2-crossed complex

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

##Idea#
[[crossed complex|Crossed complexes]] are a useful extension of [[crossed module]]s allowing not only the encoding of an algebraic model for the [[homotopy 2-type]], but also information on the '[[complex of chains on the universal cover]]'.  The category of crossed complexes is a [[monoidal closed category]] equivalent to various types of [[strict infinity-groupoid]].

To model the [[homotopy 3-type]] of a space, we can use either a [[2-crossed module]] or a [[crossed square]] (or various other algebraic models to be added some time in the future). A [[crossed complex]] is a 'hybrid', part [[crossed module]] but with a 'tail' which is a [[chain complex]]. What would be the 'hybrid' between a 2-crossed module and a chain complex? Are there examples that are easily constructed? What sort of information do they encode? Are they easy to analyse, understand, ... and useful?

##Definition##
 A __2-crossed complex__ is a [[normal complex   of  groups]]

 $$\ldots \to C_n \stackrel{\partial_n}{\longrightarrow} C_{n-1} \longrightarrow \ldots \longrightarrow C_0,$$

 together with a 2-crossed module structure  given on $C_2\to C_1\to C_0$ by a Peiffer lifting function $\{ -,-\} : C_1\times C_1 \to C_2$, such that, on writing $\pi = Coker(C_1\to C_0)$,

1. each $C_n$, $n\geq 3$ and $Ker \,\partial_2$ are  $\pi$-modules and the $\partial_n$ for $n\geq 4$, together with the codomain restriction of $\partial_3$, are $\pi$-module homomorphisms;

1. the   $ \pi $-module structure on $Ker \partial_2$ is the action  induced from the  $C_0$-action on $C_2$ for which the action of $\partial_1 C_1$ is trivial.

 A  2-crossed complex morphism is defined in the obvious way, being compatible with all the actions, the pairings and Peiffer liftings.

We will denote by $2 Crs$, the corresponding category.

##Examples:#

* Any [[2-crossed module]] clearly gives a 2-crossed complex (with trivial 'tail').

* _From simplicial groups to 2-crossed complexes_.  If $G$ is a simplicial group, then

$$ \ldots \to C(G)_3 \to  \frac{\mathcal{N}G_2}{d_0(\mathcal{N}G_3\cap D_3)} \to \mathcal{N}G_1\to \mathcal{N}G_0,$$

has the structure of a 2-crossed complex, where $\mathcal{N}G$ is the [[Moore complex]] of $G$, $D_n$ is the subgroup of $G_n$ generated by the degenerate elements, and, for $n\gt2$,

$${C}(G)_{n} = \frac{\mathcal{N}G_n}{(\mathcal{N}G_n\cap D_n)d_0(\mathcal{N}G_{n+1}\cap D_{n+1})},$$

 is the $n$-dimensional term of the crossed complex, $C(G)$, associated to the simplicial group $G$ as in the entry [[crossed complex]] (in the section **From simplicial group(oid)s to crossed complexes**.)

(There is an obvious extension of the group based definition above to a groupoid based one, and of this construction to one which takes as input a simplicially enriched groupoid.)

The Moore complex of a simplicial group $G$ has the structure of a 2-crossed complex if and only if for each $n\gt 2$, $\mathcal{N}G_n\cap D_n$ is trivial. This means that the axioms of a [[group T-complex]] are almost satisfied, but not necessarily in  dimension 2.

* A quadratic chain complex as defined by H.J. Baues is a special case of a 2-crossed complex, satisfying additional (pre-crossed module) nilpotency condition at the level of the underlying pre-crossed module. (In fact the category of quadratic chain complexes is a reflective subcategory of the category of 2-crossed complexes.) In Baues' book referenced below, there is the construction of the fundamental quadratic chain complex of a pointed CW-complex. The reflection (or cotruncation) of this to the category of quadratic modules (i.e. 3-truncated quadratic chain complexes) faithfully represents  the homotopy 3-type of a CW-space (at the level of spaces and maps between them).

* Graham Ellis defined the fundamental squared complex of a CW-complex from triad homotopy groups and generalised Whitehead products, and showed how Baues fundamental quadratic chain complex of a CW-complex can be obtained from it. A homotopy 2-crossed complex of a CW-complex can also be defined is the same way, see the work of João Faria Martins below.

##Crossed complexes and 2-crossed complexes.
Any [[crossed complex]] can be given the structure of a 2-crossed complex simply by defining a trivial Peiffer lifting, $\{-,-\}$. As the Peiffer lifting covers the Peiffer commutators in $C_1$, and these are trivial (since the bottom of the crossed complex is a crossed module), this trivial Peiffer lifting works and gives a 2-crossed complex structure.  This defines a functor from the category of crossed complexes to that of 2-crossed complexes.

Any 2-crossed complex which has a Peiffer lifting that is trivial $\{x,y\} = 1$, for all $x,y \in C_1$) is isomorphic to a crossed complex in this sense.

This functor, from $Crs$ to $2-Crs$, has a left adjoint which is the identity on the subcategory of $2-Crs$ with trivial Peiffer liftings, so $Crs$ is equivalent to a reflective subcategory of $2-Crs$

##References##

* See the [[Crossed Menagerie]], chapter 5.

* [[H.-J. Baues]], Combinatorial Homotopy and 4-Dimensional Complexes , de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).

* [[Graham Ellis]], Crossed squares and combinatorial homotopy. Mathematische Zeitschrift
Volume 214, Number 1, 93-110, DOI: 10.1007/BF02572393

* [[Joao Faria Martins]], _The fundamental 2-crossed complex of a reduced CW-complex_,  Homology Homotopy Appl. 13(2): 129-157 (2011) (web [pdf] (https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-13/issue-2/The-fundamental-2-crossed-complex-of-a-reduced-CW-complex/hha/1335806746.full).)

[[!redirects 2-crossed complexes]]


2-crossed module

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

#Contents#
* table of contents
{:toc}

##Idea#

A _2-crossed module_ encodes a semistrict 3-group -- a [[Gray-group]] -- in generalization of how a [[crossed module]] encodes a [[strict 2-group]].

A [[simplicial group]] whose [[Moore complex]] has length $1$ (that is, at most stuff in dimensions $0$ and $1$) will be the internal [[nerve]] of a strict $2$-[[2-group|group]] and the Moore complex will be the corresponding [[crossed module]].  What if we have a simplicial group whose Moore complex has at most stuff in dimensions $0$, $1$, and $2$; can we describe its structure in some similar way?  Yes, and Conduché provided a neat description of the structure involved.  From the structure one can rebuild a simplicial group, a type of internal $2$-[[nerve]] construction.

In other words, a $2$-crossed module _is_ the Moore complex of a $2$-[[truncated]] simplicial group.

## Definition

A __$2$-crossed module__ is a [[normal complex of groups]]
$$L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,$$
together with an [[action]] of $N$ on all three groups and a mapping
$$\{ - ,- \} : M\times M \to L$$
such that

1.  the action of $N$ on itself is by conjugation, and $\partial_2$ and $\partial_1$ are $N$-equivariant;

1.  for all $m_0,m_1 \in M$,
    $$\partial_2\{m_0,m_1\} = \,^{\partial_1 m_0}m_1 . m_0m_1^{-1}m_0^{-1};$$

1.  if $\ell_0,\ell_0 \in L$, then
    $$\{\partial_2\ell_0,\partial_2\ell\} = [\ell_1,\ell_0];$$

1.  if $\ell \in L$ and $m\in M$, then
    $$\{m,\partial \ell\}\{\partial \ell,m\} = \,^{\partial m}\ell.\ell^{-1};$$

1.  for all $m_0,m_1,m_2 \in M$,
    * $\{m_0,m_1m_2\} = \{m_0,m_1\}\{ \partial \{m_0,m_2\},(m_0m_1m_0^{-1})\}\{m_0,m_2\}$;
    * $\{m_0m_1,m_2\} = \,^{\partial m_0}\{m_1,m_2\}\{m_0,m_1m_2m_1^{-1}\}$;

1. if $n\in N$ and $m_0,m_1 \in M$, then
    $$ \,^{n} \{m_0,m_1\} = \{ \,^{n}m_0, \,^{n}m_1\}.$$

The pairing $\{ - ,- \} : M\times M \to L$ is often called the **Peiffer lifting**  of the $2$-crossed module.

## Remarks

*  In a $2$-crossed module as above the structure $\partial_2: L \to M$ is a [[crossed module]], but $\partial_1: M\to N$ may not be one, as the Peiffer identity need not hold.  The _[[Peiffer commutator]]_, which measures the failure of that identity, may not be trivial, but it will be a boundary element and the Peiffer lifting gives a structured way of getting an element in $L$ that maps down to it.

*  It is sometimes useful to consider a [[crossed module]] as being a [[crossed complex]] of length 1 (i.e. on possibly non-trivial morphism only).  Likewise one can consider a 2-crossed module as a special case of a [[2-crossed complex]]. Such a gadget is intuitively a 2-crossed module with a 'tail', which is a chain complex of modules over the $\pi_0$ of the base 2-crossed module, much as a crossed complex is a crossed module together with a 'tail'.

* A quadratic module, as developed by [[H.-J. Baues]], is a special case of a 2-crossed module, satisfying nilpotency conditions at the level of the underlying pre-crossed module (which is a close to being a crossed module as possible.)  The fundamental quadratic module of a CW-complex yields an equivalence of categories between the category of pointed 3-types and the category of quadratic modules.

* A functorial fundamental 2-crossed module of a CW-complex can also be defined, by using Graham Ellis fundamental crossed square of a CW-complex; this is explained in the article of  João Faria Martins, below. We can also define this fundamental 2-crossed module of a CW-complex, by using Kan's fundamental simplicial group of a CW-complex, and by applying the usual reflection from simplicial groups to simplicial groups of Moore complex of lenght two, known to be equivalent to 2-crossed modules.

* The homotopy theory of 2-crossed modules can be addressed by noting that 2-crossed modules, inducing a reflective subcategory of the category of simplicial groups, inherit a natural Quillen model structure, as explored in the article of Cabello and Garzon below. A version very close to the usual homotopy theory of crossed complexes was developed in the article of Joao Faria Martins below in a parallel way to the homotopy theory of [[quadratic module]]s and [[quadratic complex]]es as introduced by [[H. J. Baues]].

## Examples

Any [[crossed module]],
$
  G_2 \stackrel{\delta }{\to}{G_1}
$ gives a 2-crossed module, $L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,$ by setting $L = 1$, the trivial group, and, of course, $M = G_2$, $N = G_1$. Conversely any 2-crossed module having  trivial top dimensional group ($L=1$) 'is' a crossed module. This gives an inclusion of the category of crossed modules into that of 2-crossed modules, as a [[reflective subcategory]].

The reflection is given by noting that, if
$$L\stackrel{\partial_2}{\longrightarrow} M \stackrel{\partial_1}{\longrightarrow}N$$
is a 2-crossed module, then $Im\, \partial_2$ is a normal subgroup of $M$, and then there is an obvious induced crossed module structure on
$$\partial_1 : \frac{M}{Im\, \partial_2} \to N.$$

But we can do better than this.  More generally, let
$$\ldots \to 1 \to 1 \to C_3\stackrel{\partial_3}{\longrightarrow} C_2 \stackrel{\partial_2}{\longrightarrow}C_1,$$
be a truncated [[crossed complex]] (of groups) in which all higher dimensional terms are trivial, then taking $L = C_3$, $M = C_2$ and $N = C_1$, with trivial Peiffer lifting, gives one a 2-crossed complex. Conversely suppose we have a 2-crossed module with trivial Peiffer lifting: $\{m_1,m_2\} = 1$ for all $m_1$, $m_2 \in M$, axiom 3 then shows that $L$ is an Abelian group, and similarly the other axioms can be analysed to show that the result is a truncated crossed complex.

This gives:

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

The category $Crs_{2]}$ of crossed complexes of length 2 is equivalent to the full subcategory of $2-CMod$ given by those 2-crossed modules with trivial Peiffer lifting.
=--

Of course, the resulting 'inclusion' has a left adjoint, which is quite fun to check out! (You kill off the subgroup of $L$ generated by the Peiffer lifting, .... is that all?)

### From simplicial groups to 2-crossed modules

If $G$ is a simplicial group then

$$\frac{\mathcal{N}G_2}{d_0(\mathcal{N}G_3)} \to \mathcal{N}G_1\to \mathcal{N}G_0,$$

is a 2-crossed module. (You are invited to find the Peiffer lifting!)

