2-monad

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## Definition

### 2-Monad

A **2-monad** is a [[monad]] on a [[2-category]], or more generally a monad _in_ a [[3-category]].  This concept manifests at varying levels of strictness:

* For a _strict 2-monad_ (which classically is called a simply a "2-monad"), the 2-category $K$ is a [[strict 2-category]], the functor $T:K\to K$ is a [[strict 2-functor]], and the transformations $\mu$ and $\eta$ are [[strict 2-natural transformation]]s and satisfy their laws strictly.  This is the same as a $Cat$-[[enriched category theory|enriched]] monad.  Strict 2-monads live naturally in [[strict 3-category|strict 3-categories]].

* For a fully _weak 2-monad_, $K$ is a weak 2-category (such as a [[bicategory]]), $T$ is a weak (aka pseudo) 2-functor, and $\mu$ and $\eta$ are pseudo natural transformations that satisfy their laws up to specified isomorphisms satisfying coherence conditions.  Weak 2-monads live naturally in fully weak 3-categories (or [[tricategory|tricategories]])

* In between we have various notions that are sometimes called _pseudomonads_.  For instance, we could require $K$ to be a strict 2-category and $T$ a strict 2-functor, but $\mu$ and $\eta$ only pseudo natural.  This sort of pseudomonad lives naturally in a [[Gray-category]].

### Algebras and pseudoalgebras over a 2-monad

One can consider various 2-categories of algebras/modules for a 2-monad, depending on whether the algebras satisfy their laws strictly or weakly, and whether the morphisms commute with the algebra structure strictly or weakly.

At the strictest level, for a strict 2-monad $T$ we can consider the $Cat$-enriched [[Eilenberg-Moore category]], which consists of strict algebras (see [[algebra over a monad]]), strict morphisms, and strict transformations between these.  In 2-categorical literature, it is usually denoted $T Alg_s$.  Many common types of structure on categories are specified by strict algebras for a strict 2-monad, but usually the strict morphisms are too strict.

There are three types of weak morphism: _pseudo_ (which preserve the structure up to a specified coherent isomorphism), _lax_ (which preserve it up to a noninvertible transformation) and _colax_ or _oplax_ (for which the transformation goes the other direction).  See [[lax morphism]] for further discussion.  With strict algebras and these various types of morphism, we obtain 2-categories $T Alg$ (the pseudo case), $T Alg_l$, and $T Alg_c$ for lax and colax respectively.  One can also define an [[F-category]] of strict and lax morphisms together (or strict and pseudo, or pseudo and lax), and a [[double category]] which includes both lax and colax morphisms.

For example, ordinary (non-strict) [[monoidal category|monoidal categories]] are the _strict_ algebras for a strict 2-monad $T_{MC}$ on $Cat$, but usually we care about pseudo, lax, and oplax [[monoidal functor]]s rather than strict ones.  _Strict_ monoidal categories are the strict algebras for a _different_ strict 2-monad $T_{StrMC}$ on $Cat$.

We can, however, also consider *pseudo* algebras for a 2-monad; see [[pseudoalgebra for a 2-monad]].  If the 2-monad is not strict, then this is usually the only sensible course.  Pseudoalgebras for a strict 2-monad $T$ usually give an "unbiased" weaker notion of the structure specified by $T$.  For example, the pseudoalgebras for $T_{StrMC}$ are, not ordinary monoidal categories, but [[bias|unbiased]] monoidal categories.  (It is true, however, that the 2-category $Ps T_{StrMC} Alg$ of unbiased monoidal categories and strong monoidal functors is strictly 2-equivalent, i.e. Cat-enriched equivalent, to the 2-category of ordinary biased monoidal categories and strong monoidal functors.)

There are also 2-monads that specify [[property-like structure]].  For instance, there is a 2-monad whose algebras are categories with finite products.  Actually, its algebras are categories equipped with _specified_ finite products, the strict morphisms of these algebras preserve these specified finite products on the nose, and the pseudo morphisms preserve them in the usual sense of "preserving finite products."  In this case, _every_ functor between algebras is an oplax morphism, since there is always a canonical comparison map $F(A\times B) \to F(A)\times F(B)$.

For limit and colimit properties of algebras, see [[2-limits and 2-colimits in 2-categories of 2-algebras]].

## Properties

### Relation to doctrines

2-monads (particularly on [[Cat]]) are also sometimes called _[[doctrines]]_, with the intuition in mind that they are an "algebraic theory" of structure on a category just as a monad (on $Set$) is an [[algebraic theory]] of structure on a set.  However, this use of terminology is arguably at variance with the original intuitive meaning of "doctrine."

### Relation of strict 2-monads to 1-monads

A strict 2-monad $T$ has an underlying [[monad]] $T_0$, such that strict $T$-algebras and strict $T$-morphisms are the same as $T_0$-algebras and $T_0$-morphisms.  (This is a special case of the the general theory of underlying ordinary categories for [[enriched categories]].)  Moreover, if a strict 2-category $A$ admits [[powers]] or [[copowers]] with the [[interval category]], then any monad on its underlying ordinary category $A_0$ has at most one enrichment to a strict 2-monad.  Thus, in this case "being a 2-monad" is a mere [[property]] of a monad; see the "unicity" paper of John Power below.

