eta)$ are equal,
*  for each $f\colon a \to b$, the vertical composites $\lambda_f \bullet \bar{\lambda}_f\colon f \Rightarrow f$ and $\bar{\lambda}_f \bullet \lambda_f\colon \id_b \circ f \Rightarrow \id_b \circ f$ equal the appropriate identity $2$-morphisms,
*  for each $f\colon a \to b$, the vertical composites $\rho_f \bullet \bar{\rho}_f\colon f \Rightarrow f$ and $\bar{\rho}_f \bullet \rho_f\colon f \circ \id_a \Rightarrow f \circ \id_a$ equal the appropriate identity $2$-morphisms,
*  for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, the vertical composites $\bar{\alpha}_{h,g,f} \bullet \alpha_{h,g,f}\colon (h \circ g) \circ f \Rightarrow (h \circ g) \circ f$ and $\alpha_{h,g,f} \bullet \bar{\alpha}_{h,g,f}\colon h \circ (g \circ f) \Rightarrow h \circ (g \circ f)$ equal the appropriate identity $2$-morphisms,
*  for each $a \overset{f}\to b \overset{g}\to c$, the vertical composite $(\rho_g \triangleright f) \bullet \bar{\alpha}_{g,\id_b,f}$ equals the whiskering $g \triangleleft \lambda_f$, and
*  for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d \overset{i}\to e$, the vertical composites $((\bar{\alpha}_{i,h,g} \triangleright f) \bullet \bar{\alpha}_{i,h \circ g,f}) \bullet (i \triangleleft \bar{\alpha}_{h,g,f})$ and $\bar{\alpha}_{i \circ h,g,f}\bullet \bar{\alpha}_{i,h,g \circ f} $ are equal.

It is quite possible that there are errors or omissions in this list, although they should be easy to correct.  The point is not that one would *want* to write out the definition in such elementary terms (although apparently I just did anyway) but rather that one *can*.

## Examples

* Any [[strict 2-category]] is a bicategory in which the unitors and associator are identities.  This includes [[Cat]], [[MonCat]], the algebras for any strict [[2-monad]], and so on, at least as classically conceived.

* A [[monoidal category]] $M$ may be regarded as a bicategory $B M$ with a single object $\bullet$. The objects $A$ of $M$ become 1-cells $[A]: \bullet \to \bullet$ of $B M$; these are composed across the 0-cell $\bullet$ using the definition $[A] \circ_0 [B] = [A \otimes B]$, using the monoidal product $\otimes$ of $M$. The identity 1-cell $\bullet \to \bullet$ is $[I]$, where $I$ is the monoidal unit of $M$. The morphisms $f: A \to B$ become 2-cells $[f]: [A] \to [B]$ of $B M$. The associativity and unit constraints of the monoidal category $M$ transfer straightforwardly to associativity and unit data of the bicategory $B M$. The construction is a special case of [[delooping]] (see there).

* Categories, [[anafunctor]]s, and natural transformations, which is a more appropriate definition of [[Cat]] in the absence of the [[axiom of choice]], form a bicategory that is not a strict 2-category.  Indeed, without the axiom of choice, the proper notion of bicategory is [[anabicategory]].

* [[ring|Rings]], [[bimodule]]s, and bimodule homomorphisms are the prototype for many similar examples.  Notably, we can generalize from rings to [[enriched category|enriched categories]].

* Objects, [[span]]s, and morphisms of spans in any category with [[pullback]]s also form a bicategory.

* The [[fundamental 2-groupoid]] of a space is a bicategory which is not necessarily strict (although it can be made strict fairly easily when the space is Hausdorff by quotienting by [[thin homotopy]], see [[path groupoid]] and [[fundamental infinity-groupoid]]). When the space is a CW-complex, there are easier and more computationally amenable equivalent strict 2-categories, such as that arising from the fundamental [[crossed complex]].

## Coherence theorems
{#Coherence}

One way to state the [[coherence theorem]] for bicategories is that every bicategory is [[equivalence of 2-categories|equivalent]] to a [[strict 2-category]]. This "[[rectification]]" is not obtained naively by forcing [[composition]] to be [[associativity|associative]], but (at least in one construction) by freely adding new composites which are strictly associative.  Another way to state the coherence theorem is that every formal diagram of the constraints ([[associators]] and [[unitors]]) [[commuting diagram|commutes]].

Note that $n=2$ is the greatest value of $n$ for which every weak $n$-category is equivalent to a fully strict one; see [[semi-strict infinity-category]] and [[Gray-category]].

The [[proof]] of the coherence theorem is basically the same as the proof of the [[coherence theorem for monoidal categories]].  An abstract approach can be found in [Power 1989](#Power89). For a related statement see at *[[Lack's coherence theorem]]*.

The [[rectification]] [[adjunction]] between [[bicategories]] and [[strict 2-categories]] can be expressed in terms of a [[coreflection|coreflective]] [[triadjunction]] between [[tricategories]]; see [Campbell](#Campbell18).

\begin{tikzcd}
	{2\text{-}\mathrm{Cat}} & {\mathrm{Bicat}}
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{str}}"', shift right=2, from=1-2, to=1-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, shift right=2, from=1-1, to=1-2]
	\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=0, to=1]
\end{tikzcd}

Consequently, for any bicategory $B$ and 2-category $A$:
$$2\text{-}\mathrm{Cat}(\mathrm{str} B, A) \simeq \mathrm{Bicat}(B, A)$$

## Terminology

Classically, "2-category" meant [[strict 2-category]], with "bicategory" used for the weak notion.  This led to the more general use of the prefix "2-" for strict (that is, strictly [[Cat]]-enriched) notions and "bi-" for weak ones.  For example, classically a "2-adjunction" means a Cat-enriched adjunction, consisting of two strict 2-functors $F,G$ and a strictly Cat-natural isomorphism of categories $D(F X, Y)\cong C(X, G Y)$, while a "biadjunction" means the weak version, consisting of two weak 2-functors and a pseudo natural equivalence $D(F X, Y)\simeq C(X, G Y)$.  Similarly for "2-equivalence" and "biequivalence," and "2-limit" and "bilimit."

We often use "2-category" to mean a strict or weak 2-category without prejudice, although we do still use "bicategory" to refer to the particular classical algebraic notion of weak 2-category.  We try to avoid the more general use of "bi-" meaning "weak," however.  For one thing, it is confusing; a "biproduct" could mean a weak [[2-limit]], but it could also mean an object which is both a product and a coproduct (which happens quite frequently in [[additive category|additive categories]]).

Moreover, in most cases the prefix is unnecessary, since once we know we are working in a bicategory, there is usually no point in considering strict notions at all.  Fully weak limits are really the only sensible ones to ask for in a bicategory, and likewise for fully weak adjunctions and equivalences.  Even in a strict 2-category, while we might need to say "strict" sometimes to be clear, we don\'t need to say "$2$-", since we know that we are not working in a mere category.  (Max Kelly pushed this point.)