### From crossed squares to 2-crossed modules

Both [[crossed square|crossed squares]] and 2-crossed modules model all connected homotopy 3-types so one naturally asks how to pass from one description to the other.  Going from crossed squares to 2-crossed modules is easy, so will be given here (going back is harder).

Let
$$\array{& L & {\to}^\lambda & M & \\
         \lambda^\prime & \downarrow &&\downarrow & \mu\\
          &N & {\to}_{\nu}& P & \\
}$$
be a crossed square then $N$ acts on $M$ via $P$, so ${}^n m := {}^{\nu(n)}m$, and so we can form $M\rtimes N$ and the sequence
$$L\stackrel{((\lambda')^{-1},\lambda)}{\longrightarrow}M\rtimes N\stackrel{\mu\nu}{\longrightarrow}P$$
is then a 2-crossed complex.

(And, yes, these are actually group homomorphisms: $(\mu,\nu)(m,n) = \mu(m)\nu(n)$, the product of the two elements! Try it!)

The full result and an explanation of what is going on here is given in

* D. Conduché, _Simplicial Crossed Modules and Mapping Cones_, Georgian Math. J., 10, (2003),
623--636

## Related concepts

* [[group]]

* [[2-group]], [[crossed module]], [[differential crossed module]]

* [[3-group]], **2-crossed module** / [[crossed square]], [[differential 2-crossed module]]

* [[n-group]]

* [[∞-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]

[[!redirects 2-crossed modules]]

## References:

* [[H. J. Baues]]: _Combinatorial homotopy and $4$-dimensional complexes._ With a preface by Ronald Brown. de Gruyter Expositions in Mathematics, 2. Walter de Gruyter \& Co., Berlin, 1991.

* Julia  G. Cabello, [[Antonio R. Garzón]]: _Quillen's theory for algebraic models of $n$-types_.  Extracta Math.  9  (1994),  no. 1, 42--47. ([EuDML](https://eudml.org/doc/38395))

* [[P. Carrasco]] and [[T. Porter]], _Coproduct of 2-crossed modules. Applications to a definition of a tensor product for 2-crossed complexes_, Collectanea Mathematica, DOI:10.1007/s13348-015-0156-9.

*  [[Daniel Conduché]], _Modules croisés généralisés de longueur $2$_, in:
Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983). J. Pure Appl. Algebra 34 (1984), no. 2-3, 155--178.

* [[Joao Faria Martins]], _The fundamental 2-crossed complex of a reduced CW-complex_,  Homology Homotopy Appl. 13(2): 129-157 (2011) (web [pdf](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-13/issue-2/The-fundamental-2-crossed-complex-of-a-reduced-CW-complex/hha/1335806746.full).)

* [[Graham Ellis]], _Crossed squares and combinatorial homotopy_, Math. Z.  214  (1993),  no. 1, 93--110.


2-dinatural transformation

# 2-dinatural transformations

* table of contents
{: toc}

## Idea

A *2-dinatural transformation* is a [[categorification]] of a [[dinatural transformation]].  It is more complicated since there are three kinds of [[opposite 2-category]], and also because it could be [[lax natural transformation|lax]], colax, or pseudo as well as strict.

## Definition

Let $C$ and $D$ be [[2-categories]] and let
$$F,G: C^{coop}\times C^{op}\times C^{co} \times C\to D$$
be [[2-functors]].  A **lax 2-dinatural transformation** $\alpha:F\to G$ consists of

1. For each $x\in C$, a 1-morphism component
   $$ \alpha_x : F(x,x,x,x) \to G(x,x,x,x) $$

2. For each 1-morphism $f:x\to y$ in $C$, a 2-morphism component from the composite
   $$ F(y,y,x,x) \xrightarrow{F(f,f,1,1)} F(x,x,x,x) \xrightarrow{\alpha_x} G(x,x,x,x) \xrightarrow{G(1,1,f,f)} G(x,x,y,y) $$
   to the composite
   $$ F(y,y,x,x) \xrightarrow{F(1,1,f,f)} F(y,y,y,y) \xrightarrow{\alpha_y} G(y,y,y,y) \xrightarrow{G(f,f,1,1)} G(x,x,y,y).$$
   (This is a generalization of the hexagon identity for an ordinary [[dinatural transformation]].)

3. For any 2-morphism $\mu:f\to g:x\to y$ in $C$, the two composites
   $$ G(1,1,g,f)\circ \alpha_x \circ F(g,f,1,1)
   \xrightarrow{G(1,1,\mu,1) \ast 1 \ast F(\mu,1,1,1)}
   G(1,1,f,f) \circ \alpha_x \circ F(f,f,1,1)
   \xrightarrow{\alpha_f}
   G(f,f,1,1) \circ \alpha_y \circ F(1,1,f,f)
   \xrightarrow{G(1,\mu,1,1) \ast 1 \ast F(1,1,1,\mu)}
   G(f,g,1,1) \circ \alpha_y \circ F(1,1,f,g)
   $$
   and
   $$ G(1,1,g,f)\circ \alpha_x \circ F(g,f,1,1)
   \xrightarrow{G(1,1,1,\mu) \ast 1 \ast F(1,\mu,1,1)}
   G(1,1,g,g) \circ \alpha_x \circ F(g,g,1,1)
   \xrightarrow{\alpha_g}
   G(g,g,1,1) \circ \alpha_y \circ F(1,1,g,g)
   \xrightarrow{G(\mu,1,1,1) \ast 1 \ast F(1,1,\mu,1)}
   G(f,g,1,1) \circ \alpha_y \circ F(1,1,f,g)
   $$
   are equal.

4. Each 2-morphism component $\alpha_{1_x}$ is the identity $1_{\alpha_x}$.

5. For any composable 1-morphisms $x\xrightarrow{f} y \xrightarrow{g} z$, the composite of $\alpha_f$ and $\alpha_g$ (together with two functoriality commutative squares for $F$ and $G$) is equal to $\alpha_{g f}$.

## Examples

If one of $F$ and $G$ is constant and the other depends only on $C^{op}\times C$, we obtain a notion of **2-extranatural transformation**.

If $F$ and $G$ each depend on only one factor in $C^{coop}\times C^{op}\times C^{co} \times C$, we obtain a notion of strict 2-natural transformation from a 2-functor of any arbitrary variance to a 2-functor of any other arbitrary variance.  For example, if $F:C^{co}\to D$ and $G:C^{op}\to D$, then a lax 2-dinatural transformation $\alpha:F\to G$ consists of an $\ob(\mathbf{C})$-indexed family of $1$-morphisms $FX\stackrel{\alpha_X}{\to} GX$ in $\mathbf{D}$, and for each two objects $X,Y$ of $\mathbf{C}$, an $\ob[X,Y]$-indexed family of $2$-morphisms $\alpha_f$, so that for every $2$-morphism $f\stackrel{\gamma}{\Rightarrow} g$, we have the commutative diagram of $2$-morphisms in $\mathbf{D}$:
$$
\array{
&&FX&\stackrel{Ff}{\rightarrow}&FY\\
&&\alpha_X\downarrow&\stackrel{\alpha_f}{\Rightarrow}&\downarrow\alpha_Y&&FX\\
&&GX&\stackrel{Gf}{\leftarrow}&GY&&\downarrow Ff\\
FX&\stackrel{id}{\neArrow}&&&& \stackrel{G\gamma.(\alpha_Y\circ Ff)}{\seArrow}&FY\\
\alpha_X\downarrow&&&&&&\downarrow\alpha_Y\\
FY&\stackrel{id}{\seArrow}&&&&\stackrel{(Gg\circ\alpha_Y).F\gamma}{\neArrow}&GY\\
&&FX&\stackrel{Fg}{\rightarrow}&FY&&\downarrow Gg\\
&&\alpha_X\downarrow&\stackrel{\alpha_g}{\Rightarrow}&\downarrow\alpha_Y&&GX\\
&&GX&\stackrel{Gg}{\leftarrow}&GY
}$$
where . is whiskering/horizontal composition. Furthermore, given composable $1$-morphisms $X\stackrel{f}{\rightarrow}Y\stackrel{h}{\rightarrow} Z$, the $2$-morphisms $\alpha_X\stackrel{\alpha_f}{\Rightarrow}Gf\circ\alpha_Y\circ Ff$ and $\alpha_Y\stackrel{\alpha_h}{\Rightarrow}Gh\circ\alpha_Z\circ Fh$ are related via the formula $\alpha_{h\circ f}=(Gf.\alpha_h.Ff)\circ\alpha_f$, which says that the [[pasting diagram]] of $2$-morphisms:
$$
\array{
FX&\stackrel{Ff}{\rightarrow}&FY&\stackrel{Fh}{\rightarrow}&FZ\\
\alpha_X\downarrow&\stackrel{\alpha_f}{\Rightarrow}&\downarrow\alpha_Y&\stackrel{\alpha_h}{\Rightarrow}&\downarrow\alpha_Z\\
GX&\stackrel{Gf}{\leftarrow}&GY&\stackrel{Gh}{\leftarrow}&GZ
}$$
reduces to
$$
\array{
FX&\stackrel{Fh\circ Ff}{\rightarrow}&FZ\\
\alpha_X\downarrow&\stackrel{\alpha_{h\circ f}}{\Rightarrow}&\downarrow\alpha_Z\\
GX&\stackrel{Gh\circ Gf}{\leftarrow}&GZ
}$$

This sort of transformation appears in the [[category of V-enriched categories]], which is a $2$-category which comes with a [[unit enriched category]] $\mathcal{I}$ and either a lax natural transformation $[\mathcal{I},-]^{op}\Rightarrow[[\mathcal{I},-],V_0]$ (in the case of $\mathcal{V}$ a [[monoidal category|monoidal structure]] on $V$), or a lax natural transformation $[\mathcal{I},-]^{op}\Rightarrow[-,V^e]$ (in the case of $\mathcal{V}$ a [[closed category|closed structure]] on $[\mathcal{I},V^e]\cong V_0$).

[[!redirects 2-dinatural transformations]]
[[!redirects strict 2-dinatural transformation]]
[[!redirects strict 2-dinatural transformations]]
[[!redirects lax 2-dinatural transformation]]
[[!redirects lax 2-dinatural transformations]]
[[!redirects pseudo 2-dinatural transformation]]
[[!redirects pseudo 2-dinatural transformations]]
[[!redirects colax 2-dinatural transformation]]
[[!redirects colax 2-dinatural transformations]]


2-functor

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

#Contents#
* table of contents
{:toc}

## Idea

A __$2$-functor__ is the [[categorification]] of the notion of a [[functor]] to the setting of [[2-category|2-categories]]. At the 2-categorical level there are several possible versions of this notion one might want depending on the given setting, some of which collapse to the standard definition of a functor between categories when considered on $2$-categories with discrete hom-categories (viewed as $1$-categories). The least restrictive of these is a [[lax functor]], and the strictest is (appropriately) called a [[strict 2-functor]].

For the various separate definitions that do collapse to standard functors, see:

*  [[strict 2-functor]]
*  [[pseudo functor]]

There is also a notion of '[[lax functor]]', however this notion does not necessarily yield a standard functor when considered on discrete hom-categories.

For the generalisation of this to higher categories, see [[semistrict higher category]].

## Definition

Here we present explicitly the definition for the middling notion of a pseudofunctor, and comment on alterations that yield the stronger and weaker notions.

#### Pseudofunctor between strict $2$-categories

Let $\mathfrak{C}$ and $\mathfrak{D}$ be strict [[2-categories]]. A pseudofunctor $P:\mathfrak{C}\to\mathfrak{D}$ consists of

* A function $P:Ob_\mathfrak{C}\to Ob_\mathfrak{D}$.

* For each pair of objects $A,B\in Ob_\mathfrak{C}$ a functor

$$

P_{A,B}:\mathfrak{C}(A,B)\to\mathfrak{D}(P(A),P(B)).

$$

We will generally write the function and functors as $P$.