## Related concepts

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

* [[monad]]

  * **2-monad**/ [[doctrine]]

    * [[Kleisli 2-category]]

  * [[(∞,1)-monad]]

* [[operad]] / [[(∞,1)-operad]]

## References

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

* F. Marmolejo, _Doctrines whose structure forms a fully faithful adjoint string_, Theory and Applications of Categories 3 (1997), 23--44. ([TAC](http://www.tac.mta.ca/tac/volumes/1997/n2/3-02abs.html))

* [[Stephen Lack]], _A coherent approach to pseudomonads_, Advances in Mathematics 152 (2000), 179--202. ([doi:10.1006/aima.1999.1881](https://doi.org/10.1006/aima.1999.1881))

* [[Max Kelly]] and [[Steve Lack]], _On property-like structures_, Theory and Applications of Categories 3 (1997) 213--250.  ([TAC](http://www.tac.mta.ca/tac/volumes/1997/n9/3-09abs.html))

* [[Stephen Lack]], _Homotopy-theoretic aspects of 2-monads_ ([arXiv:math/0607646](http://arxiv.org/abs/math/0607646))

* [[Stephen Lack]], _Codescent objects and coherence_, JPAA 175 (2002), 223--241.

* I. J. Le Creurer, F. Marmolejo, E. M. Vitale, _Beck's theorem for pseudo-monads_, J. Pure Appl. Algebra 173 (2002), no. 3, 293--313.
* [[John Power]], *Unicity of enrichment over Cat or Gpd*, Appl. Categ. Struct. 2009, 1--7.

On the relation to [[symmetric operads]]:

* {#Kock12} [[Joachim Kock]], _Data types with symmetries and polynomial functors over groupoids_,  28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); in Electronic Notes in Theoretical Computer Science.  ([arXiv:1210.0828](http://arxiv.org/abs/1210.0828))

* {#Weber14} [[Mark Weber]], _Operads as polynomial 2-monads_ ([arXiv:1412.7599](http://arxiv.org/abs/1412.7599))

On the [[extension system]]-perspective (as for [[monads in computer science]]) generalized to pseudomonads:

Generalizing the perspective of [[extension systems]] to [[pseudomonads]]:

* [[Francisco Marmolejo]], [[Richard J. Wood]], *No-iteration pseudomonads*, [[TAC]] **28** 14 (2013) 371-402 [[tac:28-14](http://www.tac.mta.ca/tac/volumes/28/14/28-14abs.html)]

and relating to [[Kan extensions]]:

* [[Francisco Marmolejo]], [[Richard J. Wood]], *Kan extensions and lax idempotent pseudomonads*, [[TAC]] **26** 1 (2012) 1-29 [[26-01](http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html)]

[[!redirects strict 2-monad]]
[[!redirects weak 2-monad]]
[[!redirects pseudo monad]]
[[!redirects pseudo-monad]]
[[!redirects pseudomonad]]
[[!redirects 2-monads]]
[[!redirects pseudomonads]]
[[!redirects pseudo-monads]]
[[!redirects strict 2-monads]]
[[!redirects 2-monadic]]

[[!redirects 2-comonad]]
[[!redirects 2-comonads]]


2-morphism

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* [[object]]

* [[morphism]]

* **2-morphism**

* [[3-morphism]]

* [[k-morphism]]

***

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## Definition

A **2-morphism** in an [[n-category]] is a [[k-morphism]] for $k = 2$: it is a higher morphism between ordinary 1-[[morphism]]s.

So in the hierarchy of $n$-categories, the first step where 2-morphisms appear is in a [[2-category]]. This includes cases such as [[bicategory]], [[2-groupoid]] or [[double category]].

## Shapes

There are different [[geometric shapes for higher structures]]: [[globe]]s, [[simplex|simplices]], [[cube]]s, etc. Accordingly, 2-morphisms may appear in different guises:

A **globular** $2$-morphism looks like this:
$$
a\mathrlap{\begin{matrix}\begin{svg}
<svg width="76" height="37" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="79929">
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  <title>Layer 1</title>
  <path marker-end="url(#se_marker_end_svg_79929_2)" id="svg_79929_2" d="m2,18.511721c31.272522,-14.782231 42.439789,-16.425501 71.625,-1.25" stroke="#000000" fill="none"/>
  <path id="svg_79929_13" marker-end="url(#se_marker_end_svg_79929_2)" d="m2,24.511721c33.286949,14.464769 40.259941,16.4624 71.500008,1.75" stroke="#000000" fill="none"/>
 </g>
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  <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_end_svg_79929_2">
   <path stroke-width="10" stroke="#000000" fill="#000000" d="m100,50l-100,40l30,-40l-30,-40l100,40z" id="svg_79929_3"/>
  </marker>
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</svg>
\end{svg}\includegraphics[width=56]{curvearrows}\end{matrix}}{\phantom{a}\space{0}{0}{13}\Downarrow\space{0}{0}{13}\phantom{a}} b
$$

A **simplicial** $2$-morphism looks like this:

$$
\begin{matrix}
    && b
    \\
    & \nearrow &\Downarrow& \searrow
    \\
    a &&\to&& c
\end{matrix}
$$

A **cubical** $2$-morphism looks like this:

$$
\begin{matrix}
  &          & b \\
  & \nearrow &            & \searrow \\
a &          & \Downarrow &          & d \\
  & \searrow &            & \nearrow \\
  &          & c
\end{matrix}
$$

Of course, using [[identity morphisms]] and [[composition]], we can turn one into the other; which is more fundamental depends on which shapes you prefer.

## Examples

* In the [[2-category]] [[Cat]], 2-morphisms are [[natural transformation]]s between functors.

* In a [[path 2-groupoid]] 2-morphisms are certain surfaces or images of surfaces in a space, going between paths in that space.

[[!redirects 2-morphism]]
[[!redirects 2-morphisms]]
[[!redirects 2-cell]]
[[!redirects 2-cells]]


2-morphism - SVG

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category: svg


2-morphisms > history

< [[2-morphisms]]
[[!redirects 2-morphisms -- history]]


2-periodic Morava K-theory

## Idea

The 2-[[periodic ring spectrum]]-version of [[Morava K-theory]].