When we do have a strict 2-category, however, other strict notions can be quite technically useful, even if our ultimate interest is in the weak ones.  This is somewhat analogous to the use of strict structures to model weak ones in [[homotopy theory]]; see [here](http://arxiv.org/abs/math/0702535) and [here](http://arxiv.org/abs/math/0607646) for good introductions to this sort of thing.

## Related concepts

* [[category]]

* **bicategory**

  [[(infinity,2)-category]]

* [[double bicategory]] (and at [[double category#double_bicategories|double category]])

* [[tricategory]]

* [[tetracategory]]

* [[n-category]]

* [[(infinity,n)-category]]

* [[pseudofunctor]]

* [[locally univalent bicategory]]

* [[univalent bicategory]]

* [[monoidal bicategory]]

* [[closed bicategory]]

* [[initial object in a bicategory]]

* [[terminal object in a bicategory]]

* [[2-Grothendieck construction]]

Discussion about the use of the term "weak enrichment" above is at _[[weak enrichment]]_.

## References

See also the references at *[[2-category]]*.

* [[Jean Bénabou]], *Introduction to Bicategories*, Lecture Notes in Mathematics **47** Springer (1967), pp.1-77 ([doi:10.1007/BFb0074299](http://dx.doi.org/10.1007/BFb0074299))

* {#Power89} [[A. John Power]],  _A general coherence result._ J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. [doi:10.1016/0022-4049(89)90113-8](http://dx.doi.org/10.1016/0022-4049%2889%2990113-8) [MR0985657](http://www.ams.org/mathscinet-getitem?mr=985657)

* [[Saunders MacLane]], §XII.6 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Campbell18} [[Alexander Campbell]], _How strict is strictification?_, [arxiv](https://arxiv.org/abs/1802.07538)

Formalization in [[homotopy type theory]] (see also at [[internal category in homotopy type theory]]):

* [[Benedikt Ahrens]], Dan Frumin, Marco Maggesi, Niels van der Weide, _Bicategories in Univalent Foundations_ ([arXiv:1903.01152](https://arxiv.org/abs/1903.01152))

[[!redirects bicategory]]
[[!redirects bicategories]]
[[!redirects bicategorical]]

[[!redirects coherence theorem for bicategories]]


bicategory of fractions

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Much has been said about **inverting a class of morphisms** in a category (see [[localization]]), and there are many different settings in which one wants to, and can, do this. Homotopical algebra is largely concerned with how to compute the [[homotopy category]] so it is locally small. One the other hand, we have [[simplicial localization]] which retains all the homotopy information and returns an $(\infty,1)$-category.

If we have a 2-category with a notion of weak equivalence, one could localize the underlying 1-category in a way hopefully compatible with the 2-arrows, or extend the result fully into the 2-dimensional setting. In general this will require bicategories, and is the subject of [Pronk 96](#Pronk96).

## Definition

Let $B$ be a  [[bicategory]] with a class $W$ of 1-[[morphism|cells]]. $W$ is said to **admit a right calculus of fractions** if it satisfies the following conditions

* [2CF1.] $W$ contains all equivalences

* [2CF2.]
   * a) $W$ is closed under composition
   * b) If $a\in W$ and a iso-2-cell $a \stackrel{\sim}{\Rightarrow} b$ then $b\in W$

* {#2CF3} [2CF3.] For all $w: A' \to A$, $f: C \to A$ with $w\in W$ there exists a 2-commutative square
$$
\begin{matrix}
  P& \stackrel{g}{\to} & A' \\
  v \downarrow&\Rightarrow &\, \downarrow w\\
  C &\underset{f}{\to} & A
\end{matrix}
$$
with $v\in W$.

* [2CF4.] If $\alpha: w \circ f \Rightarrow w \circ g$ is a 2-cell and $w\in W$ there is a 1-cell $v \in W$ and a 2-cell $\beta: f\circ v \Rightarrow g \circ v$ such that $\alpha\circ v = w \circ \beta$. Moreover: when $\alpha$ is an iso-2-cell, we require $\beta$ to be an isomorphism too; when $v'$ and $\beta'$ form another such pair, there exist 1-cells $u,\,u'$ such that $v\circ u$ and $v'\circ u'$ are in $W$, and an iso-2-cell $\epsilon: v\circ u \Rightarrow v' \circ u'$ such that the following diagram commutes:
$$
\begin{matrix}
f \circ v \circ u & \stackrel{\beta\circ u}{\Rightarrow} & g\circ v \circ u \\
f\circ \epsilon \Downarrow \simeq &&
\simeq \Downarrow g\circ \epsilon \\
\\
f\circ v' \circ u' &\underset{\beta'\circ u'}{\Rightarrow}& g\circ v' \circ u'
\end{matrix}
$$

If $B$ is a category, then these axioms reduce to the ones of Gabriel and Zisman for a [[calculus of fractions]].

Given such a setup, Pronk constructs the [[localization]] of $B$ at $W$ and the universal functor sending elements of $W$ to equivalences.

## Example

Let $S$ be a category with binary products and pullbacks together with a class of [[davidroberts:class of admissible maps|admissible maps]] $E$.

+-- {: .num_theorem}
######Theorem:
The 2-categories $Cat(S)$ and $Gpd(S)$ of categories and groupoids internal to $S$ admit [[bicategory of fractions|bicategories of fractions]] for the class of $E$-[[davidroberts:weak equivalence|equivalences]].
=--

The resulting localization is equivalent to the bicategory of [[anafunctor|anafunctors]] in $S$. For details, see [Roberts (2012)](#Roberts12).

\begin{example}
**(Grothendieck toposes as a bicategory of fractions of localic groupoids)**\linebreak

The category of etale-complete [[localic groupoids]] (with open source and target maps) admits a [[bicategory of fractions]] at open essentially surjective fully faithful functors.

The resulting bicategory is equivalent to the bicategory of [[Grothendieck toposes]], [[geometric morphisms]], and [[natural isomorphisms]].

See Theorem 7.7 in [Moerdijk](#Moerdijk).
\end{example}

## Related entries

* [[localization of a 2-category]]

* [[category of fractions]]

* [[localization]]

* [[bicategory]]

* [[anafunctor]]

* [[orbifold]]

* [[stack]]

* [[étendue]]

## References

* {#Moerdijk} [[Ieke Moerdijk]], _The classifying topos of a continuous groupoid.  I_, Transactions of the American Mathematical Society, Volume 310, Number 2, December 1988.