* For each triplet of objects $A,B,C\in Ob_\mathfrak{C}$, a natural isomorphism

\begin{centre}

\begin{xymatrix@C20mm}

\mathfrak{C}(B,C)\times\mathfrak{C}(A,B) \rtwocell^{P\circ\Gamma}_{\Gamma\circ(P\times P)}{\gamma} & \mathfrak{D}(P(A),P(C))

\end{xymatrix}

\end{centre}

whose components are $2$-cell isomorphisms $\gamma_{f,g}:P(f\circ g) \Rightarrow P(f)\circ P(g)$ as below

\begin{centre}

\begin{xymatrix@C20mm}

P(A) \rtwocell_{P(f)\circ P(g)}^{P(f\circ g)}{\;\;\;\;\gamma_{f,g}} & P(C)

\end{xymatrix}

\end{centre}

* For each object object $A\in Ob_\mathfrak{C}$, a natural isomorphism

\begin{centre}

\begin{xymatrix@C20mm}

1 \rtwocell^{P\circ id_A\;\;\;\;\;\;}_{id_{P(A)}\;\;\;\;\;\;}{\iota\;\;\;\;\;\;\;\;\;\;} & \mathfrak{D}(P(A),P(A))

\end{xymatrix}

\end{centre}

where $1$ denotes the terminal category and $id_A$ is the identity-selecting functor at $A$. Its component is a $2$-cell isomorphism $\iota_{_*}:P(1_A)\Rightarrow 1_{P(A)}$ as below

\begin{centre}

\begin{xymatrix@C20mm}

P(A) \rtwocell_{1_{P(A)}}^{P(1_A)}{\;\;\;\;\iota_*} & P(A)

\end{xymatrix}

\end{centre}

These are subject to the following axioms:

1. For any composable triplet of $1$-cells $(f,g,h)\in Ob_{\mathfrak{C}(C,D)}\times Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$ we have that
$$
(\gamma_{f,g}\star 1_{P(h)})\circ\gamma_{f\circ g,h}=(1_{P(f)}\star\gamma_{g,h})\circ\gamma_{f,g\circ h},
$$
where $\circ$ denotes vertical composition and $\star$ denotes horizontal composition, as illustrated by the following commutative $2$-cell diagram in $\mathfrak{D}(P(A),P(D))$:

\begin{centre}

\begin{xymatrix@R20mm@C20mm}

P(f)\circ P(g)\circ P(h) & \ar@2{->}[l]_{1_{P(f)}\star\gamma_{g,h}} P(f)\circ P(g\circ h) \\ P(f\circ g)\circ P(h) \ar@2{->}[u]^{\gamma_{f,g}\star 1_{P(h)}} & \ar@2{->}[l]^{\gamma_{f\circ g,h}} \ar@2{->}[u]_{\gamma_{f,g\circ h}} P(f\circ g\circ h)

\end{xymatrix}

\end{centre}

2. For any composable $1$-cells $(f,g)\in Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$ we have that
$$
\iota_B\star 1_{P(g)}=\gamma_{1_B,g}^{-1},
$$
$$1_{P(f)}\star\iota_B=\gamma_{f,1_B}^{-1},
$$
as illustrated by the commutative $2$-cell diagrams below

\begin{centre}

\begin{xymatrix@R30mm@C35mm}

P(A)\rtwocell^{P(g)}_{P(g)}{\;\;\;\;\;\;1_{P(g)}} \drtwocell<5.5>_{P(g)\;\;\;\;\;}^{\;\;\;\;\;\;\;\;\;\;\;\;\;\; P(1_B)\circ P(g)}{\;\;\;\;\;\;\;\gamma_{1_B,g}^{-1}} & P(B) \dtwocell^{\;\;\;\;\;\;P(1_B)}_{1_{P(B)\;\;\;\;\;\;}}{\iota_B} \\ & P(B)

\end{xymatrix}              \begin{xymatrix@R30mm@C35mm}

P(B) \rtwocell^{P(1_B)}_{1_{P(B)}}{\;\;\;\;\iota_B} \drtwocell<5.5>_{P(f)\;\;\;\;\;}^{\;\;\;\;\;\;\;\;\;\;\;\;\;\; P(f)\circ P(1_B)}{\;\;\;\;\;\;\;\gamma_{f,1_B}^{-1}} & P(B) \dtwocell<4>^{\;\;\;\;P(f)}_{P(f)\;\;\;\;}{1_{P(f)}} \\ & P(C)

\end{xymatrix}

\end{centre}

#### Lax 2-Functor

To obtain the notion of a __lax functor__ we only require that the coherence morphisms $\gamma_{f,g}$ and $\iota_A$ be $2$-cells, not necessarily $2$-cell isomorphisms.  This prevents us from going back and forth between preimages and images of identity $1$-cells and horizontally composed $1$-cells/$2$-cells.  Similarly, to obtain an __oplax functor__ we reverse the direction of these 2-cells.

#### Strict 2-Functor

To obtain the notion of a strict $2$-functor we require that $\gamma_{f,g}$ and $\iota_A$ are identity arrows, so horizontal composition and $1$-cell identities literally factor through each functor in the same way vertical composition and $2$-cell identities do.

\begin{remark}
There is a notion of a 'weak 2-category', however it usually doesn\'t make sense to speak of strict $2$-functors between weak $2$-categories[^1], but it does make sense to speak of lax (or 'weak') $2$-functors between strict $2$-categories.  Indeed, the weak $3$-[[3-category|category]] [[Bicat]] of bicategories, pseudofunctors, [[pseudonatural transformations]], and [[modifications]] is  [[equivalence of categories|equivalent]] to its full sub-3-category spanned by the strict 2-categories.  However, it is not equivalent to the $3$-category [[Str2Cat]] of strict $2$-categories, *strict* $2$-functors, transformations, and modifications. (For discussion of the terminological choice "$2$-functor" and $n$-functor in general, see [[higher functor]].)
\end{remark}

## Properties

\begin{prop}\label{CharacterizationOfEquivalencesOf2Categories}
**(recognition of [[equivalences of 2-categories]] assuming the [[axiom of choice]])**
\linebreak
  Assuming the [[axiom of choice]], a 2-functor $F \,\colon\, \mathcal{C} \xrightarrow{\;} \mathcal{D}$ is an [[equivalence of 2-categories]] precisely if it is

1. **essentially surjective**:

   [[surjection|surjective]] on [[equivalence in a 2-category|equivalence]] [[equivalence classes|classes]] of [[objects]]: $\pi_0(F) \;\colon\; \pi_0(\mathcal{C}) \twoheadrightarrow \pi_0(\mathcal{D})\;$,

1. **fully faithful** (e.g. [Gabber & Ramero 2004, Def. 2.4.9 (ii)](#GabberRamero04)):

   for each [[pair]] of [[objects]] $X,\, Y \in \mathcal{C}$ the component functor is an [[equivalence of categories|equivalence of]] [[hom-categories]] $F_{X,Y} \,\colon\, \mathcal{C}(X,Y) \xrightarrow{\simeq} \mathcal{D}\big(F(X), F(Y)\big)$,

   which by the analogous theorem for 1-functors ([this Prop.](equivalence+of+categories#ViaEssentiallySurjectiveAndFullyFaithful)) means equivalently that $F$ is (e.g. [Johnson & Yau 2020, Def. 7.0.1](#JohnsonYau20))

   1. **essentially full on 1-cells**:

      namely that each component functor $F_{X,Y}$ is an [[essentially surjective functor]];

   1. **fully faithful on 2-cells**:

      namely that each component functor $F_{X,Y}$ is a [[fully faithful functor]].

\end{prop}

This is classical [[folklore]]. It is made explicit in, e.g. [Gabber & Ramero 2004, Cor. 2.4.30](#GabberRamero04); [Johnson & Yau 2020, Thm. 7.4.1](#JohnsonYau20).

## Related concepts

* [[function]]

* [[functor]]

* **2-functor** / [[pseudofunctor]] / [[(2,1)-functor]]

* [[n-functor]]

* [[(∞,1)-functor]]

* [[(∞,n)-functor]]

[[!include properties of functors -- contents]]

## References

Textbook accounts:

* {#GabberRamero04} [[Ofer Gabber]], [[Lorenzo Ramero]], Def. 2.1.14 in: *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584))

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

[^1]: Although there are certain contexts in which it does.  For instance, there is a [[model structure]] on the category of [[bicategories]] and strict 2-functors between them, which models the homotopy theory of bicategories and weak 2-functors.

[[!redirects 2-functors]]
[[!redirects strict 2-functor]]
[[!redirects strict 2-functors]]


2-gerbe

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

#Contents#
* table of contents
{:toc}

## Definition

### Nonabelian 2-gerbes

For $\mathcal{X}$ an [[(∞,1)-topos]], a **2-gerbe** $P$ in $\mathcal{X}$ is an object which is

1. [[1-connective]];

1. [[2-truncated]].

The first condition says that it is an [[(∞,1)-sheaf]] with values in [[2-groupoid]]s. The second says that $P \to *$ is an [[effective epimorphism]] and that the [[categorical homotopy groups in an (∞,1)-topos|0-th homotopy sheaf]] is the terminal sheaf. In the literature this is often stated as saying that $P$ is a) _locally connected_ and b) _locally non-empty_ .

### Abelian 2-gerbes

For $\mathcal{X}$ an [[(∞,1)-topos]], an  **abelian 2-gerbe** $P$ in $\mathcal{X}$ is an object which is

1. [[2-truncated]];

1. [[2-connected]].

## Related concepts

* [[principal bundle]] / [[torsor]] / [[associated bundle]]

* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]

* [[principal 3-bundle]] / **2-gerbe** / [[bundle 2-gerbe]]

* [[principal ∞-bundle]] / [[associated ∞-bundle]]  / [[∞-gerbe]]

## References

A comprehensive discussion of nonabelian 2-gerbes is in

* [[Lawrence Breen]], _On the classification of 2-gerbes and 2-stacks_ , Astérisque 225 (1994) ([numdam:AST_1994__225__1_0](http://www.numdam.org/item/?id=AST_1994__225__1_0))

A more expository discussion is in

* [[Lawrence Breen]], _Notes on 1- and 2-gerbes_ in [[John Baez]], [[Peter May]] (eds.) _[[Towards Higher Categories]]_ ([arXiv:math/0611317](http://arxiv.org/abs/math/0611317)).
  {#Breen}

Abelian 2-gerbes are a special case (see [[∞-gerbe]]) of the discussion in section 7.2.2 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

See also

* [[Ettore Aldrovandi]], _2-Gerbes bound by complexes of gr-stacks, and cohomology_ Journal of Pure and Applied Algebra 212 (2008), 994–103 ([pdf](http://www.math.fsu.edu/~aluffi/archive/paper271.pdf))

[[!redirects 2-gerbes]]


2-Giraud theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _2-Giraud theorem_ is the generalization of [[Giraud's theorem]] from [[topos theory]] to [[2-topos theory]].

## Statement

The following theorem, which generalizes the classical [[Giraud theorem]], is due to [[StreetCBS]].

+-- {: .num_theorem #Giraud}
###### Theorem
For a [[2-category]] $K$, the following are equivalent.

* $K$ is equivalent to the 2-category of [[2-sheaves]] on a small [[2-site]].
* $K$ is an infinitary [[2-pretopos]] with a small [[eso-generator]].
* $K$ is a [[reflective sub-2-category]] of a category $[C^{op},Cat]$ of [[2-presheaves]] with left-exact reflector.
=--

In fact, it is not hard to prove the same theorem for [[n-categories]], for any $1\le n\le 2$.

+-- {: .num_theorem #Giraud}
###### Theorem
For an [[n-category]] $K$, the following are equivalent.

* $K$ is equivalent to the $n$-category of [[n-sheaves]] on a small [[n-site]].
* $K$ is an infinitary [[n-pretopos]] with a small [[eso-generator]].
* $K$ is a reflective sub-$n$-category of a category $[C^{op},n Cat]$ of $n$-presheaves with left-exact reflector.
=--

For $n=2$ this is Street's theorem; for $n=1$ it is the classical theorem.  The other values included are of course $n=(1,2)$ and $n=(2,1)$.

## References

* [[Mike Shulman]], _[[michaelshulman:2-Giraud theorem]]_


2-Grothendieck construction

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

#Contents#
* table of contents
{:toc}

## Idea

Generalization of the [[Grothendieck construction]] from [[categories]] to [[2-categories]] and [[bicategories]].

## Related concepts

* [[(infinity,1)-Grothendieck construction|$(\infty,1)$-Grothendieck construction]]

## References

Definition of a bicategory of elements construction for pseudo-distributors/pseudo-profunctors:

* [[Ross Street]], §1.10 of: _Fibrations in bicategories_, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]], **21** 2 (1980) 111-160 [[numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1980__21_2_111_0)]

More exhaustive references:

* [[Igor Baković]], *Grothendieck construction for bicategories* [[pdf](https://www2.irb.hr/korisnici/ibakovic/sgc.pdf)]

* [[Igor Baković]], *Fibrations of bicategories* [[pdf](https://www2.irb.hr/korisnici/ibakovic/groth2fib.pdf)]

* [[Mitchell Buckley]]. _Fibred 2-categories and bicategories_, Journal of pure and applied algebra **218** 6 (2014) 1034-1074 [[doi](https://doi.org/10.1016/j.jpaa.2013.11.002)]

See also:

* [[Luca Mesiti]], *The 2-Set-enriched Grothendieck construction and the lax normal conical 2-limits* [[arXiv:2302.04566](https://arxiv.org/abs/2302.04566)]

[[!redirects 2-Grothendieck constructions]]

[[!redirects Grothendieck construction for 2-categories]]
[[!redirects Grothendieck constructions for 2-categories]]

[[!redirects Grothendieck construction for bicategories]]
[[!redirects Grothendieck constructions for bicategories]]


2-group

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of  _2-group_ is a [[vertical categorification]] of the notion of _[[group]]_.