Let $C$ be any [[elliptic curve]] over the [[prime field]] $\mathbb{F}_p$.  This will have a [[formal group]] of either [[height of a formal group law|height]] $h=1$ ("ordinary curve") or height $h=2$ ("[[supersingular elliptic curve]]").  For any such $C$, there is a [[ring spectrum]] $K_C$ with coefficient ring $\pi_* K_C=\mathbb{F}_p[u,u^{-1}]$, with $u\in \pi_2$, which is [[complex orientable cohomology theory|complex orientable]], whose formal group is the formal group of $C$. This $K_C$ is the "2-periodic Morava $K$-theory" associated to the formal group.

(grapped from [this MO comment](http://mathoverflow.net/a/201436/381) by [[Charles Rezk]]).

[[!redirects periodic Morava K-theory]]


2-periodic sphere spectrum

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## Idea

The 2-[[periodic ring spectrum]]-version of the [[sphere spectrum]]. It is formed by taking the [[Thom spectrum]] of the [[E2-operad|$E_2$]] map

$$
  \mathbb{Z}
    \;\simeq\;
  \Omega^2 B U(1)
    \longrightarrow
  \Omega^2 B U
    \longrightarrow
  B U \times \mathbb{Z}
  \,,
$$

where the last map is specified by [[Bott periodicity]] ([Lurie, Rotation Invariance, Remark 3.5.13](#LurieRotation)). It cannot be equipped with an [[E-infinity operad|$E_{\infty}$]] structure.

As a spectrum it is given by the [[direct sum]] ([[wedge sum]])
$\bigoplus_{n \in \mathbb{Z}} \mathbb{S}^{-2n}$
of all even-degree [[suspensions]] of the plain [[sphere spectrum]].

## References

* {#LurieRotation} [[Jacob Lurie]], _Rotation Invariance in Algebraic K-Theory_, ([pdf](https://people.math.harvard.edu/~lurie/papers/Waldhaus.pdf))

* [[Mohammed Abouzaid]], [[Andrew J. Blumberg]], A.2.3 in: _Arnold Conjecture and Morava K-theory_ ([arXiv:2103.01507](https://arxiv.org/abs/2103.01507)).


2-plectic geometry

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## Idea

_2-Plectic geometry_ is the higher generalization of [[symplectic geometry]], the special case of  [[n-plectic geometry]] ([[multisymplectic geometry]]) for $n = 2$. This is the input for [[higher prequantum geometry]] in degree 2.

As symplectic geometry naturally describes [[classical mechanics]] and, via [[geometric quantization]], [[quantum mechanics]], hence 1-dimensional [[quantum field theory]], so 2-plectic geometry naturally describes 2-dimensional [[classical field theory]] and, via its [[higher geometric quantization]], 2-dimensional [[QFT]].

## Examples

* A [[semisimple Lie group]] is canonically a 2-plectic manifold, with the canonical 3-form $\langle -, [-,-]\rangle$ on the [[Lie algebra]] (the canonical [[Lie algebra cohomology|Lie algebra cocycle]]), extended to a [[left invariant differential form]].

* A [[G2-manifold]] is manifold of [[dimension]] 7 characterized by carrying a 2-plectic form.

* A (pre-)[[symplectic groupoid]] is a [[Lie groupoid]] equipped with a (pre-)2-plectic structure.

## Related concepts

[[!include geometric quantization extensions - table]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

* [[Nambu mechanics]]

## References

### General

Over [[smooth manifolds]], the general setup is discussed in

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))

and considered in the general context of [[higher differential geometry]]/[[extended prequantum field theory]] in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]],

  _[[schreiber:Higher geometric prequantum theory]]_,

  _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_

### Application to the string $\sigma$-model
 {#ReferencesStringSigmaModel}

Applications to the 2-dimensional [[string]] [[sigma-model]] (see at [[string Lie 2-algebra]]) are discussed in

* {#BaezRogers10} [[John Baez]], [[Chris Rogers]], _Categorified Symplectic Geometry and the String Lie 2-Algebra_, Homology Homotopy Appl. Volume 12, Number 1 (2010), 221-236. ([arXiv:0901.4721](http://arxiv.org/abs/0901.4721),[euclid](https://projecteuclid.org/euclid.hha/1296223828)).

* [[John Baez]], [[Alexander Hoffnung]], [[Chris Rogers]], _Categorified Symplectic Geometry and the Classical String_ ([arXiv:0808.0246](http://arxiv.org/abs/0808.0246/))

A survey of some (potential) applications of 2-plectic geometry in [[string theory]] and [[M2-brane]] models is in section 2 of

* [[Christian Saemann]], [[Richard Szabo]], _Groupoid quantization of loop spaces_ ([arXiv:1203.5921](http://arxiv.org/abs/1203.5921))

and in

* [[Christian Saemann]], [[Richard Szabo]], _Quantization of 2-Plectic Manifolds_ ([arXiv:1106.1890](http://arxiv.org/abs/1106.1890))

[[!redirects 2-plectic manifold]]
[[!redirects 2-plectic manifolds]]


2-plethory

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## Idea

A **2-plethory** ([Baez-Moeller-Trimble](#BMT21)) is a categorification of a rig-[[plethory]]. Just as a rig-plethory is defined as a monoid in $(\mathsf{Birig}, \odot)$, a 2-plethory is a [[pseudomonoid]] in $(2\mathsf{Birig}, \odot)$.