* O. Abbad, [[Enrico Vitale|E. M. Vitale]], _Faithful calculus of fractions_ , Cah. Top. Géom. Diff. Catég. **54** No. 3 (2013) 221-239. ([preprint](perso.uclouvain.be/enrico.vitale/FrazioniFedeli.pdf))

* E. Vitale, _Bipullbacks and categories of fractions_, [pdf](https://perso.uclouvain.be/enrico.vitale/BPBc.pdf)

* [[Dorette A. Pronk]], _Etendues and stacks as bicategories of fractions_, Comp. Math. **102** 3 (1996) pp.243-303. ([numdam:CM_1996__102_3_243_0](http://www.numdam.org/item/?id=CM_1996__102_3_243_0))

* {#Roberts12} [[David Roberts]], _Internal categories, anafunctors and localisations_, TAC **26** (2012) pp.788-829. ([pdf](http://www.tac.mta.ca/tac/volumes/26/29/26-29.pdf)) {#Roberts12}

* M. Tommasini, _A bicategory of reduced orbifolds from the point of view of differential geometry_ , arXiv:1304.6959 (2013). ([pdf](http://arxiv.org/pdf/1304.6959))

* M. Tommasini, _Some insights on bicategories of fractions I_ , arXiv:1410.3990 (2014). ([pdf](http://arxiv.org/pdf/1410.3990))

* M. Tommasini, _Some insights on bicategories of fractions II_ , arXiv:1410.5075 (2014). ([pdf](http://arxiv.org/pdf/1410.5075))

* M. Tommasini, _Some insights on bicategories of fractions III_ , arXiv:1410.6395 (2014). ([pdf](http://arxiv.org/pdf/1410.6395))

See also:

* {#Renaudin06} [[Olivier Renaudin]], Section 1.2 of: *Plongement de certaines théories homotopiques de Quillen dans les dérivateurs*, Journal of Pure and Applied Algebra Volume 213, Issue 10, October 2009, Pages 1916-1935
([arXiv:math/0603339](https://arxiv.org/abs/math/0603339), [doi:10.1016/j.jpaa.2009.02.014](https://doi.org/10.1016/j.jpaa.2009.02.014))

[[!redirects bicategories of fractions]]
[[!redirects bicategorical localization]]
[[!redirects bicategorical localizations]]


bicategory of maps

# Bicategory of maps
* table of contents
{: toc}

## Definition

If $K$ is a [[bicategory]], then a [[morphism]] $f \colon a \to b$ is called a **map** if it has a [[right adjoint]] $f^* \colon b \to a$.  (This is in slight contrast to the common usage of "map" to denote simply a [[morphism]] in any category.)

The bicategory $Map K$ is the [[locally full sub-2-category]] of $K$ determined by the maps.

## Examples

* In the bicategory [[Rel]] of [[sets]] and [[relations]], a relation is a map if and only if it is the [[graph of a function]].  Consequently, $Map Rel$ is equivalent to [[Set]].

* Similarly, if $C$ is a category with finite [[limits]], then there is a bicategory $Span C$ of [[spans]] in $C$.  The bicategory $Map Span C$ is equivalent to $C$.

* In the bicategory [[Prof]] of [[categories]] and [[profunctors]] (perhaps [[enriched category|enriched]]), if $B$ is a [[Cauchy complete category]], then a profunctor $A\to B$ is a map if and only if it is represented by a functor $A\to B$.  If $B$ is not Cauchy complete, then maps $A\to B$ correspond to functors from $A$ to the Cauchy completion of $B$.

## Properties

If every map in $K$ is [[comonadic morphism|comonadic]] and $Map K$ has a terminal object, then $Map K$ is equivalent to a $1$-category.  If in addition $K$ is a [[cartesian bicategory]] and every [[comonad]] in $K$ has an [[Eilenberg--Moore object]], then $K$ is [[biequivalence|biequivalent]] to $Span Map K$, $Map K$ having finite limits.  The converse is true if pullback squares in $Map K$ satisfy the Beck--Chevalley condition in $K$, i.e. if their [[mates]] are invertible (see [\[LWW10\]](#LWW10)).

$Map K$ is a [[regular category]] if and only if $K$ is a unitary tabular [[allegory]], equivalently a [[bicategory of relations]] in which every [[coreflexive morphism]] [[split idempotent|splits]].  In that case $Rel Map K \simeq K$.

Similarly, $Map K$ is a [[topos]] if and only if $K$ is a unitary tabular power allegory.

## Maps and equipments

A [[2-category equipped with proarrows]] is, by definition, a bijective-on-objects [[pseudofunctor]] $K\to M$ such that the image of every arrow in $K$ is a map in $M$.  Equivalently, therefore, it is a bijective-on-objects pseudofunctor $K\to Map M$.

Hence the inclusion $Map M \to M$ is the "universal" proarrow equipment that can be constructed with a given bicategory $M$ as its bicategory of proarrows.  More precisely, there is a forgetful functor from $Equip$ to $Bicat$ which remembers only the bicategory $M$ of proarrows, and the assignment of $M$ to $Map M \to M$ is its right adjoint.

+--{: .query}
[[Mike Shulman]]: This is obviously morally true, but I can't be bothered right now to check which 1-, 2-, or 3-categories of equipments and bicategories one has to use to make it precisely correct.
=--

A lot of work in bicategories that makes use of maps can easily be reformulated in a proarrow equipment. One advantage of proarrow equipments over bicategories of maps is they can distinguish between a category and its Cauchy completion (as objects of [[Prof]]).

## See also

* [[category of maps]]

## References

* Carboni, Walters, _Cartesian bicategories I_, JPAA 49, 1987.
{: #CW87 }

* Lack, Walters, Wood, _Bicategories of spans as cartesian bicategories_, TAC 24(1), 2010.
{: #LWW10 }

[[!redirects bicategories of maps]]
[[!redirects map in a bicategory]]
[[!redirects maps in a bicategory]]


bicategory of relations

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

# Bicategories of relations
* table of contents
{: toc}

## Idea

A *bicategory of relations* is a [[(1,2)-category]] which behaves like the 2-category of internal [[relations]] in a [[regular category]].  The notion is due to Carboni and Walters.

## Definition

[Note: in this article, the direction of composition is diagrammatic (i.e., "anti-Leibniz").

A **bicategory of relations** is a [[cartesian bicategory]] which is [[locally posetal 2-category|locally posetal]] and moreover in which for every object $X$, the [[diagonal morphism|diagonal]] $\Delta\colon X\to X\otimes X$ and [[codiagonal]] $\nabla\colon X\otimes X\to X$ satisfy the [[Frobenius condition]]:
$$\nabla\Delta = (1\otimes \Delta)(\nabla \otimes 1).$$

Of course, in the locally posetal case the definition of cartesian bicategory simplifies greatly: it amounts to a symmetric monoidal [[Pos]]-[[enriched category]] for which every object carries a commutative [[comonoid]] structure, for which the structure maps $\Delta_X: X \to X \otimes X$, $\varepsilon_X: X \to 1$ are [[left adjoint]] 1-morphisms, and for which every morphism $R: X \to Y$ is a colax morphism of comonoids in the sense that the following inequalities hold:

$$R \Delta_Y \leq \Delta_{X \otimes X} (R \otimes R), \qquad R \varepsilon_Y \leq \varepsilon_X.$$

We remark that such structure is unique when it exists (being a cartesian bicategory is a property of, as opposed to structure on, a bicategory). The [[tensor product]] $\otimes$ behaves like the ordinary product of [[relation]]s. Note this is not a cartesian product in the sense of the usual universal property; nevertheless, it is customary to write it as a product $\times$, and we follow this custom below. It does become a cartesian product if one restricts to the subcategory of left adjoints (called _maps_), which should be thought of as functional relations.