It is the special case of an [[n-group]] for $n=2$, equivalently an [[∞-group]] which is [[n-truncated|1-truncated]]. Under the [[looping and delooping]]-equivalence, 2-groups are equivalent to [[pointed object|pointed]] [[n-connected|connected]] [[homotopy n-type|homotopy 2-types]].

Somewhat more precisely, a _$2$-group_ is a [[group object]] in the [[(2,1)-category]] of [[groupoids]].  Equivalently, it is a [[monoidal groupoid]] in which the [[tensor product]] with any [[object]] has an [[inverse]] up to [[isomorphism]].  Also equivalently, by the [[looping and delooping]]-equivalence, it is a [[pointed object|pointed]] [[2-groupoid]] with a single [[equivalence class]] of objects.

Like other notions of [[higher category theory]], $2$-groups come in weak and strict forms, depending on how you interpret the above.

### Strict $2$-groups

The earliest version studied is that of [[strict 2-group]]s.

A __strict $2$-group__ consists of:

* a collection of [[group]] homomorphisms of the form
  $$
    C_1 \stackrel{s,t}{\to} C_0 \stackrel{i}{\to} C_1
  $$
such that the composites $s\cdot i$ and $t\cdot i$ are the identity morphisms on $C_0$, and
  such that, writing  $C_1 \times_{t,s} C_1$ for the pullback,
  $$
    \array{
      C_1 \times_{t,s} C_1 &\to& C_1
      \\
      \downarrow && \downarrow^{t}
      \\
      C_1 &\stackrel{s}{\to}& C_0
    }
  $$
  there is, in addition,  a homomorphism
  $$
     C_1 \times_{t,s} C_1 \stackrel{comp}{\to} C_1
  $$
  "respecting $s$ and $t$";

* such that the _composition_ $comp$ is associative
  and unital with respect to $i$ "in the obvious way".

See [[strict 2-group]] for further discussion and examples.

### Weak $2$-groups

A __weak $2$-group__, or simply __$2$-group__, is a (weak) [[monoidal category]] where every morphism is invertible and *such that*:

*  given any object $x$, there exists an object $x^{-1}$ such that the monoidal products $x \otimes x^{-1}$ and $x^{-1} \otimes x$ are each [[isomorphism|isomorphic]] to the monoidal unit $1$.

A __coherent $2$-group__ is a monoidal category where every morphism is invertible and *equipped with*:

*  for each object $x$ a specific object $x^{-1}$ and specific [[isomorphism]]s from $x \otimes x^{-1}$ and $x^{-1} \otimes x$ to $1$ which form an [[adjoint equivalence]].

A theorem in HDA V (see references) shows that every weak $2$-group may be made coherent.  For purposes of [[internalization]], one probably wants to use the coherent version.

## Definition

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

The [[(2,1)-category]] $2Grp$ of **2-groups** is equivalently

* the [[full sub-(∞,1)-category|full sub-(2,1)-category]] of that of [[monoidal categories]] and [[strong monoidal functors]] on those that are [[groupoids]] and whose [[tensor product]] has weak [[inverses]] for each object;

* the [[full sub-(∞,1)-category]] of that of [[∞-groups]] on the [[n-truncated|1-truncated]] objects;

* the [[full sub-(∞,1)-category]] of that of [[group object in an (∞,1)-category|group objects]] in [[∞Grpd]] on the [[n-truncated|1-truncated]] objects;

* the [[full sub-(∞,1)-category]]

  $$
    \infty Grpd_{1 \leq \bullet \leq 2}^{*/}
    \hookrightarrow
    \infty Grpd^{*/}
  $$

  of [[∞Grpd]]$^{*/}$ on those objects which are both [[n-connected|connected]] as well as [[n-truncated|2-truncated]].

=--

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

The last equivalent characterization is related to the previous ones by the [[looping and delooping]]-equivalence

$$
  \array{
    Grp(\infty Grpd)
    &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
    \infty Grpd^{*/}_{\geq 1}
    \\
    \uparrow^{\mathrlap{full\;inc.}}
    &&
    \uparrow^{\mathrlap{full\;inc.}}
    \\
    2 Grp
    &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
    2Grpd_{\geq 1}^{*/}
  }
  \,.
$$

Here $(-)^{*/}$ denotes taking [[pointed objects]], hence the [[over-(∞,1)-category|slice under]] the point, and $(-)_{\geq}$ denotes the full [[sub-(∞,1)-category|full inclusion]] on [[n-connected|connected]] objects.

=--

By replacing in the last of these equivalent characterizations the ambient [[(∞,1)-topos]] [[∞Grpd]] with any other one, to be denoted $\mathbf{H}$, obtains notions of 2-groups with extra structure. For instance for $\mathbf{H} = $ [[Smooth∞Grpd]] the $(\infty,1)$-topos of [[smooth ∞-groupoids]] one obtains:

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

The [[(2,1)-category]] $Smooth2Grp$ of **smooth 2-groups** is

$$
  \array{
    Grp(Smooth \infty Grpd)
    &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
    Smooth\infty Grpd^{*/}_{\geq 1}
    \\
    \uparrow^{\mathrlap{full\;inc.}}
    &&
    \uparrow^{\mathrlap{full\;inc.}}
    \\
    Smooth 2 Grp
    &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
    Smooth2Grpd_{\geq 1}^{*/}
  }
  \,.
$$

=--

Below in [presentation by crossed modules](#PresentationByCrossedModules) are discussed more explict presentations of $2Grp$ and $Smooth2Grpd$ etc. by explicit algebraic data.

## Properties

### Presentation by crossed modules
 {#PresentationByCrossedModules}

By the discussion there, every _[[∞-group]]_ has a _presentation_ by a [[simplicial group]]. More precisely, the [[(∞,1)-category]], $\infty Grp$, is [[presentable (∞,1)-category|presented]] by the [[model structure on simplicial groups]] (for instance under [[simplicial localization]])

$$
  \infty Grpd \simeq L_W Grp^{\Delta^{op}}
  \,.
$$

Moreover, if $G \in Grp^{\Delta^{op}}$ is an [[n-group]], then it is equivalent to a [[coskeleton|n-coskeletal]] simplicial group. For $n = 2$ one finds that these are naturally identified with _[[crossed modules]]_ of groups (see there for more details).

In conclusion, this means that

+-- {: .num_prop #PresentationOf2GrpByCrossedModules}
###### Proposition

The [[(2,1)-category]] $2Grp$ of 2-groups is [[equivalence of (infinity,1)-categories|equivalent]] to the [[simplicial localization]] of the [[category with weak equivalences]] whose

* objects are [[crossed modules]]

* morphisms are [[homomorphisms]] of crossed modules;

* weak equivalences are those morphisms of crossed modules which correspond to [[weak homotopy equivalences]] of the corresponding simplicial groups.

=--

A straightforward analysis shows that

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

For $(G_1 \stackrel{\delta}{\to} G_0, G_0 \stackrel{\alpha}{\to} Aut(G_1))$ a [[crossed module]], the [[homotopy groups]] of the corresponding [[2-group]]/[[simplicial group]] are

* $\pi_0 = G_0 / im(\delta)$ (the [[quotient]] of $G_0$ by the [[image]] of $\delta$, which is necessarily a [[normal subgroup]] of $G_0$);

* $\pi_1 = ker(\delta)$ (the [[kernel]] of $\delta$).

Accordingly, a weak equivalence of crossed modules $f : G \to H$ is a morphism of crossed modules which induces an [[isomorphism]] of kernel and cokernel of $\delta_G$ with that of $\delta_H$.

=--

Similar statements hold for 2-groups with extra structure. For instance the $(2,1)$-category $Smooth2Grp$ of smooth 2-groups is equivalent to the [[simplicial localization]] of the category whose

* objects are [[sheaves]] of [[crossed modules]] on [[CartSp]]${}_{smooth}$;

* weak equivalences are those morphisms of sheaves of crossed modules which on every [[stalk]] induce weak equivalences of crossed modules as above.

(See the discussion at [[Smooth∞Grpd]] for more on this.)

## Examples

### Specific examples

#### Picard 2-group

* [[Picard 2-group]]

#### Automorphism 2-groups

For $C$ any [[2-category]] and $c \in C$ any object of it, the category $Aut_C(c) \subset Hom_C(c,c)$ of auto-equivalences of $c$ and invertible 2-morphisms between these is naturally a 2-group, whose group product comes from the horizontal composition in $C$.

If $C$ is a [[strict 2-category]] there is the notion of strict [[automorphism 2-group]]. See there for more details on that case.

For instance if $C = Grp_2 \subset Grpd$ is the 2-category of [[group]] obtained by regarding groups as one-object [[groupoid]]s, then for $H \in Grp$ a group, its automorphism 2-group obtained this way is the strict 2-group

$$
  AUT(H) := Aut_{Grp_2}(H)
$$

corresponding to the [[crossed module]] $(H \stackrel{Ad}{\to} Aut(H))$, where $Aut(H)$ is the ordinary [[automorphism group]] of $H$.

#### Inner automorphism 2-groups

See [[inner automorphism 2-group]].

#### String 2-group

See [[string 2-group]].

#### Platonic 2-group

See _[[Platonic 2-group]]_

### Equivalences of 2-groups
 {#ExamplesForEquivalencesOf2Groups}

We discuss some weak equivalences in the [[category with weak equivalences]] of [[crossed modules]] and crossed module homomorphisms, which presents $2Grp$ by the discussion [above](#PresentationByCrossedModules).

#### From inclusions of normal subgroups
 {#FromInclusionsOfNormalSubgroups}

Let $G$ be a [[group]] and $N \hookrightarrow G$ the inclusion of a [[normal subgroup]]. Equipped with the canonical [[action]] of $G$ on $N$ by [[conjugation]], this inclusion constitutes a [[crossed module]]. There is a canonical morphism of crossed modules from $(N \hookrightarrow G)$ to $(1 \to G/N)$, hence to the ordinary [[quotient]] group, regarded as a crossed module.

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

The morphism $(N \hookrightarrow G) \to G/N$ is a weak equivalence of crossed modules,  prop. \ref{PresentationOf2GrpByCrossedModules}. Accordingly, it presents an [[equivalence in an (infinity,1)-category|equivalence]] of 2-groups.

=--

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

The canonical morphism in question is given by the commuting diagram of groups

$$
  \array{
    N &\stackrel{f_1}{\to}& 1
    \\
    \downarrow && \downarrow
    \\
    G &\stackrel{f_0}{\to}& G/N
  }
  \,.
$$

By prop. \ref{HomotopyGroupsOfCrossedModule} we need to check that this induces an isomorphism on the [[kernel]] and [[cokernel]] of the vertical morphisms.

The kernel of the left vertical morphism is the trivial group, because $N \hookrightarrow G$ is an inclusion, by definition. Clearly also the kernel of the right vertical morphisms is the trivial group. Hence $f_1$ restricted to the kernels is the unique morphism from the trivial group to itself, hence is an isomrphism.

Moreover, the cokernel of the left vertical morphism is of course the quotient $G/N$ and $f_0$, being the quotient map, is manifestly an isomorphism on cokernels.

=--

This class of weak equivalence plays an important role as constituting [[∞-anafunctor|2-anafunctors]] that exhibit long [[fiber sequence]] extensions of [[short exact sequences]] of [[central extensions]] of groups.

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

Let $A \to \hat G$ be the inclusion of a [[center|central]] subgroup, exhibiting a [[central extension]] $A \to \hat G \to G$ with $G := \hat G/A$. Then this [[short exact sequence]] of groups extends to a long [[fiber sequence]] of 2-groups

$$
  A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A
  \,,
$$

where $\mathbf{B}A$ denotes the 2-group given by the [[crossed module]] $(A \to 1)$, and similarly for the other cases.