The property that polynomials in $\mathbb{N}[x]$ can be composed is a consequence of the fact that  $\mathbb{N}[x]$ is the free rig in one generator, and hence represents the identity functor $\mathsf{Rig}\to \mathsf{Rig}$. Similarly, the category of [[Schur functor]]s $\mathsf{Schur}$ can be seen as a 2-rig (in the sense below) representing the identity 2-functor $2\mathsf{Rig}\to 2\mathsf{Rig}$. As a consequence, $\mathsf{Schur}$ has the structure of a 2-plethory. Taking isomorphism classes, one recovers the rig-[[plethory]] structure on the positive [[symmetric function]]s, justifying the term "categorified [[plethysm]]".

## Definitions
Let $k$ be a field of characteristic zero, and let $\mathsf{Rig}$ be the category of [[rig]]s and rig morphisms.
Let $U:\mathsf{Rig} \to \mathsf{Set}$ be  the [[forgetful functor]].

A birig is equivalently defined as:

* A rig object in $\mathsf{Rig}^\text{op}$.

* A rig $B$ and a lift of the representable functor $\mathsf{Rig}(B,-):\mathsf{Rig}\to \mathsf{Set}$ to a functor $\Phi_B:\mathsf{Rig}\to \mathsf{Rig}$ such that $U \circ \Phi_B=\mathsf{Rig}(B,-)$. Observe that any two representable 2-functors $\Phi_B$, $\Phi_B'$ can be composed. Denote the resulting object by $B\odot B'$, so that $\Phi_B \circ \Phi_B'=\Phi_{B' \circ B}$.

* A functor $\Phi:\mathsf{Rig}\to \mathsf{Rig}$ that is a right adjoint.

Denote by $(\mathsf{Birig}, \odot)$ the monoidal category of birigs, viewed as a [[full subcategory]] of $\mathsf Rig$.

A rig-[[plethory]] is equivalently defined as:

* A monoid in $(\mathsf{Birig}, \odot)$.

* A birig $B$ such that $\Phi_B$ is a comonad.

The categorified versions of the definitions above are as follows.

The category $2\mathsf{Rig}$ has:

* Objects: [[symmetric monoidal]] $k$-linear categories for which the tensor product is bilinear on hom-spaces and which are [[Cauchy complete]].

* 1-morphisms: [[symmetric monoidal]] [[linear functor]]s.

* 2-morphisms: [[symmetric monoidal]] linear natural transformations.

Let $\mathsf{U}:\mathsf{2Rig} \to \mathsf{Cat}$ be  the forgetful [[2-functor]].

A **2-birig** is a 2-rig $\mathsf B$ and a lift of the 2-functor $\mathsf{2Rig}(B,-):\mathsf{2Rig}\to \mathsf{Cat}$ to a 2-functor $\Phi_\mathsf{B}:\mathsf{2Rig}\to \mathsf{2Rig}$ such that $\mathsf U \circ \Phi_\mathsf{B}=\mathsf{2Rig}(\mathsf{B},-)$. Again, any two representable 2-functors $\Phi_\mathsf{B}$, $\Phi_\mathsf{B'}$ can be composed. Denote the resulting object by $\mathsf{B}\odot \mathsf{B'}$, so that $\Phi_\mathsf{B} \circ \Phi_\mathsf{B'}=\Phi_{\mathsf{B'}\odot \mathsf{B}}$.

Denote by $(\mathsf{2Birig}, \odot)$ the [[monoidal 2-category]] of 2-birigs, viewed as a full [[sub-2-category]] of $\mathsf{2Rig}$.

A **2-plethory** can be equivalently defined as

* A [[pseudomonoid]] in $(2\mathsf{Birig}, \odot)$.

* A 2-comonad $\Phi:\mathsf{2Rig}  \to  \mathsf{2Rig}$ whose underlying 2-functor is a right 2-adjoint.

## Example: Schur functors
For any 2-plethory $\Phi$ with left 2-adjoint $\Psi$, the composition $\mathsf{U}\Phi$ is representable, with representing object $\Psi(\overline{k\mathsf{S}})$, where $\overline{k\mathsf{S}}$ is the [[Cauchy completion]] of the $k$-linearization of the [[permutation groupoid]] $\mathsf{S}$. In turn, the 2-rig $\overline{k\mathsf{S}}$ is equivalent to $\mathsf{Schur}$ made into a 2-rig with the pointwise tensor product.

In particular, the identity $1: \mathsf{2Rig} \to \mathsf{2Rig}$ is a 2-plethory with underlying 2-rig $\overline{k\mathsf{S}}$. The set isomorphism classes $J(\overline{k\mathsf{S}})$ has a commutative monoid structure coming from the decategorifications of $\oplus$ and $\otimes$. Then, $J(\overline{k\mathsf{S}})$ is isomorphic to the monoid of positive [[symmetric functions]] $\Lambda_+$. The 2-plethory structure induces the well-known rig-plethory structure on $\Lambda_+$. In turn, the [[Grothendieck group]] of $\mathsf{Schur}$ is $K(\overline{k\mathsf{S}})=\mathbb{Z}\otimes_\mathbb{N} J(\overline{k\mathsf{S}})$, and the rig-plethory  structure on $J(\overline{k\mathsf{S}})$ extends to a ring-plethory structure on $K(\overline{k\mathsf{S}})$.

## Related concepts

* [[biring]]

* [[plethory]]

* [[plethysm]]

* [[Schur functor]]

##References

* {#TW70} D. Tall and [[Gavin Wraith|G. Wraith]], Representable functors and operations on rings, _Proc. London Math. Soc._ **3** (1970), 619--643.