## Properties

We record a few consequences of this notion.

+-- {: .num_prop}
######Proposition
(Separability condition) $\Delta\nabla = 1$.
=--

+-- {: .proof}
######Proof
In one direction, we have the 2-cell $1 \leq \Delta\nabla$ which is the unit of the adjunction $\Delta \dashv \Delta_* = \nabla$. In the other direction, there is an adjunction $\varepsilon \dashv \varepsilon_*$ between the counit and its dual, and the unit of this adjunction is a 2-cell $1 \leq \varepsilon\varepsilon_*$, from which we derive

$$\Delta\nabla = \Delta\Delta_* \leq \Delta (1 \times \varepsilon)(1 \times \varepsilon_*)\Delta_* = 1$$

where the last equation follows from the equation $\Delta(1 \times \varepsilon) = 1$ and its dual $(1 \times \varepsilon_*)\Delta_* = 1$.
=--

+-- {: .num_prop}
######Proposition
(Dual Frobenius condition) $\nabla \Delta = (\Delta \times 1)(1 \times \nabla)$.
=--

+-- {: .proof}
######Proof
The locally full subcategory whose 1-cells are _maps_ (= left adjoints) has finite products, in particular a symmetry involution $\sigma$ for which

$$\Delta_X \sigma_{X X} = \Delta_X$$

Additionally, the right adjoint $\sigma_{X Y *}$ is the inverse $\sigma_{X Y}^{-1} = \sigma_{Y X}$. Hence the equation above is mated to $\sigma_{X X} \nabla_X = \nabla_X$, and we calculate

$$\array{
\nabla_X \Delta_X & = & \sigma_{X X}\nabla_X \Delta_X \sigma_{X X} \\
 & = & \sigma_{X X} (1 \times \Delta_X)(\nabla_X \times 1) \sigma_{X X} \\
 & = & (\Delta_X \times 1_X) \sigma_{X \times X, X} \sigma_{X \times X, X} (1_X \times \nabla_X) \\
 & = & (\Delta_X \times 1_X) (\sigma_{X X} \times 1) (1 \times \sigma_{X X}) (1 \times \nabla_X) \\
 & = & (\Delta_X \times 1_X)(1_X \times \nabla_X)
}$$

which gives the dual equation.
=--

+-- {: .num_theorem}
######Theorem
In a bicategory of relations, each object is dual to itself, making the bicategory a compact closed bicategory.
=--

+-- {: .proof}
######Proof
Both the unit and counit of the desired adjunction $X \dashv X$ are given by equality predicates:

$$\eta_X = (1 \stackrel{\varepsilon_*}{\to} X \stackrel{\Delta}{\to} X \times X) \qquad \theta_X = (X \times X \stackrel{\nabla}{\to} X \stackrel{\varepsilon}{\to} 1)$$

One of the triangular equations follows from the commutative diagram

$$\array{
X & \stackrel{1 \times \varepsilon_*}{\to} & X \times X & \stackrel{1 \times \Delta}{\to} & X \times X \times X \\
 & {}_1\searrow & \downarrow \mathrlap{\nabla} & & \downarrow \mathrlap{\nabla \times 1} \\
 & & X & \overset{\Delta}{\to} & X \times X \\
 & &  & {}_{1}\searrow & \downarrow \mathrlap{\varepsilon \times 1} \\
 & & & & X
}$$

where the square expresses a Frobenius equation. The other triangular equation uses the dual Frobenius equation.
=--

The compact closure allows us to define the opposite of a relation $R: X \to Y$ as the 1-morphism mate:

$$R^{op} = (Y \stackrel{1_Y \times \eta_X}{\to} Y \times X \times X \stackrel{1_Y \times R \times 1_X}{\to} Y \times Y \times X \stackrel{\theta_Y \times 1_X}{\to} X)$$

In this way, a bicategory of relations becomes a [[dagger-compact category|†-compact]] $Pos$-enriched category.

Recall (again) that a **map** in the bicategory of relations is the same as a 1-cell that has a right adjoint.

+-- {: .num_prop #strict}
###### Proposition
If $f \colon X \to Y$ is a map, then $f$ is a strict morphism of comonoids.
=--

+-- {: .proof}
###### Proof
If $g$ is the right adjoint of $f$, we have $\Delta_X g \leq (g \times g)\Delta_Y$ since $g$, like any morphism, is a lax comonoid morphism. But this 2-cell is mated to a 2-cell $(f \times f)\Delta_X \to \Delta_Y f$, inverse to the 2-cell $\Delta_Y f \to (f \times f)\Delta_X$ that comes for free. So $f$ preserves comultiplication strictly. A similar argument shows $f$ preserves the counit strictly.
=--

+-- {: .num_prop}
###### Proposition
If $f \colon X \to Y$ is a map, then $f^{op}: Y \to X$ equals the right adjoint $f_\ast$.
=--

+-- {: .proof}
###### Proof
The proof is much more perspicuous if we use [[string diagram|string diagrams]]. But the key steps are given in two strings of equalities and inequalities. The first gives a counit for $f \dashv f^{op}$, and the second gives a unit. We have

$$\array{
f f^{op} & \stackrel{0}{=} & f \circ (\varepsilon_Y \times 1)(\nabla_Y\times 1)(1 \times f \times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
& \stackrel{1}{=} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(1 \times f \times f)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
 & \stackrel{2}{=} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(1 \times \Delta_Y)(1 \times f)(1 \times \varepsilon_X_\ast)\\
 & \stackrel{3}{\leq} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(1 \times \Delta_Y)(1 \times \varepsilon_Y_\ast) \\
 & \stackrel{4}{=} & (\varepsilon_Y \times 1)\Delta_Y \nabla_Y(1 \times \varepsilon_Y_\ast) \\
 & \stackrel{5}{=} & 1_Y
}$$

where (0) uses the definition of $f^{op}$, (1) uses properties of monoidal categories, (2) uses the fact that $f$ strictly preserves comultiplication, (3) is mated to the fact that $f$ laxly preserves the counit, (4) is a Frobenius condition, and (5) uses comonoid and dual monoid laws. We can also "almost" run the same calculation backwards to get the unit:

$$\array{
1_X & \stackrel{0}{=} & (\varepsilon_X \times 1)\Delta_X \nabla_X(1 \times \varepsilon_X_\ast) \\
 & \stackrel{1}{=} & (\varepsilon_X \times 1)(\nabla_X \times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
 & \stackrel{2}{=} & (\varepsilon_Y \times 1)(f \times 1)(\nabla_X\times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
 & \stackrel{3}{\leq} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(f \times f \times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \\
 & \stackrel{4}{=} & (\varepsilon_Y \times 1)(\nabla_Y \times 1)(1 \times f \times 1)(1 \times \Delta_X)(1 \times \varepsilon_X_\ast) \circ f \\
 & \stackrel{5}{=} & f^{op} f
}$$

where (0) uses comonoid and dual monoid laws, (1) uses a Frobenius condition, (2) uses the fact that $f$ preserves the counit, (3) is mated to the fact that $f$ laxly preserves comultiplication, (4) uses properties of monoidal categories, and (5) uses the definition of $f^{op}$.
=--