Here the [[connecting homomorphism]] $G \to \mathbf{B}A$ is presented in the category of crossed modules by a zig-zag / [[anafunctor]] whose left leg is the above weak equivalence:

$$
  (1 \to G) \stackrel{\simeq}{\leftarrow}
  (A \to \hat G)
   \to
  (A \to 1)
  \,.
$$

=--

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

For smooth 2-groups, useful examples of the above are smooth refinements of various [[universal characteristic classes]]:

* the second [[Stiefel-Whitney class]]

  $$
    w_2 : \mathbf{B}SO \to \mathbf{B}^2\mathbb{Z}_2
  $$

  is induced this way from the central extension $\mathbb{Z}_2 \to Spin \to SO$ of the [[special orthogonal group]] by the [[spin group]];

* the first [[Chern class]]

  $$
    c_1 : \mathbf{B}U(1) \to \mathbf{B}^2 \mathbb{Z}
  $$

  induced from the central extension $\mathbb{Z} \to \mathbb{R} \to U(1)$.

=--

## Related concepts

* [[group]]

  * [[groupoid]], [[monoidal groupoid]]

  * **2-group**, [[crossed module]], [[differential crossed module]]

    * [[braided 2-group]], [[symmetric 2-group]]

    * [[quantum 2-group]]

  * [[3-group]], [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]

  * [[n-group]]

  * [[∞-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]

  * [[group stack]]

    [[smooth 2-group]]

* [[higher gauge symmetry]]

* [[generalized global symmetry]]

[[!include homotopy n-types - table]]

## References
 {#References}

### Original articles
 {#OriginalArticles}

The notion of 2-groups first appears in

* {#Solian72} [[Alexandru Solian]], *Groupe dans une catégorie*, C. R. Acad. Sc. Paris **275** (1972) [[ark:/12148/bpt6k57310477/f7](https://gallica.bnf.fr/ark:/12148/bpt6k57310477/f7.item)]

* {#Solian80} [[Alexandru Solian]], *Coherence in categorical groups*,  Communications in Algebra **9** 10 (1980) 1039-1057 [[doi:10.1080/00927878108822631](https://doi.org/10.1080/00927878108822631)]

and in

* {#Sinh73} [[Hoàng Xuân Sính]], _Gr_-catégories, PhD thesis, Hanoi (1973, 1975) [[web](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html), [pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh_I.pdf), [[Sinh_Gr-Categories.pdf:file]]]

[Sinh (1973)](#Sinh73) was supervised by [[Alexander Grothendieck]] and showed that 2-groups are classified, up to equivalence, by quadruples consisting of:

* a group $G$
* an abelian group $A$
* an action of $G$ as automorphisms of $A$
* an element of $H^3(G,A)$, often called the *Sinh invariant*.

She later published two papers on the subject:

* {#Sinh78} [[Hoàng Xuân Sính]],  *Gr-catégories strictes*, Acta Mathematica Vietnamica **3** 2 (1978) 47-59 [[web](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html), [pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Gr_categories_strictes.pdf)]

* {#Sinh83} [[Hoàng Xuân Sính]], *Catégories de Picard restreintes*, Acta Mathematica Vietnamica **7** 1 (1983) 117-122 [[pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Categories_de_Picard_restreintes.pdf), [web](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html)]

[Sinh 1978](Sinh78) appears to prove that every 2-group is equivalent to a [[strict 2-group]] arising from a [[crossed module]].  The second calls a [[symmetric 2-group]] (i.e. a [[symmetric monoidal category]] with all objects and morphisms invertible) a *[[Picard category]]*, and calls a Picard category *restrained* if the [[braiding]] $B_{x,x} \colon x \otimes x \to x \otimes x$ is the [[identity morphism|identity]] for all objects $x$. The article then proves that every Picard category is equivalent to one arising from a 2-term [[chain complex]] of abelian groups.

Computational enumeration of geometrically [[discrete group|discrete]] 2-groups using the computer program [XMod](http://pages.bangor.ac.uk/~mas023/chda/xmod/xmod244.html):

* Murat Alp, Christopher Wensley, _Enumeration of $Cat^1$-groups of low order_,  Int. J. Algebra Comput. 10, 407 (2000) [[doi:10.1142/S0218196700000170](http://www.worldscientific.com/doi/abs/10.1142/S0218196700000170)]

* [[Graham Ellis]], Luyen Van Le, *Homotopy 2-types of Low order*, Experimental Mathematics **23** 4 (2014)  [[doi:10.1080/10586458.2014.912059](https://doi.org/10.1080/10586458.2014.912059), [pdf](http://hamilton.nuigalway.ie/preprints/2t.pdf)]

### Review

An early textbook account on [[strict 2-groups]] and explaining the relation to [[crossed modules]]:

* [[Saunders MacLane]], §XII.8 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)]

Exposition of general 2-groups as [[monoidal categories]] with all objects and morphisms invertible (sometimes called [[Picard 2-groups]]):

*  {#BaezLauda03} [[John Baez]], [[Aaron Lauda]], _HDA V: 2-Groups_, Theory and Applications of Categories **12** (2004) 423-491 [[arXiv:math.QA/0307200](http://arxiv.org/abs/math.QA/0307200), [tac:12-14](http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html)]

### Geometric 2-groups

Beware that most of the above discussion is about [[discrete infinity-group|geometrically discrete]] 2-groups.

Discussion of geometrically structured 2-groups (notably [[smooth 2-groups]], hence "Lie 2-groups"):

* {#Schommer-Pries11} [[Chris Schommer-Pries]], *Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group* Geometry & Topology **15** (2011) 609-676 [[arXiv:0911.2483](http://arxiv.org/abs/0911.2483), [doi:10.2140/gt.2011.15.609](https://doi.org/10.2140/gt.2011.15.609)]

and via [[group stacks]]:

* [[Urs Schreiber]], §2.6.5 & §3.4.2 of: _[[schreiber:differential cohomology in a cohesive topos]]_ [[arXiv:1310.7930](https://arxiv.org/abs/1310.7930)]

For more on this see the references at *[[string 2-group]]*.

### Examples

A key example due to its universality for [[higher central extensions]] of [[compact Lie groups]] by the [[circle 2-group]] to [[smooth 2-groups]] is

* the *[[string 2-group]]* -- see [there](string+2-group#References) for references.

Further on [[2-group]]-[[higher central extensions|extensions]] by the [[circle 2-group]]:

of [[tori]] (see also at *[[T-duality 2-group]]*):

* [[Nora Ganter]], _Categorical Tori_, SIGMA 14 (2018), 014, 18 ([arXiv:1406.7046](https://arxiv.org/abs/1406.7046))

of [[finite subgroups of SU(2)]] (to [[Platonic 2-groups]]):

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192), [hs:30](https://journals.mq.edu.au/index.php/higher_structures/article/view/30))

{#ReferencesInCondensedMatterPhysics} Arguments that 2-groups play a role in [[symmetry protected topological phases of matter]]:

* [[Anton Kapustin]], [[Ryan Thorngren]], *Higher symmetry and gapped phases of gauge theories*, in *Algebra, Geometry, and Physics in the 21st Century*, Progress in Mathematics, **324** (2017) 177-202 [[arXiv:1309.4721](https://arxiv.org/abs/1309.4721), [doi:10.1007/978-3-319-59939-7_5](https://doi.org/10.1007/978-3-319-59939-7_5)]

* {#BBCW19} [[Maissam Barkeshli]], [[Parsa Bonderson]], [[Meng Cheng]], [[Zhenghan Wang]], *Symmetry Fractionalization, Defects, and Gauging of Topological Phases*, Phys. Rev. B **100** 115147 (2019) [[arXiv:1410.4540](https://arxiv.org/abs/1410.4540), [doi:10.1103/PhysRevB.100.115147](https://doi.org/10.1103/PhysRevB.100.115147), [talk pdf](http://helper.ipam.ucla.edu/publications/stq2015/stq2015_12401.pdf)]

In this vein, 2-groups are now discussed by a vocal group of physicists under the term "[[generalized global symmetry|generalized symmetries]]" or similar.

[[!redirects 2-groups]]

[[!redirects weak 2-group]]
[[!redirects weak 2-groups]]


2-groupoid

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

# Contents
* table of contents
{:toc}

## Idea

A **$2$-groupoid** is

* a [[(n,r)-category|(2,0)-category]]

* a [[2-category]] in which all morphisms are [[equivalence]]s

* an [[n-groupoid]] for $n = 2$

* a $2$-[[truncated]] [[∞-groupoid]].

## Definition

Fix a meaning/model of [[∞-groupoid]], however weak or strict you wish. Then a __$2$-groupoid__ is an $\infty$-groupoid such that all [[parallel pair]]s of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 3$. Thus, up to [[equivalence of categories|equivalence]], there is no point in mentioning anything beyond $2$-morphisms, except whether two given parallel $2$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $2$-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of $2$-morphisms as [[equality]].

## Specific models

There are various objects that model the abstract notion of $2$-groupoid.

### Bigroupoids

A **[[bigroupoid]]** is a [[bicategory]] in which all morphisms are [[equivalences]].

Bigroupoids may equivalently be thought of in terms of their [[Duskin nerve]]s. These are precisely the [[Kan complex]]es that are 2-[[hypergroupoid]]s.

### 2-Hypergroupoids

A $2$-[[hypergroupoid]] is a model for a 2-groupoid. This is a [[simplicial set]], whose vertices, edges, and 2-[[simplices]] we identify with the [[object]]s, [[morphism]]s and [[2-morphism]]s of the form

$$
  \array{
     && y
     \\
     & \nearrow &\Downarrow& \searrow
      \\
     x &&\stackrel{}{\to}&& z
  }
$$

in the 2-groupoid, respectively.

Moreover, the 3-[[simplices]] in the simplicial set encode the composition operation: given three composable 2-simplex faces of a tetrahedron (a 3-[[horn]])

$$
  \array{
    y &\to& &\to& z
    \\
    \uparrow &\seArrow& &\nearrow&  \downarrow
    \\
    \uparrow &\nearrow& &\Downarrow& \downarrow
    \\
    x &\to&&\to& w
  }
  \;\;\;
  \;\;\;
  \array{
    y &\to& &\to& z
    \\
    &\searrow& &\swArrow&  \downarrow
    \\
    && &\searrow& \downarrow
    \\
    &&&& w
  }
$$

the unique composite of them is is a fourth face $\kappa$ and a 3-cell $comp$ filling the resulting hollow tetrahedron:

$$
  \array{
    y &\to& &\to& z
    \\
    \uparrow &\seArrow& &\nearrow&  \downarrow
    \\
    \uparrow &\nearrow& &\Downarrow& \downarrow
    \\
    x &\to&&\to& w
  }
  \;\;\;
  \stackrel{comp}{\to}
  \;\;\;
  \array{
    y &\to& &\to& z
    \\
    \uparrow &\searrow& &\swArrow&  \downarrow
    \\
    \uparrow &{}_\kappa\Downarrow& &\searrow& \downarrow
    \\
    x &\to&&\to& w
  }
  \,.
$$

The 3-[[coskeletal]]-condition says that every boundary of a 4-[[simplex]] made up of five such tetrahedra has a unqiue filler. This is the [[associativity]] [[coherence law]] on the comoposition operation:

$$
 \array{\arrayopts{\rowalign{center}}
 \array{\begin{svg}
 [[!include monoidal category > pentagonator]]
 \end{svg}}
 }
$$

This says that any of the possible ways to use several of the 3-simpleces to compose a bunch of compsable 2-morphisms are actually equal.

### Homotopy 2-types

More generally one may consider a [[Kan complex]] that are just [[homotopy equivalent]] to a $3$-[[coskeletal]] one as a $2$-groupoid -- precisely: as representing the same [[homotopy type]], namely a [[homotopy 2-type]].

## Strict $2$-groupoids

The general notion of $2$-groupoid above is also called __weak $2$-groupoid__ to distinguish from the special case of **[[strict 2-groupoids]]**. A strict $2$-groupoid is a [[strict 2-category]] in which all morphisms are strictly invertible. This is equivalently a certain type of [[Grpd]]-[[enriched category]].

## Examples

* For $A$ an [[abelian group]], there is its double [[delooping]] 2-groupoid $\mathbf{B}^2 A$. This is given by the [[strict]] 2-groupoid that comes from the [[crossed complex]] $A \to 0 \stackrel{\to}{\to} 0$. As a [[Kan complex]] this is the image under the [[Dold-Kan correspondence]] of the [[chain complex]] $[A \to 0 \to 0]$.

* The [[fundamental 2-groupoid]] of a [[topological space]].