* {#BW05} [[James Borger]], [[Ben Wieland]],  Plethystic algebra, _Advances in Mathematics_ **194** (2005), 246--283.  ([web](http://wwwmaths.anu.edu.au/~borger/papers/03/paper03.html))

* {#BMT21} [[John Baez]], [[Joe Moeller]], [[Todd Trimble]], Schur functors and categorified plethysm, [arXiv:2106.00190](https://arxiv.org/abs/2106.00190)


2-polycategory

A **2-polycategory** is a [[polycategory]] enriched over [[Cat]].  It can be identified with a [[double polycategory]] with only identity vertical arrows.


2-poset

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# $2$-posets
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A **2-poset** is any of several concepts that generalize ([[categorify]]) the notion of [[partial order|posets]] one step in [[higher category theory]]. One does not usually hear about $2$-posets by themselves but instead as special cases of [[2-category|$2$-categories]], such as the [[locally posetal 2-category|locally posetal]] ones.

$2$-posets can also be called **(1,2)-categories**, being a special case of [[(n,r)-categories]].  The concept generalizes to [[n-poset|$n$-posets]].

## Definition

### Explicit definition

A **2-poset** is a [[category]] $C$ such that

1. For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, there is a binary relation $R \leq_{A, B} S$
2. For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $R \leq_{A, B} R$.
3. For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, $T:Hom(A, B)$, $R \leq_{A, B} S$ and $S \leq_{A, B} T$ implies $R \leq_{A, B} T$.
4. For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, $R \leq_{A, B} S$ and $S \leq_{A, B} R$ implies $R = S$.

$C$ is only a **2-proset** if $C$ only satisfies 1-3.

### From infinity-categories

Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$2$-poset__ is an $\infty$-category such that all parallel pairs of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 2$. Thus, up to [[equivalence of categories|equivalence]], there is no point in mentioning anything beyond $2$-morphisms, not even whether two given parallel $2$-morphisms are equivalent. This definition may give a concept more general than a locally posetal $2$-category for your preferred definition of $2$-category, but it will be equivalent if you ignore irrelevant data.

## Examples

* [[Pos]]
* [[Rel]]
* [[simplex category]]
* [[Lat]]
* [[DistLat]]
* [[Frm]]
* [[Locale]]
* [[HeytAlg]]
* [[BoolAlg]]

* [[allegory]]
* [[bicategory of relations]]
* [[first-order hyperdoctrine with equality]]

Just as the motivating example of a $2$-category is the $2$-category [[Cat]] of categories, so the motivating example of a $2$-poset is the $2$-poset [[Pos]] of posets.

## Related concepts

* [[dagger 2-poset]]

* [[n-poset]]

  * [[poset]]

  * **2-poset**

[[!redirects 2-posets]]
[[!redirects 2-proset]]
[[!redirects 2-prosets]]
[[!redirects (1,2)-category]]
[[!redirects (1,2)-categories]]
[[!redirects locally ordered 2-category]]
[[!redirects locally ordered 2-categories]]
[[!redirects locally ordered]]


2-poset of partial maps

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## Contents ##
* table of contents
{:toc}

## Definition ##

Given a [[dagger 2-poset]] $A$, the **2-poset of partial maps** $Map_\bot(A)$ is the [[subobject|sub]]-[[2-poset]] whose [[objects]] are the objects of $A$ and whose [[morphisms]] are the [[functional morphism in a dagger 2-poset|functional morphisms]] of $A$.

## Examples ##

* For the dagger 2-poset [[Rel]] of [[sets]] and [[relations]], the [[2-poset]] of partial maps $Map_\bot(Rel)$ is equivalent to the [[category]] of sets and [[partial functions]] $Set_\bot$.

## See also ##

* [[partial map]]
* [[category of maps]]
* [[category of monic maps]]


2-presheaf

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#### $(\infty,2)$-Topos theory
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## Idea

The generalization of the notion of _[[presheaf]]_ from [[category theory]] to [[2-category theory]] is an [[indexed category]].

[[!redirects 2-presheaves]]


2-pretopos

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#Contents#
* table of contents
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## Idea

The generalization of the notion of _[[pretopos]]_ from [[category theory]] to [[2-category theory]].

## Definition

Let $n$ be 2, (2,1), (1,2), or 1.

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

An **$n$-pretopos** is an $n$-[[exact 2-category|exact]] [[(n,r)-category|n-category]] which is also [[extensive 2-category|extensive]].  An **infinitary $n$-pretopos** is an $n$-pretopos which is infinitary-extensive.

=--

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

As remarked [[coherent 2-category|here]], regularity plus extensivity implies coherency.  Thus an $n$-pretopos is, in particular, a [[coherent 2-category|coherent]] $n$-category.  Conversely, we have:

=--

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

An $n$-category is an $n$-pretopos if and only if it is coherent and every (finitary) $n$-[[2-polycongruence|polycongruence]] is a kernel.

=--

## Examples

* [[Cat]] is a 2-pretopos.  Likewise, [[Gpd]] is a (2,1)-pretopos and [[Pos]] is a (1,2)-pretopos.

* A 1-category is a 1-pretopos precisely when it is a [[pretopos]] in the usual sense.  Note that, as remarked for [[exact 2-category|exactness]], a 1-category is unlikely to be an $n$-pretopos for any $n\gt 1$.

* Since no nontrivial [[(0,1)-categories]] are extensive, the definition as phrased above is not reasonable for $n=(0,1)$.  However, for some purposes (such as the [[2-Giraud theorem|n-Giraud theorem]]), it is convenient to define an (infinitary) **(0,1)-pretopos** to simply be an (infinitary) coherent (0,1)-category (exactness being automatic).