In fact, what this proof really proves is a converse of the earlier [proposition](#strict):

+-- {: .num_prop}
###### Proposition
If $f$ is a strict comonoid morphism, then $f$ has a right adjoint: $f \dashv f^{op}$.
=--

+-- {: .num_prop}
######Proposition
If $f, g: X \to Y$ are maps and $f \leq g$, then $f = g$. Thus, the locally full subcategory whose morphisms are maps is locally discrete (the hom-posets are discrete).
=--

+-- {: .proof}
######Proof
A 2-cell inequality $\alpha: f \leq g$ is mated to a inequality $\alpha_*: g_* \leq f_*$. On the other hand, whiskering $1_Y \times \alpha \times 1_X$ with $1_Y \times \eta_X$ and $\theta_Y \times 1_X$, as in the construction of opposites above, gives $\alpha^{op}: f^{op} \leq g^{op}$. Since $f^{op} = f_*$ and $g^{op} = g_*$, we obtain $f^{op} = g^{op}$, and because $f^{op op} = f$, we obtain $f = g$.
=--

## The Beck-Chevalley condition

Bicategories of relations $\mathbf{B}$ satisfy a Beck-Chevalley condition, as follows. Let $Prod(B_0)$ denote the free category with finite products generated by the set of objects of $\mathbf{B}$. According to the results at [[free cartesian category]], $Prod(B_0)$ is finitely complete.

Since $Map(\mathbf{B})$ has finite products, there is a product-preserving functor $\pi: Prod(B_0) \to \Map(\mathbf{B})$ which is the identity on objects. Again, according to the results of [[free cartesian category]], we have the following result.

+-- {: .num_lemma}
###### Lemma
If a diagram in $Prod(B_0)$ is a pullback square, then application of $\pi$ to that diagram is a pullback square in $Map(\mathbf{B})$.
=--

Let us call pullback squares of this form in $Map(\mathbf{B})$ _product-based_ pullback squares.

+-- {: .num_prop}
######Proposition
Given a product-based pullback square
$$\array{
W & \stackrel{h}{\to} & X \\
k \downarrow & & \downarrow f \\
Y & \underset{g}{\to} & Z
}$$
in $Map(\mathbf{B})$, the Beck-Chevalley condition holds: $h_* k = f g_*$.
=--

See Brady-Trimble for further details. The critical case to consider is the pullback square

$$\array{
X & \overset{\Delta}{\to} & X \times X \\
\mathllap{\Delta} \downarrow & & \downarrow \mathrlap{\Delta \times 1} \\
X \times X & \underset{1 \times \Delta}{\to} & X \times X \times X
}$$

where the Beck-Chevalley condition is exactly the Frobenius condition.

One way of interpreting this result is by viewing $\mathbf{B}$ as a hyperdoctrine or monoidal fibration over $Prod(B_0)$, where the fiber over an object $B$ is the local hom-poset $\hom(B, 1)$. Each $f: A \to B$ in the base induces a pullback functor, by precomposing $R: B \to 1$ with $\pi(f): A \to B$ in $Map(\mathbf{B})$. Existential quantification is interpreted by precomposing with right adjoints $\pi(f)_*$. The Beck-Chevalley condition exerts compatibility between quantification and pullback/substitution functors.

## Relation to allegories

Any bicategory of relations is an [[allegory]]. Recall that an allegory is a $Pos$-enriched $\dagger$-category whose local homs are meet-semilattices, satisfying Freyd's modular law:

$$R S \cap T \leq (R \cap T S^{op})S$$

A proof of the modular law is given in the blog post by R.F.C. ("Bob") Walters referenced below.

In fact, we may prove a little more:

+-- {: .num_theorem}
######Theorem
The notion of bicategory of relations is equivalent to the notion of unitary pretabular allegory.
=--

+-- {: .proof}
######Proof
A bicategory of relations has a unit $1$ in the sense of allegories:

* $1$ is a partial unit: we have $\varepsilon_1 = id_1: 1 \to 1$, and for any $R: 1 \to 1$ we have $R = R\varepsilon_1 \leq \varepsilon_1 = id_1$.

* Any object $X$ is the source of a map $\varepsilon_X: X \to 1$, which being a map is entire. Thus $1$ is a unit.

A bicategory of relations is also pretabular, for the maximal element in $\hom(X, Y)$ is tabulated as $(\pi_X)_* \pi_Y$, where $\pi_X$, $\pi_Y$ are the product projections for $X \times Y$.

In the other direction, suppose $\mathbf{A}$ is a unitary pretabular allegory.  There is a faithful embedding, which preserves the unit, of $\mathbf{A}$ into its coreflexive splitting, which is unitary and tabular, and hence equivalent to $Rel$ of the regular category $Map(\mathbf{A})$.  By the proposition below, then, $\mathbf{A}$ is a full sub-2-category of a bicategory of relations.  Because the product of two Frobenius objects is again Frobenius, it suffices to show that $\mathbf{A}$ is closed under products in its coreflexive splitting.  The inclusion of $\mathbf{A}$ into the latter preserves the unit and the property of being a map, and hence preserves top elements of hom sets, while any allegory functor must preserve tabulations.  So the tabulation $(\pi^{X Y}_X, \pi^{X Y}_Y)$ of $\top_{X Y}$ in $\mathbf{A}$ is a tabulation of $\top_{1_X 1_Y}$ in the coreflexive splitting, and hence $1_{X \times Y} \cong 1_X \times 1_Y$.
=--

+-- {: .num_prop}
###### Proposition
If $C$ is a [[regular category]], then the bicategory $Rel C$ of [[Rel|relations]] in $C$ is a bicategory of relations.
=--
+-- {: .proof}
###### Proof
By theorem 1.6 of Carboni--Walters, $\mathbf{A}$ is a [[Cartesian bicategory]] if:

* $Map(\mathbf{A})$ has finite products.

* $\mathbf{A}$ has local finite products, and $id_1$ is the top element of $\mathbf{A}(1,1)$.

* The tensor product defined as
$$ R \otimes S =  (p R p_*) \cap (p' S p'_*) $$
is functorial, where $p$ and $p'$ are the appropriate product projections.