* The [[fundamental infinity-groupoid|fundamental $\infty$-groupoid]] of a [[topological space]] that is itself a [[homotopy 2-type]].

* The [[path 2-groupoid]] of a [[smooth manifold]].

## Related concepts

* [[Grpd-enriched category]]

[[!include homotopy n-types - table]]

[[!redirects 2-groupoid]]
[[!redirects 2-groupoids]]

[[!redirects weak 2-groupoid]]
[[!redirects weak 2-groupoids]]


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

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

## Idea

For $\mathfrak{g}$ a [[Lie 2-algebra]] the **2-groupoid of $\mathfrak{g}$-valued forms** is the [[2-groupoid]] whose objects are [[differential form]]s with values in $\mathfrak{g}$, whose morphisms are [[gauge transformation]]s between these, and whose 2-morphisms are _higher order gauge transformations_ of those.

This naturally refines to a non-[[concrete sheaf|concrete]] [[Lie 2-groupoid]] is the 2-[[truncated]] [[∞-Lie groupoid]] whose $U$-parameterized smooth families of objects are smooth [[differential form]]s with values in a [[Lie 2-algebra]], and whose morphisms are gauge transformations of these.

This is the [[higher category theory|higher category]] generalization of the [[groupoid of Lie-algebra valued forms]].

A [[cocycle]] with coefficients in this 2-groupoid is a [[connection on a 2-bundle]].

## Definition

### For strict Lie 2-algebras

Consider a Lie [[strict 2-group]] $G$ corresponding to a Lie [[crossed module]] $(G_2 \stackrel{\delta}{\to} G_1)$ with action $\alpha : G_1 \to Aut(G_2)$.  Write $\mathbf{B}G$ for the corresponding [[delooping]] 2-groupoid, the one coming from the [[crossed complex]]

$$
  [\mathbf{B}G]
  =
  (G_2 \stackrel{\delta}{\to} G_1 \stackrel{\to}{\to} *)
  \,.
$$

Write $[\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1]$ for the corresponding [[differential crossed module]] with action $\alpha_* : \mathfrak{g}_1 \to der(\mathfrak{g}_2)$

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

The 2-groupoid of Lie 2-algebra valued forms is defined to be the 2-stack

$$
  \bar \mathbf{B}G : CartSp{}^{op} \to 2Grpd
$$

which assigns to $U \in CartSp$ the following 2-groupoid:

* An [[object]] is a pair

  $$
    A \in \Omega^1(U,\mathfrak{g}_1)\,, \;\;\;
    B \in \Omega^2(U,\mathfrak{g}_2)
    \,.
  $$

* A 1-[[morphism]] $(g,a) : (A,B) \to (A',B')$ is a pair

  $$
    g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)
  $$

  such that

  $$
    A' = Ad_{g^{-1}}\left( A + \delta_* a \right) + g^{-1} d g
  $$

  and

  $$
    B' = \alpha_{g^{-1}}(
      B + d a + [a \wedge a] + \alpha_*(A \wedge a)
    )
    \,.
  $$

  The composite of two 1-morphisms

  $$
    (A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to}
    (A'', B'')
  $$

  is given by the pair

  $$
    (g_1 g_2, a_1 + (\alpha_{g_2})_* a_2)
    \,.
  $$

* a [[2-morphism]] $f : (g,a) \Rightarrow (g', a'):(A,B)\to (A',B')$ is a function

  $$
    f \in C^\infty(U,G_2)
  $$

  such that

  $$
    g' = \delta(f)^{-1} \cdot g
  $$

  and

  $$
    a' = Ad_{f^{-1}} \left(a  + (r_f^{-1} \circ \alpha_f)_*(A)\right) + f^{-1} d f
  $$

and composition is defined as follows: vertical composition is given by pointwise multiplication ([[David Roberts|DR]]: the order still needs sorting out!) and horizontal composition is given as horizontal composition in the one-object 2-groupoid $\mathbf{B}G)$.

=--

### For general Lie 2-algebras
 {#ForGeneralLie2Algebras}

We consider now $\mathfrak{g}$ a general [[Lie 2-algebra]].

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

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

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

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

with these generators.

We thus have

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

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

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

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

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

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

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

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

This is given by a 1-form

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

and a 2-form

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

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

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

and whose 3-form component is

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

## Properties

+-- {: .un_def }
###### Proposition
**(flat Lie 2-algebra valued forms)**

The full sub-2-groupoid on _flat_ Lie 2-algebra valued forms, i.e. those pairs $(A,B)$ for which the 2-form [[curvature]]

$$
  \delta_* B - d A + [A \wedge A] = 0
$$

and the 3-form [[curvature]]

$$
  d B + [A \wedge B] = 0
$$

vanishes is a resolution of the underlying discrete Lie 2-groupoid $\mathbf{\flat} \mathbf{B}G$ of the Lie 2-groupoid $\mathbf{B}G$.

=--

This is discussed at [[∞-Lie groupoid]] in the section strict Lie 2-groups -- differential coefficients.

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

Let $\mathbf{\Pi}_2 : CartSp \to 2LieGrpd$ be the smooth 2-[[fundamental groupoid]] functor and let $P_2 : CartSp \to 2LieGrpd$ be the [[path n-groupoid|path 2-groupoid]] functor, taking values in the 2-catgeory $2Grpd(Difeol)$ of 2-groupoids [[internal category|internalization]] to [[diffeological space]]s. Then

* the 2-groupoid of Lie 2-algebra valued forms for which both 2- and 3-form curvature vanish is canonically equivalent to

  $$
    Hom_{2Grpd(Diffeol)}(\Pi_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd
    \,;
  $$

* the 2-groupoid of Lie 2-algebra valued forms for which the 2-form curvature vanishes is canonically equivalent to

  $$
    Hom_{2Grpd(Diffeol)}(P_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd
    \,;
  $$

=--

The equivalence is given by 2-dimensional [[parallel transport]].
A proof is in [SchrWalII](http://arxiv.org/abs/0802.0663).

The following proposition asserts that the Lie 2-groupoid of Lie 2-algebra valued forms is the coefficient object for for _differential nonabelian cohomology_ in degree 2, namely for _connections_ on [[principal 2-bundle]]s and in particular on [[gerbe]]s.

+-- {: .un_def }
###### Proposition
**(2-bundles with connection)**

For $X$ a [[paracompact space|paracompact]] [[smooth manifold]] and $\{U_i \to X\}$ a [[good open cover]] the 2-groupoid, let $X \stackrel{\simeq}{\leftarrow} C(\{U_i\})$ be the corresponding [[Cech nerve]] smooth 2-groupoid. Then

$$
  Hom_{2Grpd(Diffeol)}( C(\{U_i\}), \bar \mathbf{B}G)
$$

is equivalent to the [[2-groupoid]] of $G$-[[principal 2-bundle]]s with [[connection on a 2-bundle|2-connection]].

=--

This is discussed and proven in
[SchrWalII](http://arxiv.org/abs/0802.0663) for the case where the 2-form curvature is restricted to vanish. In this case the above can be written as

$$
  Hom( C(\{U_i\}), Hom(P_2(-), \mathbf{B}G))
  \simeq
  Hom(P_2(C(\{\U_i\})), \mathbf{B}G)
  \,,
$$

where $P_2(C(\{\U_i\}) \in 2LieGrpd$ is a resolution of the [[path 2-groupoid]] of $X$.

## Related concepts

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

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

  * [[nonabelian Stokes theorem]]

* [[3-groupoid of Lie 3-algebra valued forms]]

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

## References

The 2-groupoid of Lie 2-algebra valued forms described in [definition 2.11](http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.0663v3.pdf#page=27) of

* Schreiber, Waldorf, _Smooth functors versus differential forms_ (web). {#SchWalII}

There are many possible conventions. The one reproduced above is supposed to describe the _bidual_ [[opposite 2-category]] of the 2-groupoid as defined in that article, with the direction of 1- and 2-morphisms reversed.

See also

differential cohomology in an (∞,1)-topos -- survey - connections on 2-bundles.

[[!redirects Lie 2-algebra valued 2-form]]

[[!redirects Lie 2-algebra valued forms]]
[[!redirects Lie 2-algebra valued differential forms]]
[[!redirects Lie 2-algebra valued form]]


2-Hilbert space

#Contents#
* table of contents
{:toc}

## Idea

The concept of $2$-Hilbert space is supposed to be a [[categorification]] of that of [[Hilbert space]], or at least of [[finite-dimensional vector space|finite dimensional]] such: [[inner product spaces]].

One way to define this is as a [[Kapranov-Voevodsky 2-vector space]] where the [[hom-functor]] plays the role of the [[categorification|categorified]] [[inner product]] ([Baez 96](#Baez96)).

More generally this works for [[semisimple categories]] (...)

## Related concepts

* [[Schur's lemma]]

* [[2-vector space]]

## References

See

* {#Baez96} [[John Baez]]; _Higher-Dimensional Algebra II: 2-Hilbert Spaces_ ([arXiv](http://arxiv.org/abs/q-alg/9609018))

for one approach.

[[!redirects 2-Hilbert spaces]]


2-isomorphism

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

\tableofcontents

\section{Idea}

In a [[higher category theory|higher category]], [[inverse|invertibility]] of [[n-morphisms]] in its highest dimension is always considered strictly. Thus in an ordinary [[category]], we have [[1-morphisms]] which can be [[isomorphism|isomorphisms]]. In a [[2-category]], it is the [[2-morphisms]] for which, when it comes to invertibility, we always ask for a strict inverse, whereas for [[1-morphisms]] we typically ask only for an [[equivalence]]. These 2-morphisms which admit an inverse are known as _2-isomorphisms_.

\section{Definition}

\begin{defn} Let $\mathcal{A}$ be a [[2-category]]. A _2-isomorphism_ in $\mathcal{A}$ is a 2-arrow $\phi : f \rightarrow f'$ of $\mathcal{A}$ which admits a (strict) inverse, that is to say, there is a 2-arrow $\phi^{-1}: f' \rightarrow f$ of $\mathcal{A}$ such that $\phi^{-1} \circ \phi = id(f)$ and $\phi \circ \phi^{-1} = id\left(f' \right)$. \end{defn}

\section{Related concepts}

A 2-category in which every 2-arrow is a 2-isomorphism is known as a [[(2,1)-category]].

[[!redirects 2-isomorphisms]]


2-Lawvere theory

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _2-Lawvere theory_ or _Lawvere 2-theory_ is the [[categorification]] of [[Lawvere theory]].

See also at _[[doctrine]]_.

## Related concepts

* [[algebraic theory]] / [[Lawvere theory]] /  **2-Lawvere theory** [[(∞,1)-algebraic theory]]

*[[enriched Lawvere theory]]

* [[monad]] / [[2-monad]] / [[(∞,1)-monad]]

* [[operad]] / [[(∞,1)-operad]]

## References

* [[Noson Yanofsky]], _The syntax of coherence_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 2000. [Cahiers pdf](http://www.numdam.org/article/CTGDC_2000__41_4_255_0.pdf), [arXiv](https://arxiv.org/abs/math/9910006)

* [[John Gray]], _2-algebraic theories and triples_, in Colloques sur l’algebre des categories. Amiens 1973. Resumes des conferences vol. 14, pp. 178–180.

* [[John Bourke]], [[Nick Gurski]], _The Gray tensor product via factorisation_, Applied Categorical Structures, Volume 25, Issue 4, pp 603-624, (2017). [ACS](https://link.springer.com/article/10.1007/s10485-016-9467-6), [arXiv](https://arxiv.org/abs/1508.07789)

* [[Jonathan Cohen]], _Coherence for rewriting 2-theories_, PhD thesis, Australian National University, (2009). [arXiv](https://arxiv.org/abs/0904.0125)

* [[John Power]], [[Stephen Lack]], _Lawvere 2-theories_, 2007. ([slides](http://www.mat.uc.pt/~categ/ct2007/slides/lack.pdf))

* [[Noson Yanofsky]], _Coherence, Homotopy and 2-Theories_, K-Theory 23: Pgs 203 - 235. (2001) ([arXiv:math.CT/0007033](https://arxiv.org/abs/math/0007033))

[[!redirects 2-Lawvere theories]]

[[!redirects Lawvere 2-theory]]
[[!redirects Lawvere 2-theories]]

[[!redirects 2-algebraic theory]]
[[!redirects 2-algebraic theories]]


2-limit

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

# Contents
* table of contents
{: toc}

## Idea

A **2-limit** is the type of [[limit]] that is appropriate in a (weak) [[2-category]]. (Since general 2-categories are often called _[[bicategories]]_, 2-limits are often called _[[bilimits]]_.)