## Properties

### Colimits

An $n$-pretopos has [[2-coproducts]] and [[2-quotients]] of $n$-congruences, which are an important class of colimits.  However, it can fail to admit all finite colimits, for essentially the same reason as when $n=1$: namely, some ostensibly "finite" colimits secretly involve infinitary processes.  In a 1-category, this manifests in the construction of arbitrary coequalizers and pushouts, where we must first _generate_ an equivalence relation by an infinitary process and then take its quotient.

For 2-categories it is even easier to find counterexamples: the 1-pretopos $FinSet$ does in fact have all finite colimits, but the 2-pretopos [[FinCat]] of finite categories (that is, finitely many objects and finitely many morphisms) does not have coinserters, coinverters, or coequifiers.  (The category $FPCat$ of finitely _presented_ categories does have finite colimits, but fails to have finite limits.)

However, it is natural to conjecture that just as in the case $n=1$, once an $n$-pretopos is also countably-coherent, it does become finitely cocomplete.  See [[colimits in an n-pretopos]].

## References

This is due to

* [[Mike Shulman]], _[[michaelshulman:2-pretopos]]_

based on

* [[Ross Street]], _[[StreetCBS]]_

[[!redirects 2-pretopoi]]
[[!redirects 2-pretoposes]]
[[!redirects n-pretopos]]
[[!redirects n-pretopoi]]
[[!redirects n-pretoposes]]
[[!redirects infinitary 2-pretopos]]
[[!redirects infinitary 2-pretopoi]]
[[!redirects infinitary 2-pretoposes]]
[[!redirects infinitary n-pretopos]]
[[!redirects infinitary n-pretopoi]]
[[!redirects infinitary n-pretoposes]]


2-pro-object

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# Contents
* table of contents
{:toc}

## Idea

In the context of [[2-category theory]], the concept of a **2-pro-object** is the [[categorification]] of the concept of a [[pro-object]]. A 2-pro-object in a 2-category, $\mathcal{C}$, is a 2-functor (or diagram) indexed by a 2-cofiltered 2-category. These 2-pro-objects form a 2-category, $2Pro(\mathcal{C})$, which is closed under small 2-cofiltered pseudolimits.

Pre-composition with the inclusion $c:\mathcal{C} \to 2Pro(\mathcal{C})$ is an [[equivalence of 2-categories]]:
$$
c^{\ast}: Hom(2Pro(\mathcal{C}),Cat)_+ \to Hom(\mathcal{C},Cat),
$$
where $Hom(2Pro(\mathcal{C}),Cat)_+$ is the full subcategory whose objects are those 2-functors that preserve small 2-cofiltered pseudolimits ([Descotte & Dubuc, Thrm 2.4.2](#DescotteDubuc)).

## References

* {#DescotteDubuc} [[Maria Emilia Descotte]], [[Eduardo Dubuc]], _A theory of 2-pro-objects (with expanded proofs)_, ([arXiv:1406.5762](https://arxiv.org/abs/1406.5762))

* [[Maria Emilia Descotte]], _A theory of 2-pro-objects, a theory of 2-model 2-categories and the 2-model structure for 2-Pro(C)_, ([arXiv:2010.10636](https://arxiv.org/abs/2010.10636))


2-pullback

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# $2$-pullbacks
* table of contents
{: toc}

An ordinary [[pullback]] is a [[limit]] over a [[diagram]] of the form $A \to C \leftarrow B$.  Accordingly, a __2-pullback__ (or **2-fiber product**) is a [[2-limit]] over such a diagram.

## Definition

Saying that "a 2-pullback is a 2-limit over a [[cospan]]" is in fact a sufficient definition, but we can simplify it and make it more explicit.

A __$2$-pullback__ in a [[2-category]] is a square

$$
  \array{
    P
     & \overset{p}{\to}
     &
    A
    \\
     \mathllap{^q}\big\downarrow
     & \cong &
     \big\downarrow\mathrlap{^f}
     \\
     B
     & \underset{g}{\longrightarrow} & C
  }
$$

which commutes up to [[2-isomorphism]], and which is [[universal property|universal]] among such squares in a [[2-category theory|2-category theoretic]] sense.  This means that

1. given any other such square

   $$
     \array{
       Z & \overset{v}{\longrightarrow} &A
       \\
       \mathllap{^w}\big\downarrow
       & \cong &
       \big\downarrow\mathrlap{^f}
       \\
       B & \underset{g}{\longrightarrow} &
     C
     }
   $$

   which commutes up to [[2-isomorphism]], there exists a morphism $u \colon Z\to P$ and isomorphisms $p u \cong v$ and $q u \cong w$ which are [[coherence|coherent]] with the given ones above, and

1. given any [[pair]] of morphisms $u,t \colon Z\to P$ and [[2-morphisms]] $\alpha \colon p u \to p t$ and $\beta  \colon q u \to q t$ such that $f \alpha = g \beta$ (modulo the given isomorphism $f p \cong g q$), there exists a unique 2-morphism $\gamma \colon u\to t$ such that $p \gamma = \alpha$ and $q \gamma = \beta$.

## Equivalence of definitions

The simplification in the above explicit definition has to do with the omission of an unnecessary structure map.  Note that an ordinary pullback of $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ comes equipped with maps $P\overset{p}{\to} A$, $P\overset{q}{\to} B$, and $P\overset{r}{\to} C$, but since $r = f p$ and $r = g q$, the map $r$ is superfluous data and is usually omitted.  In the 2-categorical case, where identities are replaced by isomorphisms, it is, strictly speaking, different to give merely $p$ and $q$ with an isomorphism $f p \cong g q$, than to give $p$, $q$, and $r$ with isomorphisms $r \cong f p$ and $r \cong g q$.  However, when 2-limits are considered as only defined up to equivalence (as is the default on the nLab), the two resulting notions of "2-pullback" are the same.  In much of the 2-categorical literature, the version with $r$ specified would be called a __bipullback__ and the version with $r$ not specified would be called a __bi-iso-comma-object__.