The first two are obvious, and for the third we may reason in the [[internal language]] of $C$.  Clearly
$$ 1 \otimes 1 = [x = x \wedge x' = x'] = 1  $$
The formula whose meaning is by $R S \otimes R' S'$ is
$$ (\exists y. R x y \wedge S y z) \wedge (\exists y'. R' x' y' \wedge S' y' z') $$
and we may use [[Frobenius reciprocity]] to get
$$
\begin{aligned}
  & \exists y. (R x y \wedge S y z \wedge y^* \exists y'. R' x' y' \wedge S' y' z') \\
  & \equiv \exists y. (R x y \wedge S y z \wedge  \exists y'. y^* (R' x' y'   \wedge S' y' z')) \\
  & \equiv \exists y, y'. R x y \wedge S y z \wedge R' x' y' \wedge S' y' z' \\
\end{aligned}
$$
which is the meaning of $(R \otimes R')(S \otimes S')$.  Finally, the Frobenius law is
$$ [\exists x'. (x_1, x_2) = (x', x') = (x_3, x_4)]
= [\exists x'. (x_1, x') = (x_3, x_3) \wedge (x_2, x_2) = (x_4, x')]  $$
which follows from transitivity and symmetry of equality.
=--

## See also

Other attempted axiomatizations of the same idea "something that acts like the category of relations in a regular category" include:

* [[allegories]]

* [[1-category equipped with relations]]

## Related concepts

* [[category of correspondences]]

* [[(infinity,n)-category of correspondences]]

## References

* [[Aurelio Carboni]], [[Bob Walters]], _Cartesian Bicategories, I_, [article](https://doi.org/10.1016/0022-4049%2887%2990121-6)

* [[Aurelio Carboni]], [[Max Kelly]], [[Bob Walters]], [[Richard Wood]], _Cartesian Bicategories II_, ([arXiv:0708.1921](https://arxiv.org/abs/0708.1921))

* [[Bob Walters]], [blog post](http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html), showing that any bicategory of relations is an [[allegory]]. Indeed, a bicategory of relations is equivalent to a unitary pretabular allegory.

* [[Evan Patterson]], _Knowledge Representation in Bicategories of Relations_, ([arXiv:1706.00526](https://arxiv.org/abs/1706.00526))

This article shows how one can model RDF (Resource Description Framework) and parts of OWL (Ontology Web Language) in bicategories of relations, whose internal logic is [[regular logic]]. He ends by showing how one can extend these to [[distributive bicategories of relations]] whose internal logic is [[coherent logic]], which is equivalent in expressivity to first order logic.

[[!redirects bicategories of relations]]


BICEP2

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

#Contents#
* table of contents
{:toc}

## Idea

_BICEP2_  (for "Background Imaging of Cosmic Extragalactic Polarization") is the name of an astrophysical [[experiment]] (their [webpage](http://www.cfa.harvard.edu/CMB/bicep2/)) which released its data in March 2014. The experiment claims to have detected a pattern called the "B-mode" in the polarization of the [[cosmic microwave background]] (CMB).

### Claims

This data, if confirmed, would have been likely due to a [[gravitational wave]] mode created during the period of [[cosmic inflation]] by a [[vacuum fluctuation]] in the [[field (physics)|field]] of [[gravity]] which then at the era of decoupling left the characteristic B-mode imprint on the CMB. This fact alone was regarded as further strong evidence for the already excellent experimental evidence for [[cosmic inflation]] as such (competing models did not predict such gravitational waves to be strong enough to be detectable in this way).

What would have singled out the BICEP2 result over previous confirmations of cosmic inflation is that the data also gives a quantitative value for the [[energy]] scale at which cosmic inflation happened (the [[mass]] of the hypothetical [[inflaton]]), namely at around $10^{16}$[[GeV]]. This is noteworthy as being only two order of magnituded below the [[Planck scale]], and hence 12 or so orders of magnitude above energies available in current accelerator [[experiments]] (the [[LHC]]). Also, it is at least a curious coincidence that this is precisely the hypothetical [[GUT]] scale.

It was thought that this value rules out a large number of variant models of [[cosmic inflation]] and favors the model known as _[[chaotic inflation]]_.

### Problems

Further measurement by the [Planck collaboration](#Planck) showed that much of the signal claimed by BICEP2 is naturally explained just by galactic dust forground, see [below](#Resonaances). Further measurements seem to be needed to clarify the situation.

## Related entries

## References

* [BICEP2 webpage](http://www.cfa.harvard.edu/CMB/bicep2/)

* [general BICEP webpage](http://bicep.caltech.edu/public/

* {#Planck} [Planck collaboration webpage](http://sci.esa.int/planck/)

* {#Resonaances} Resonaances, _[BICEP: what was wrong and what was right](http://resonaances.blogspot.de/2014/09/bicep-what-was-wrong-and-what-was-right.html)_

* _A Joint Analysis of BICEP2/Keck Array and Planck Data_ ([arXiv:1502.00612](http://arxiv.org/abs/1502.00612))

[[!redirects bicep2]]

[[!redirects BICEP2 experiment]]
[[!redirects bicep2 experiment]]


bicharacteristic flow

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

#Contents#
* table of contents
{:toc}

## Definition

Let $X$ be a [[smooth manifold]] and let $D$ be a [[differential operator]] on (smooth [[sections]] of) the [[trivial line bundle]] over $X$ (or more generally a [[properly supported pseudo-differential operator]]). Then the [[principal symbol]] $q(D)$ of $D$ is equivalently a [[smooth function]] on the [[cotangent bundle]] $T^\ast X$ (by [this example](symbol+of+a+differential+operator#BicharacteristicFlow)). With the [[cotangent bundle]] canonically regarded as a [[symplectic manifold]], let

$$
  v_{q(D)} \in \Gamma\left(T\left(T^\ast X\right) \right)
$$

be the corresponding [[Hamiltonian vector field]].

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

The _bicharacteristic flow_ of $D$ is the [[Hamiltonian flow|Hamiltonian]] [[flow of a vector field|flow]] of the [[Hamiltonian vector field]] $v_{q(D)}$ inside the submanifold defined by $q = 0$. Moreover:

1. A single [[flow line]] in $T^\ast X$ is called a _bicharacteristic strip_ of $D$,

1. the [[projection]] of such to a [[curve]] in $X$ is called a _bicharacteristic curve_.

1. The [[relation]] $C$ on $T^\ast X$ given by

   $$
     \left((x_1,k_1) \sim (x_2, k_2)\right)
     \;\coloneqq\;
     \left(
       q(x_i,k_i) = 0
       \;\;\text{and}\;\;
       (x_1,k_1) \,\text{is connected to}\,
       (x_2,k_2) \,\text{by a bicharacteristic strip}
     \right)
   $$

   is called the _bicharacteristic relation_.