There are three notable changes when passing from ordinary 1-limits to 2-limits:

1. In order to satisfy the [[principle of equivalence]], the "cones" in a 2-limit are required to commute only up to [[2-morphism|2-isomorphism]].

2. The [[universal property]] of the limit is expressed by an [[equivalence of categories]] rather than a [[bijection]] of [[sets]].  This means that

   1. every other cone over the diagram that commutes up to isomorphism factors through the limit, up to isomorphism, and

   2. every transformation _between_ cones also factors through a 2-cell in the limit.  We will give some examples below.

3. Since 2-categories are [[enriched category|enriched]] over [[Cat]] (this is precise in the [[strict 2-category|strict]] case, and [[bicategory|weakly]] true otherwise), $Cat$-[[weighted limit]]s become important.  This means that both the diagrams we take limits of and the shape of "cones" that limits represent can involve $2$-cells as well as $1$-cells.

## Definition

Let $K$ and $D$ be [[2-categories]], and $J\colon D\to Cat$ and $F\colon D\to K$ be [[2-functors]].  A **$J$-weighted (2-)limit of $F$** is an object $L\in K$ equipped with a [[pseudonatural equivalence]]
$$ K(X,L) \simeq [D,Cat](J,K(X,F-)). $$
where $[D,Cat]$ denotes the 2-category of [[2-functors]] $D\to Cat$, [[pseudonatural transformations]] between them, and [[modifications]] between those.

A 2-limit in the [[opposite 2-category]] $K^{op}$ is called a **2-colimit** in $K$.  Everything below applies dually to 2-colimits, the higher analogues of [[colimits]].  (But somebody might want to make a separate page that gives appropriate examples of these.)

### Strictness and terminology
 {#Terminology}

If $K$ and $D$ are [[strict 2-categories]], $J$ and $F$ are [[strict 2-functors]], and if we replace this pseudonatural equivalence by a (strictly 2-natural) isomorphism *and* the 2-category $[D,Cat]$ by the 2-category $[D,Cat]_{strict}$ of strict 2-functors and strict 2-natural transformations, then we obtain the definition of a **[[strict 2-limit]]**.  This is precisely a Cat-weighted limit in the sense of ordinary [[enriched category]] theory.  See [[strict 2-limit]] for details.

On the other hand, if $K$, $D$, $J$, and $F$ are strict as above, and we replace the equivalence by an isomorphism but keep the weak meaning of $[D,Cat]$, then we obtain the notion of a **strict pseudolimit**.  Strict pseudolimits are, in particular, 2-limits, whereas strict 2-limits are not always (although some, such as [[PIE-limits]] and [[flexible limits]], are).  In a strict 2-category, these types of strict limits are often technically useful in constructing the "up-to-isomorphism" 2-limits we consider here.

When we know we are working in a (weak) 2-category, the only type of limit that makes sense is a (non-strict) 2-limit.  Therefore, we usually call these simply "limits."  To emphasize the distinction with the strict 2-limits in a strict 2-category, the "up-to-isomorphism" 2-limits were historically often called _bilimits_ (by analogy with [[bicategory]] for "weak 2-category").  However, this terminology is somewhat unfortunate, not only because it doesn't generalize well to $n$, but because it leads to words like "biproduct," which also has the [[biproduct|completely unrelated meaning]] of an object that is both a product and a coproduct (which is common in [[additive category|additive categories]]).

Unfortunately, we probably shouldn't use "weak limit" to emphasize the "up-to-isomorphism" nature of these limits, because that also has the [[weak limit|completely unrelated meaning]] of an object in a 1-category satisfying the existence, but not the uniqueness property of an ordinary limit.

## Examples

### 2-limits over diagrams of special shape

Any ordinary type of limit can be "2-ified" by boosting its ordinary universal property up to a 2-categorical one.  In the following examples we work in a 2-category $K$.

* A **[[terminal object]]** in $K$ is an object 1 such that $K(X,1)$ is equivalent to the [[terminal category]] for any object $X$.  This means that for any $X$ there is a morphism $X\to 1$ and for any two morphisms $f,g:X\to 1$ there is a unique morphism $f\to g$, and this morphism is an isomorphism.

* A **[[product]]** of two objects $A,B$ in $K$ is an object $A\times B$ together with a natural equivalence of categories $K(X,A\times B) \simeq K(X,A)\times K(X,B)$.  This means that we have projections $p:A\times B\to A$ and $q:A\times B\to B$ such that (1) for any $f:X\to A$  and $g:X\to B$, there exists an $h:X\to A\times B$ and isomorphisms $p h\cong f$ and $q h\cong g$, and (2) for any $h,k:X\to A\times B$ and 2-cells $\alpha:p h \to p k$ and $\beta: q h \to q k$, there exists a unique $\gamma:h \to k$ such that $p \gamma = \alpha$ and $q \gamma = \beta$.

* A **[[pullback]]** of a [[co-span]] $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ consists of an object $A\times_C B$ and projections $p:A\times_C B\to A$ and $q:A\times_C B\to B$ together with an isomorphism $\phi:f p \cong g q$, such that (1) for any $m:X\to A$ and $n:X\to B$ with an isomorphism $\psi:f m \cong g n$, there exists an $h:X\to A\times_C B$ and isomorphisms $\alpha:p h \cong m$ and $\beta:q h \cong n$ such that $g\beta . \phi h . f \alpha^{-1} = \psi$, and (2) given any two morphisms $h,k:X\to A\times_C B$ and 2-cells $\mu:p h \to p k$ and $\nu:q h \to q k$ such that $f \mu = g \nu$ (modulo the given isomorphism $f p \cong g q$), i.e., $\phi k . f\mu = g\nu . \phi h$, there exists a unique 2-cell $\gamma:h\to k$ such that $p \gamma = \mu$ and $q \gamma = \nu$.  This is sometimes called the _pseudo-pullback_ but that term more properly refers to a particular [[strict 2-limit]].

* An **[[equalizer]]** of $f,g:A\to B$ consists of an object $E$ and a morphism $e:E\to A$ together with an isomorphism $f e \cong g e$, which is universal in a sense the reader should now be able to write down.  This is sometimes called the _pseudo-equalizer_ but that term more properly refers to a particular [[strict 2-limit]].  Note that frequently, such as in the construction of all limits from basic ones, equalizers need to be replaced by [[descent object]]s.

There are also various important types of 2-limits that involve 2-cells in the diagram shape or in the weight, and hence are not just "boosted-up" versions of 1-limits.

* The **[[comma object]]** of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $(f/g)$ and projections $p:(f/g)\to A$ and $q:(f/g)\to B$ together with a transformation (not an isomorphism) $f p \to g q$.  In [[Cat]], [[comma objects]] are [[comma category|comma categories]].  Comma objects are sometimes called _lax pullbacks_ but that term more properly refers to the lax version of a pullback; see "lax limits" below.

* The **[[inserter]]** of a pair of parallel arrows $f,g:A \;\rightrightarrows\; B$ is a universal object $I$ equipped with a map $i:I\to A$ and a 2-cell $f i \to g i$.

* The **[[equifier]]** of a pair of parallel 2-cells $\alpha,\beta: f\to g: A\to B$ is a universal object $E$ equipped with a map $e:E\to A$ such that $\alpha e = \beta e$.

* The **[[inverter]]** of a 2-cell $\alpha:f\to g:A\to B$ is a universal object $V$ with a map $v:V\to A$ such that $\alpha v$ is invertible.

* The **[[power]]** of an object $A$ by a category $C$ is a universal object $A^C$ equipped with a functor $C\to K(A^C,A)$.  Of particular importance is the case when $C$ is the [[walking arrow]] $\mathbf{2}$.

### Finite 2-Limits

A 2-limit is called **finite** if its diagram shape and its weight are both "finitely presentable" in a suitable sense (defined in terms of [[computads]]; see [Street's article](#StreetLimitsIndexed) _Limits indexed by category-valued 2-functors_ ).  Pullbacks, comma objects, inserters, equifiers, and so on are all finite limits, as are powers by any finitely presented category.  All finite limits can be constructed from pullbacks, a terminal object, and powers with $\mathbf{2}$.

### $(2,1)$-limits
  {#(2,1)limit}

If the ambient [[2-category]] is in fact a [[(2,1)-category]] in that all [[2-morphism]]s are invertible then there is a rich set of tools available for handling the 2-limits in this context. We may say **$(2,1)$-limits** and **$(2,1)$-colimits** in this case.

These are then a special case of the more general [[(∞,1)-limit]]s and [[(∞,1)-colimit]]s in a [[(∞,1)-category]]. A [[(2,1)-category]] is a special case of an [[(∞,1)-category]].

Traditionally, [[(∞,1)-limit]]s are best known in terms of the presentation of $(\infty,1)$-categories by [[categories with weak equivalences]] in general and [[model categories]] in particular.  (2,1)-limits can often also be viewed in this way.  The corresponding presentation of the $(\infty,1)$-limits / $(2,1)$-limits is called **[[homotopy limit]]s** and **[[homotopy colimit]]s**.

For instance 2-limits in the [[(2,1)-category]] [[Grpd]] of [[groupoid]]s, [[functor]]s and (necessarily) [[natural isomorphism]]s. Are equivalently computed as [[homotopy limit]]s in the [[model structure on simplicial sets]] $sSet_{Quillen}$ of diagrams of [[1-truncated]] [[Kan complex]]es. (The equivalence of homotopy limits with $(\infty,1)$-limits is discussed at [[(∞,1)-limit]]).

Or for instance, more generally, the 2-limits in any [[(2,1)-sheaf]](=[[stack]]) [[(2,1)-topos]] may be computed as [[homotopy limit]]s in a [[model structure on simplicial presheaves]] over the given [[(2,1)-site]] of diagrams of [[1-truncated]] [[simplicial presheaves]]. This includes as examples [[big topos|big (2,1)-toposes]] such as those over the large sites [[Top]] or [[SmoothMfd]] where computations with [[topological groupoid]]s/[[topological stack]]s, [[Lie groupoid]]s/[[differentiable stack]]s etc. take place.

### Lax limits
{#lax}

A **lax limit** can be defined like a 2-limit, except that the triangles in the definition of a cone are required only to commute up to a specified _transformation_, not necessarily an isomorphism.  In other words, in place of the 2-category $[D,Cat]$ we use the 2-category $[D,Cat]_l$ whose morphisms are [[lax natural transformations]]; thus the lax limit $L$ of a diagram $F$ weighted by $J$ is equipped with a universal lax natural transformation $J\to K(L,F-)$.

This may look like a different concept, but in fact, for any weight $J$ there is another weight $Q_l(J)$ such that lax $J$-weighted limits are the same as $Q_l(J)$-weighted 2-limits.  Here $Q_l$ is the [[lax morphism classifier]] for 2-functors.  Therefore, lax limits are really a special case of 2-limits.  Similarly, **oplax limits**, in which we use oplax natural transformations, are also a special case of 2-limits.

There is a further simplification of lax limits in the case of "conical" lax limits where the weight $J=\Delta 1$ is constant at the [[terminal category]].  In this case, it is easy to check that a lax natural transformation $\Delta 1 \to K(X,F-)$ is the same as a lax natural transformation $\Delta X \to F$.  Thus, a conical lax limit of $F$ is a representing object for such lax transformations.

Here are some examples.

* Lax terminal objects and lax products are the same as ordinary ones, since there are no commutativity conditions on the cones.

* The **lax limit of an arrow** $f:A\to B$ is a universal object $L$ equipped with projections $p:L\to A$ and $q:L\to B$ and a 2-cell $f p \to q$.  Note that this is equivalent to a comma object $(f/1_B)$.

* The **lax pullback** of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $P$ equipped with projections $p:P\to A$, $q:P\to B$, $r:P\to C$, and 2-cells $f p \to r$ and $g q \to r$.

Note that lax pullbacks are _not_ the same as [[comma objects]].  In general comma objects are much more useful, but there are 2-categories that admit all lax limits but do not admit comma objects, so using "lax pullback" to mean "comma object" can be misleading.

A **lax colimit** in $K$ is, of course, a lax limit in $K^{op}$.  Thus, it is a representing object for lax natural transformations $J \to K(F-,L)$.  There is a subtlety here, however, because in the case $J=\Delta 1$, a lax natural transformation $\Delta 1 \to K(F-,L)$ is the same as an *oplax* natural transformation $F \to \Delta L$.  Thus, it is easy to mistakenly say "lax colimit" when one really means "oplax colimit" and vice versa.