The unsimplified definition would be: a __$2$-pullback__ in a [[2-category]] is a diagram
$$\array{P & \overset{p}{\to} &A \\
  ^q\downarrow & \searrow & \downarrow^f\\
  B& \underset{g}{\to} &C }$$

in which each triangle [[commuting square|commutes]] up to [[2-isomorphism]], and which is [[universal property|universal]] among such squares in a [[2-category theory|2-category theoretic]] sense.

This means that

1. {#AnyOther} given any other such square

   $$\array{
     Z & \overset{v}{\longrightarrow} &A
     \\
     \mathllap{^w}\big\downarrow
     & \searrow &
     \big\downarrow\mathrlap{^f}
     \\
     B & \underset{g}{\longrightarrow} &C
   }$$

   in which the triangles commute up to [[2-isomorphism]], there exists a [[1-morphism]] $u\colon Z \to P$ and 2-isomorphisms $p u \cong v$ and $q u \cong w$ which are [[coherence|coherent]] with the given ones above, and

2. given any [[pair]] pf morphisms $u,t\colon Z \to P$ and 2-morphisms $\alpha\colon p u \to p t$ and $\beta\colon q u \to q t$ such that $f \alpha = g \beta$ (modulo the given isomorphism $f p \cong g q$), there exists a unique 2-cell $\gamma\colon u \to t$ such that $p \gamma = \alpha$ and $q \gamma = \beta$.

To see that these definitions are equivalent, we observe that both assert the [[representable functor|representability]] of some [[2-functor]] (where "representability" is understood in the 2-categorical "up-to-equivalence" sense), and that the corresponding 2-functors are equivalent.

* In the simplified case, the functor $F_1\colon K^{op}\to Cat$ sends an object $Z$ to the category whose

  * objects are squares commuting up to isomorphism, i.e. maps $v\colon Z\to A$ and $w\colon Z\to B$ equipped with an isomorphism $\mu\colon f v \cong g w$, and whose

  * morphisms from $(v,w,\mu)$ to $(v',w',\mu')$ are pairs $\phi\colon v\to v'$ and $\psi\colon w\to w'$ such that $\mu' . (f \phi) = (g \psi) . \mu$.

* In the unsimplified case, the functor $F_2\colon K^{op}\to Cat$ sends an object $Z$ to the category whose

  * objects consist of maps $v\colon Z\to A$, $w\colon Z\to B$, and $x\colon Z\to C$ equipped with isomorphisms $\kappa\colon f v \cong x$ and $\lambda\colon x\cong g w$, and whose

  * morphisms from $(v,w,x,\kappa,\lambda)$ to $(v',w',x',\kappa',\lambda')$ are triples $\phi\colon v\to v'$, $\psi\colon w\to w'$, and $\chi\colon x\to x'$ such that $\kappa' . (f \phi) = \chi . \kappa$ and $\lambda' . \chi = (g \psi) . \lambda$.

We have a canonical [[pseudonatural transformation]] $F_2\to F_1$ that forgets $x$ and sets $\mu = \lambda . \kappa$.  This is easily seen to be an [[equivalence]], so that any representing object for $F_1$ is also a representing object for $F_2$ and conversely.  (Note, though, that in order to define an inverse equivalence $F_1\to F_2$ we must choose whether to define $x = f v$ or $x = g w$.)

## Variations

2-pullbacks can also be identified with [[homotopy pullbacks]], when the latter are interpreted in $Cat$-enriched homotopy theory.

### Strict 2-pullbacks
 {#StrictPullback}

If we are in a [[strict 2-category]] and all the coherence isomorphisms ($\mu$, $\kappa$, $\lambda$, etc.) are required to be identities, and $u$ in property (1) is required to be unique, then we obtain the notion of a **strict 2-pullback**.  This is an example of a [[strict 2-limit]].  Note that since we must have $x = f v = g w$, the two definitions above are still the same.  In fact, they are now even isomorphic (and determined up to isomorphism, rather than equivalence).

In literature where "2-limit" means "strict 2-limit," of course "2-pullback" means "strict 2-pullback."

Obviously not every 2-pullback is a strict 2-pullback, but also not every strict 2-pullback is a 2-pullback, although the latter is true if either $f$ or $g$ is an [[isofibration]] (and in particular if either is a [[Grothendieck fibration]]).  A strict 2-pullback is, in particular, an ordinary pullback in the underlying 1-category of our strict 2-category, but it has a stronger universal property than this, referring to 2-cells as well (namely, part (2) of the explicit definition).

### Strict weighted limits

If the coherence isomorphisms $\mu$, $\kappa$, $\lambda$ in the squares are retained, but in (1) the isomorphisms $p u \cong r$ and $q u \cong s$ are required to be identities and $u$ is required to be unique, then the simplified definition becomes that of a **[[strict iso-comma object]]**, while the unsimplified definition becomes that of a **strict pseudo-pullback**.  (Iso-comma objects are so named because if the isomorphisms in the squares are then replaced by mere morphisms, we obtain the notion of (strict) [[comma object]]).

Every [[strict iso-comma object]], and every strict pseudo-pullback, is also a (non-strict) 2-pullback.  In particular, if strict iso-comma objects and strict pseudo-pullbacks both exist, they are equivalent, but they are *not* isomorphic.  (Note that their strict universal property determines them up to isomorphism, not just equivalence.)   In many strict 2-categories, such as [[Cat]], 2-pullbacks can naturally be constructed as either strict iso-comma objects or strict pseudo-pullbacks.