=--

## Examples

### Of the Klein-Gordon operator
 {#OfTheKleinGordonOperator}

+-- {: .num_example #BicharachteristicFlowOfKleinGordonOperator}
###### Example
**(bicharacteristic curves of [[wave operator]]/[[Klein-Gordon operators]] are the [[lightlike]] [[geodesics]])**

Let $(X,g)$ be a [[Lorentzian manifold]] and let $D \coloneqq \Box_g - m^2$ be its [[wave operator]]/[[Klein-Gordon operator]].

Then the bicharacteristic curves of $D$ (def. \ref{BicharacteristicFlow}) are precisely the [[lightlike]] [[geodesics]] of $(X,e)$, and the bicharacteristic strips are precisely these geodesices with their [[cotangent vectors]].

Accordingly two cotangent vectors are bicharacteristically related $(x_1,k_1) \sim (x_2,k_2)$ precisely if there is a [[lightlike]] [[geodesic]] connecting the points, with $k_1$ and $k_2$ the corresponding cotangents, hence one the result of [[parallel transport]] of the other along the geodesic.

=--

([Radzikowski 96, prop. 4.2 and below (6)](#Radzikowski96))

Specifically on [[Minkowski spacetime]]:

+-- {: .num_example}
###### Example
**([[bicharacteristic flow]] of [[Klein-Gordon operator]] on [[Minkowski spacetime]])**

Let $\mathbb{R}^{p,1}$ be [[Minkowski spacetime]] of [[dimension]] $p+1$ consider the [[Klein-Gordon operator]]

$$
  D
  \;=\;
  \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu}
  -
  \left(\tfrac{m c}{\hbar}\right)^2
  \,.
$$

Its [[principal symbol]] is the function

$$
  \array{
    T^\ast \mathbb{R}^{p,1}
      &\overset{q}{\longrightarrow}&
    \mathbb{R}
    \\
    (x,k) &\mapsto& \eta^{\mu \nu} k_\mu k_\nu
  }
$$

Hence $q(k) = 0$ is the condition that the [[wave vector]] $k$ be [[lightlike]].

The [[Hamiltonian vector field]] corresponding to $q$ is

$$
  \begin{aligned}
    v_q
    & =
    -\tfrac{1}{2} \eta^{\mu \nu} k_\mu \partial_{x^\nu}
    \\
    & =
    -\tfrac{1}{2} k^\mu \partial_{x^\mu}
  \end{aligned}
$$

in that

$$
  \begin{aligned}
    \iota_{v_q} d k_\mu \wedge d x^\mu
    &=
    \tfrac{1}{2} \eta^{\mu \nu} k_\mu d k_\mu
    \\
    & =
    d q(k)
  \end{aligned}
$$

It follows that the [[bicharacteristic curves]] are precisely the [[lightlike]] [[curves]]

$$
  \array{
    \mathbb{R} &\overset{\gamma_k}{\longrightarrow}& \mathbb{R}^{p,1}
    \\
    \tau &\mapsto& (\gamma^\mu(0) +  \tau k^\mu)
  }
$$

and the corresponding [[bicharacteristic strips]] are these with their lightlike contangent vector constantly carried along

$$
  \array{
    \mathbb{R} &\overset{\gamma_k}{\longrightarrow}& T^\ast\mathbb{R}^{p,1}
    \\
    \tau &\mapsto& \left((\gamma^\mu(0) +  \tau k^\mu),(k_\mu)\right)
  }
$$

=--

## Properties

### Propagation of singularities

The _[[propagation of singularities theorem]]_ says that the [[wave front set]] of a [[distribution|distributional]] solution to the [[differential equation]] of a sufficiently nice [[differential operator]] (or generally of a [[properly supported pseudo-differential operator]]) is preserved by the bicharacteristic flow.

## References

* {#DuistermaatHoermander72} [[Johann Duistermaat]], [[Lars Hörmander]], _Fourier integral operators II_, Acta Mathematica 128, 183-269, 1972 ([Euclid](https://projecteuclid.org/euclid.acta/1485889724))

Review in the context of the [[free field|free]] [[scalar field]] on [[globally hyperbolic spacetimes]] (with $Q$ the [[wave operator]]/[[Klein-Gordon operator]]) is in

* {#Radzikowski96} [[Marek Radzikowski]], _Micro-local approach to the Hadamard condition in quantum field theory on curved space-time_, Commun. Math. Phys. 179 (1996), 529–553 ([Euclid](http://projecteuclid.org/euclid.cmp/1104287114))

[[!redirects bicharacteristic flows]]

[[!redirects bicharacteristic strip]]
[[!redirects bicharacteristic strips]]

[[!redirects bicharacteristic curve]]
[[!redirects bicharacteristic curves]]


bicommutant theorem

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

# Contents
* table of contents
{:toc}

## Idea

The _bicommutant theorem_ characterizes concrete [[von Neumann algebras]] as those concrete $C^*$-[[C-star algebra|algebras]] ($C^*$-algebras of [[bounded operator]]s on some [[Hilbert space]]) that are the [[commutant]]s of their own commutants.

## Statement

The __bicommutant theorem__ (as known as the _double commutant theorem_, or _von Neumann's double commutant theorem_) is the following result:

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

Let $A \subseteq B(H)$ be a sub-$*$-[[star-algebra|algebra]] of the algebra of [[bounded linear operator|bounded linear operators]] on a [[Hilbert space]] $H$. Then $A$ is a [[von Neumann algebra]] (and therefore also a $C^*$-[[C-star algebra|algebra]]) in $H$ if and only if $A = A''$, where $A'$ denotes the [[commutant]] of $A$.
=--

Notice that the condition of $A$ being a von Neumann algebra (being closed in the [[weak operator topology]]; "weak" here can be replaced by "strong", "ultrastrong", or "ultraweak" as described in [[operator topology]]), which is a [[topology|topological]] condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).

## References

* Wikipedia article: [bicommutant theorem] (http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem)

[[!redirects bicommutant theorem]]
[[!redirects double commutant theorem]]
[[!redirects von Neumann double commutant theorem]]
[[!redirects von Neumann's double commutant theorem]]
[[!redirects von Neumann's double commutant theorem]]
[[!redirects von Neumann\'s double commutant theorem]]


bicomodule

Let $C,D$ be [[comonoid]]s in a [[monoidal category]] $A = (A,\otimes,1)$. A $C$-$D$ __bicomodule__ is an object $M$ in $A$, with left $C$-[[coaction]] $\lambda_C : M\to C\otimes M$ and right $D$-coaction $\rho_D: M\to M\otimes D$ which commute in the sense that

$$
(\lambda_C\otimes id_D)\circ\rho_D = (id_C\otimes \rho_D)\circ \lambda_C.
$$

Typical cases are when $A$ is the category of $k$-[[modules]] where $k$ is a [[commutative unital ring]] (the comonoids are then $k$-[[coalgebras]]), and the more general case of bicomodules over [[corings]], where $A$ is the category of $k$-bimodules where $k$ is a possibly [[noncommutative ring]].