+-- {: .un_remark}
###### Remark
With this in mind, one might be tempted to switch the meanings of "lax colimit" and "oplax colimit", but there is a good reason not to.  Recall that a lax $J$-weighted limit is the same as an ordinary $Q_l(J)$-weighted limit.  Standard terminology in enriched category theory is that a $W$-weighted colimit in an enriched category $K$ is the same as a $W$-weighted limit in $K^{op}$, and indeed in that generality there is no other option.  Thus, a lax $J$-weighted colimit in $K$ should be an ordinary $Q_l(J)$-weighted colimit in $K$, hence a $Q_l(J)$-weighted limit in $K^{op}$, and thus a lax $J$-weighted limit in $K^{op}$.
=--

Here are some examples of lax and oplax colimits:

* A [[Kleisli object]] is a lax colimit of a [[monad]], regarded as a diagram in a 2-category.

* The [[collage]] of a [[profunctor]] is its lax colimit, regarded as a diagram in the 2-category [[Prof]].

* When $C$ is a category, the [[Grothendieck construction]] of a functor $C\to Cat$ is the same as its *oplax* colimit; see [here](http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit).

### 2-Colimits in $Cat$
 {#2ColimitsInCat}

As shown [here](http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit), if $C$ is an ordinary category and $F \colon C \to Cat$ is a [[pseudofunctor]], then the [[oplax colimit]] of $F$ is given by the [[Grothendieck construction]] $\int F$ --- and its [[pseudo-colimit]] is given by [[localization|formally inverting]] the [[cartesian morphism|opcartesian]] morphisms in $\int F$.  This yields a construction of certain pseudo 2-colimits in $Cat$.

Moreover, a similar result holds more generally when $C$ is a [[bicategory]].  In this case, $\int F$ is also a bicategory: a 2-cell from $(m \colon c \to d, f \colon m_*x \to y)$ to $(n \colon c \to d, g \colon n_*x \to y)$ is given by a 2-cell $\mu \colon m \Rightarrow n$ in $C$ such that $\mu_* x$ is a morphism $f \to g$ over $y$.

Let $\pi_*$ denote the functor that sends a bicategory $K$ to the category whose objects are those of $K$ and whose hom-sets are the [[connected category|connected components]] of the hom-categories of $K$; let also $d_*$ denote the functor that sends a category $X$ to the corresponding locally discrete bicategory.  Then there is an equivalence of categories

$$ [K, d_* X] \simeq [\pi_* K, X]  $$

It is straightforward to check that the first of the above facts extends to the bicategorical case:

$$ Lax(F, \Delta X) \simeq [{\textstyle \int} F, d_* X] $$

as does the fact that a lax transformation on the left is pseudo if and only if the corresponding functor on the right inverts the opcartesian morphisms in $\int F$.  It is almost trivial that the adjunction $\pi_* \dashv d_*$ holds when restricted to the functor $[-, -]_{S^{-1}}$ that takes two categories or bicategories to the full subcategory of functors that invert the class $S$ of morphisms.  Taking $S$ to be the opcartesian morphisms in $\int F$, then, we have

$$
  Ps(F, \Delta X) \simeq [{\textstyle \int} F, d_* X]_{S^{-1}}
  \simeq [\pi_* {\textstyle \int} F, X]_{S^{-1}}
  \simeq [(\pi_* {\textstyle \int} F)[S^{-1}], X]
$$

Hence the pseudo colimit of $F$ is got by taking its bicategory of elements, applying the 'local $\pi_0$' functor, and then inverting the (images of the) opcartesian morphisms as usual.

## Related concepts

* [[limit]]

* **2-limit**

* [[strict 2-limit]]

* [[marked 2-limit]]

* [[double limit]]

* [[2-limits and 2-colimits in 2-categories of 2-algebras]]

* [[(∞,1)-limit]]

  * [[homotopy limit]]

  * [[lax (∞,1)-colimit]]

* [[quasi-limit]]

* [[coherence for bicategories with finite limits]]

* [[representable 2-category]]

## References

* {#Street74} [[Ross Street]], *Elementary cosmoi I* (§6) in *Category Seminar*, Lecture Notes in Mathematics **420**, Springer (1974) [[doi:10.1007/BFb0063103](https://doi.org/10.1007/BFb0063103)]

* {#StreetLimitsIndexed} [[Ross Street]], _Limits indexed by category-valued 2-functors_ Journal of Pure and Applied Algebra **8**, Issue 2 (1976) pp 149-181. doi:[10.1016/0022-4049(76)90013-X](https://doi.org/10.1016/0022-4049(76%2990013-X)

* [[Max Kelly]], _Elementary observations on 2-categorical limits_, Bulletin of the Australian Mathematical Society (1989), 39: 301-317, doi:[10.1017/S0004972700002781](https://doi.org/10.1017/S0004972700002781)

* [[Ross Street]], _Fibrations in Bicategories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume **21** (1980) no. 2, pp 111-160.  [Numdam](http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0 ) and correction, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume **28** (1987) no. 1, pp 53-56 [Numdam](http://www.numdam.org/item?id=CTGDC_1987__28_1_53_0 )

Section 6, page 37 in

* [[Steve Lack]], _A 2-categories companion_. In: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications, vol **152** 2010 Springer, New York, NY. doi:[10.1007/978-1-4419-1524-5_4](https://doi.org/10.1007/978-1-4419-1524-5_4), arXiv:[math.CT/0702535](http://arxiv.org/abs/math.CT/0702535).

* G. J. Bird, [[Max Kelly]], [[John Power]], [[Ross Street]], _Flexible limits for 2-categories_, Journal of Pure and Applied Algebra **61** Issue 1 (1989) pp 1-27. doi:[10.1016/0022-4049(89)90065-0](http://dx.doi.org/10.1016/0022-4049(89%2990065-0)
Chapters 3,4,5 in

* [[Thomas Fiore]], _Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory_, Mem. Amer. Math. Soc. **182** (2006), no. 860 ([arXiv:math/0408298](http://arxiv.org/abs/math/0408298)) ([AMS page](http://bookstore.ams.org/memo-182-860), [Google Books](https://books.google.com.au/books?id=y_DUCQAAQBAJ))

[[!redirects 2-limits]]
[[!redirects 2-colimit]]
[[!redirects 2-colimits]]
[[!redirects 2-categorical limit]]
[[!redirects 2-categorical limits]]
[[!redirects 2-categorial limit]]
[[!redirects 2-categorial limits]]
[[!redirects 2-categorical colimit]]
[[!redirects 2-categorical colimits]]
[[!redirects 2-categorial colimit]]
[[!redirects 2-categorial colimits]]
[[!redirects weighted 2-limit]]
[[!redirects weighted 2-limits]]

[[!redirects bicolimit]]
[[!redirects bicolimits]]

[[!redirects lax limit]]
[[!redirects lax limits]]
[[!redirects lax colimit]]
[[!redirects lax colimits]]
[[!redirects oplax limit]]
[[!redirects oplax limits]]
[[!redirects oplax colimit]]
[[!redirects oplax colimits]]
[[!redirects colax limit]]
[[!redirects colax limits]]
[[!redirects colax colimit]]
[[!redirects colax colimits]]

[[!redirects strict lax limit]]
[[!redirects strict lax limits]]
[[!redirects strict lax colimit]]
[[!redirects strict lax colimits]]
[[!redirects strict oplax limit]]
[[!redirects strict oplax limits]]
[[!redirects strict oplax colimit]]
[[!redirects strict oplax colimits]]
[[!redirects strict colax limit]]
[[!redirects strict colax limits]]
[[!redirects strict colax colimit]]
[[!redirects strict colax colimits]]

[[!redirects pseudolimit]]
[[!redirects pseudolimits]]
[[!redirects pseudo limit]]
[[!redirects pseudo limits]]
[[!redirects pseudo-limit]]
[[!redirects pseudo-limits]]
[[!redirects pseudocolimit]]
[[!redirects pseudocolimits]]
[[!redirects pseudo colimit]]
[[!redirects pseudo colimits]]
[[!redirects pseudo-colimit]]
[[!redirects pseudo-colimits]]

[[!redirects strict pseudolimit]]
[[!redirects strict pseudolimits]]
[[!redirects strict pseudo limit]]
[[!redirects strict pseudo limits]]
[[!redirects strict pseudo-limit]]
[[!redirects strict pseudo-limits]]
[[!redirects strict pseudocolimit]]
[[!redirects strict pseudocolimits]]
[[!redirects strict pseudo colimit]]
[[!redirects strict pseudo colimits]]
[[!redirects strict pseudo-colimit]]
[[!redirects strict pseudo-colimits]]

[[!redirects (2,1)-limit]]
[[!redirects (2,1)-limits]]
[[!redirects (2,1)-colimit]]
[[!redirects (2,1)-colimits]]


2-limits and 2-colimits in 2-categories of 2-algebras

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

#Contents#
* tic
{: toc}

## Introduction

In moving from one dimension to two dimensions, there are a proliferation of concepts, with a choice of weakness for each one. For one, there is the notion of (strict) [[2-category]] and [[bicategory]], (strict) [[2-monad]] and [[pseudomonad]], and (strict) 2-algebras and pseudoalgebras. Therefore, while it is clear that "2-algebras for a 2-monad on a 2-category inherit 2-limits (and certain 2-colimits) from the base", it can be tricky to recall which level of strictness is appropriate in each case. On this page, we list the various limit and colimit creation properties for two-dimensional algebras.

## Limits

We shall use the conventional terminology of "2-" for strict concepts and "pseudo" for weak concepts to make it easier to compare with the references.

* The 2-category of 2-algebras and strict morphisms for a 2-monad on a 2-category inherits all 2-limits (this follows from $Cat$-enriched category theory).

* The 2-category of 2-algebras and pseudo morphisms for a 2-monad on a 2-category inherits all [[PIE 2-limits]] (§3 of [BKP89](#BKP89).

* The 2-category of 2-algebras and lax morphisms for a 2-monad on a 2-category inherits all [[oplax limits]], limits of strict morphisms, [[equifiers]] and [[inserters]] where one morphism is strict, [[Eilenberg–Moore objects]] for [[comonads]], [[products]] and [[powers]] (see [Lack05](#Lack05) and [LS11](#LS11)). The page [[rigged limit]] contains more details.

* The 2-category of 2-algebras and pseudo morphisms for a [[flexible 2-monad]] on a 2-category inherits all [[flexible limits]] (see Remark 7.2 of [BKPS89](#BKPS89)).

* The 2-category of pseudoalgebras and pseudo morphisms for a [[pseudomonad]] on a 2-category inherits all [[bilimits]] (see Theorem 6.3.1.6 of [Osmond21](#Osmond21)).

## Related pages

* [[2-monad]]

* [[colimits in categories of algebras]]

## References

* {#BKP89} R. Blackwell, G. M. Kelly, and A. J. Power, _Two-dimensional monad theory_, Jour. Pure Appl. Algebra 59 (1989), 1--41

* {#BKPS89} G. J. Bird, [[Max Kelly]], [[John Power]], [[Ross Street]], _Flexible limits for 2-categories_, Journal of Pure and Applied Algebra **61** Issue 1 (1989) pp 1-27. doi:[10.1016/0022-4049(89)90065-0](http://dx.doi.org/10.1016/0022-4049(89%2990065-0)

* {#Lack05} [[Stephen Lack]], _Limits for lax morphisms_, *Appl. Categ. Structures*,
13(3):189--203, 2005

* {#LS11} [[Stephen Lack]] and [[Mike Shulman]], _Enhanced 2-categories and limits for lax morphisms_, [arXiv](http://arxiv.org/abs/1104.2111).

* {#Osmond21} Axel Osmond, _A categorical study of spectral dualities_, PhD thesis, Université Paris Cité, 2021.


2-line

#Contents#
* table of contents
{:toc}

## Idea

Given a [[2-ring]] $R$, a _2-line_ over $R$ is an $R$-[[2-module]] which is [[equivalence|equivalent]] to $R$ as a 2-module.

## Related concepts

* [[line]]

* [[2-line bundle]]

[[!redirects 2-lines]]


2-logic

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

#Contents#
* table of contents
{:toc}

## Idea

As a [[category]] has an [[internal logic]] so a [[2-category]] has internal _2-logic_

## Related concepts

* [[type theory]], [[logic]]

* [[2-type theory]], **2-logic**

* [[(∞,1)-type theory]], [[(∞,1)-logic]]

## References

* [[Mike Shulman]], _[[michaelshulman:2-categorical logic]]_

  * [[michaelshulman:internal logic of a 2-category]]