There is an operation of [[cotensor product]] for bicomodules over coalgebras/corings; however it is not associative in general, unlike the [[tensor product]] of bimodules over rings!

[[!redirects bicomodule]]
[[!redirects bicomodules]]


bicompact space

# Bicompact spaces
* table of contents
{: toc}

## Definitions

A __bicompact space__ is a [[bitopological space]] both of whose underlying [[topological spaces]] are [[compact spaces]], often assumed to be [[compact Hausdorff space|Hausdorff]].

$Bi Comp$ is the [[category]] whose [[object|objects]] are bicompact spaces and whose [[morphism|morphisms]] are [[bicontinuous maps]].

## References

* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)

[[!redirects bicompact space]]
[[!redirects bicompact spaces]]
[[!redirects bicompact bitopological space]]
[[!redirects bicompact bitopological spaces]]
[[!redirects BiComp]]
[[!redirects Bi Comp]]


bicomplete category

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

# Complete categories
* table of contents
{: toc}

## Definition

A [[category]] $C$ is **bicomplete** if it is both a [[complete category]] as well as a [[cocomplete category]], hence if it has all [[small limit|small]] [[limit|limits]] and [[colimits]]: that is, if every [[small diagram|small]] [[diagram]]

$$ F: D \to C$$

where $D$ is a [[small category]] has a [[limit]] and a [[colimit]] in $C$.

## Related concepts

* [[M-complete category]]

[[!redirects bicomplete categories]]


biconditional

biconditional "$\Leftrightarrow$"  is [[conditional]] in both directions "$\Rightarrow$", "$\Leftarrow$": [[equivalence]]

[[!redirects biconditionals]]


bicovariant differential calculus

# Bicovariant differential calculus
* table of contents
{: toc}

## Definitions

Let $A$ be a $k$-algebra. A first order differential calculus $\Omega^1(A)$ is an $A$-bimodule with an
$A$-derivation $d: A\to\Omega^1(A)$ such that $\Omega^1(A)$ is $k$-spanned by elements of the form $a d(b)$ where $a,b\in A$.

Let $A = H$ be a Hopf algebra. A first order differential calculus $d:H \to \Gamma = \Omega^1(H)$
is bicovariant if $\Omega^1(H)$ is also an $H$-bicomodule (compatibly left and right comodule) where the left and right coaction $\rho^L$, $\rho^R$ are $A$-bimodule maps and $d$ is an $A$-bicomodule map. In other words, $\Gamma$ is a [[Hopf bimodule]]
and $d:A\to \Gamma$ is a Hopf bimodule map.

## Relation to Yetter--Drinfeld condition

It is well-known that the category of Hopf bimodules
${}^H_H\mathcal{M}^H_H$ is equivalent (even as a monoidal category!) to the category of [[Yetter–Drinfeld module]]s, at least for finite-dimensional
Hopf algebra $H$.

## Literature

The notion of bicovariance of a noncommutative
differential calculus is introduced in

* [[Stanisław Lech Woronowicz]], _Differential calculus on compact matrix pseudogroups (quantum groups)_ Commun. Math. Phys. __122__, 125--170 (1989) [euclid](http://projecteuclid.org/euclid.cmp/1104178320) [MR0994499](http://www.ams.org/mathscinet-getitem?mr=0994499)

* [[Shahn Majid]], _Classification of bicovariant differential calculi_, Journal of Geometry and Physics __25__:1–2, April 1998, 119--140; doi [q-alg/9608016](http://arxiv.org/abs/q-alg/9608016v2)

* Konrad Schmüdgen, Axel Schüler, _Classification of bicovariant differential calculi on quantum groups of type A, B, C and D_, Commun. Math. Phys. __167__:3, pp 635--670 (1995) [euclid](http://projecteuclid.org/euclid.cmp/1104273123)

* Ursula Carow-Watamura, Satoshi Watamura, _Complex quantum group, dual algebra and bicovariant differential calculus_, Comm. Math. Phys. __151__:3 (1993) 487--514 [euclid](http://projecteuclid.org/euclid.cmp/1104252236)

* Peter Schauenburg, _Hopf modules and Yetter--Drinfel′d modules_, J. Algebra __169__:3 (1994) 874--890 [doi](http://dx.doi.org/10.1006/jabr.1994.1314)

category: algebra

[[!redirects bicovariant differential calculus]]
[[!redirects bicovariant differential calculi]]


bicrossed product

#Idea#

The bicrossed product generalizes the [[semidirect product]] of groups.

This construction is essential to the [[quantum double construction]] of Drinfel'd.

#Definition#

Given a pair of matched groups $H$ and $K$, the bicrossed product of groups $H\times K$ on the set $H \times K$
is given by
\[ (h,k)\cdot(h',k') = (h\alpha(k,h'),\beta(k,h')k')\]
with unit $(1,1)$ and $h,h'\in H$, $k,k'\in K$, where $\alpha: K\times H\rightarrow H$, $\beta: K\times H\rightarrow K$ are left and right actions, respectively.

A pair of groups $(H,K)$ is said to be matched if there exists a left action $\alpha$ of $K$ on the set $H$ and a right action $\beta$ of the group $H$ on the set $K$ such that for all $h,h'\in H$, $k,k'\in K$, the following hold:

* $\beta(k k',h) = \beta(k,\alpha(k',h))\beta(k',h)$,
* $\alpha(k,h h') = \alpha(k,h)\alpha(\beta(k,h),h')$,
* $\alpha(k,1) = 1$,
* $\beta(1,h) = 1$.

+-- {: .query}
Need to define the bicrossed product of algebras.
=--

#References#

C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer-Verlag, New York-Berlin, 1995.


bicubical set

#Contents#
* table of contents
{:toc}

## Idea

Bicubical set are to [[cubical sets]] as [[bisimplicial sets]] are to [[simplicial sets]].

## References

On [[synthetic (∞,1)-category theory]] in [[cubical type theory]] with bicubical sets:

* {#WeaverLicata20} [[Matthew Weaver]], [[Daniel Licata]], *A Constructive Model of Directed Univalence in Bicubical Sets*, in: *Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’20*, Association for Computing Machinery (2020) 915–928 [[doi:10.1145/3373718.3394794](https://doi.org/10.1145/3373718.3394794)]

Exposition in:

* [[Matthew Weaver]], *A Constructive Model of Directed Univalence in Bicubical Sets*, talk at HoTTEST (April 2020) [[pdf](https://www.cs.princeton.edu/~mzweaver/pdfs/HoTTEST.pdf), [video](https://youtu.be/kkfNjqSx4Nw)]

[[!redirects bicubical sets]]


bicycle

A [[cycle]]/[[cocycle]] in [[bivariant cohomology]] is sometimes called a _bi-cycle_.

[[!redirects bi-cycle]]
[[!redirects bi-cycles]]