### Lax versions

Replacing the isomorphism $\mu$ in the simplified definition by a mere transformation results in a [[comma object]], while replacing $\kappa$ and $\lambda$ in the unsimplified definition by mere transformations results in a [[lax pullback]].  In a [[(2,1)-category]], any [[comma object]] or [[lax pullback]] is also a 2-pullback, but this is not true in a general 2-category.  Note that comma objects are often misleadingly called lax pullbacks.

## Related concepts

* [[comma object]]

[[!redirects 2-pullback]]
[[!redirects 2-pullbacks]]
[[!redirects 2-fiber product]]
[[!redirects 2-fiber products]]
[[!redirects bipullback]]
[[!redirects bipullbacks]]
[[!redirects bi-pullback]]
[[!redirects bi-pullbacks]]

[[!redirects pseudopullback]]
[[!redirects pseudopullbacks]]
[[!redirects pseudo-pullback]]
[[!redirects pseudo-pullbacks]]
[[!redirects pseudo pullback]]
[[!redirects pseudo pullbacks]]

[[!redirects strict pseudopullback]]
[[!redirects strict pseudopullbacks]]
[[!redirects strict pseudo-pullback]]
[[!redirects strict pseudo-pullbacks]]
[[!redirects strict pseudo pullback]]
[[!redirects strict pseudo pullbacks]]


2-representation

[[!redirects 2-representations]]

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#Contents#
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{:toc}

## Idea

A _2-representation_ of a [[2-group]] $G$ is an [[infinity-action]] of $G$ on a [[2-vector space]] $V$. The higher analog of the [[representation]] of a [[group]].

## Related concepts

* [[representation]], [[infinity-representation]]

## References

* [[Zhen Huan]], _2-Representations of Lie 2-groups and 2-Vector Bundles_ ([arXiv:2208.10042](https://arxiv.org/abs/2208.10042))

On [[2-representations]] of the [[string 2-group]] on [[2-vector spaces]] and the construction of the [[stringor bundle]]:

* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *A representation of the string 2-group*, [[arXiv:2206.09797](https://arxiv.org/abs/2206.09797)]

Reviewed in:

* [[Konrad Waldorf]], *The stringor bundle*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#WaldorfMar2023)]


2-rig

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# $2$-rigs
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{: toc}

## Idea

The notion of  _$2$-rig_ is supposed to be a [[categorification]] of that of a [[rig]].  Several inequivalent formalizations of this idea are in the literature.

Just as a rig is a multiplicative [[monoid]] whose underlying set also has a notion of addition, so a $2$-rig is a [[monoidal category]] whose underlying category also has a notion of addition, and we can describe this notion of addition in a few different ways.

Note that we don\'t expect a $2$-rig to have additive inverses; by the same argument as in the [[Eilenberg swindle]], they are unreasonable to expect.  However, in a monoidal [[abelian category]], we have as close to additive inverses as is reasonable and so a categorification of a [[ring]].

Compare also the notion of [[rig category]].

## Definitions
 {#Definitions}

Since [[categorification]] involves some arbitrary choices that will be determined by the precise intended application, there is a bit of flexibility of what exactly one may want to call a _2-ring_. We first list some immediate possibilities of classes of monoidal and enriched categories that one may want to think of as 2-rings:

* [Enriched monoidal categories](#EnrichedMonoidalCategories)

But a central aspect of an ordinary ring is the [[distributivity law]] which says that the product in the ring preserves sums. Since sums in a 2-ring are given by [[colimits]], this suggests that a 2-ring should be a cocomplete category which is compatibly [[monoidal category|monoidal]] in that the the tensor product preserves colimits:

* [Compatibly monoidal cocomplete categories](#MonoidalCompleteCateories)

But there are still more properties which one may want to enforce, notably that homomorphisms of 2-rings form a [[2-abelian group]]. This is achieved by demanding the underlying category to be not just cocomplete by [[presentable category|presentable]]:

* [Compatibly monoidal presentable categories](#CompatiblyMonoidalPresentableCategories).

### Enriched monoidal categories
 {#EnrichedMonoidalCategories}

1.  A __$2$-rig__ might be an [[Ab-enriched category]] which is [[enriched monoidal category|enriched monoidal]].

2.  A __$2$-rig__ might be an [[additive category]] which is enriched monoidal.

3.  A __$2$-rig__ might be a [[distributive monoidal category]]: a monoidal category with finite [[coproducts]] such that the monoidal product distributes over the coproducts.

4.  A __$2$-rig__ might be a [[closed monoidal category]] with finite coproducts.

5.  Finally, a __$2$-ring__ is a monoidal [[abelian category]].

Note that (2) is a special case of both (1) and (3), which are independent.  (4) is a special case of (3), by the [[adjoint functor theorem]].  (5) is a special case of (2), of course.

### Compatibly monoidal cocomplete categories
 {#MonoidalCompleteCateories}

In ([Baez-Dolan](#BaezDolan)) the following is considered:

+-- {: .num_defn #BD2Rig}
###### Definition

A _2-rig_ is a [[monoidal category|monoidal]] [[colimit|cocomplete category]] where the [[tensor product]] respects [[colimits]].

=--

One can define [[braided monoidal category|braided]] and [[symmetric monoidal category|symmetric]] 2-rigs in this sense (and indeed, also in the other senses listed above).  In particular, there is a [[2-category]] $\mathbf{Symm2Rig}$ with:

* symmetric monoidal cocomplete categories where the monoidal product distributes over colimits as objects,

* symmetric monoidal cocontinuous functors as 1-morphisms,

* symmetric monoidal natural transformations as 2-morphisms.

### Compatibly monoidal presentable categories
 {#CompatiblyMonoidalPresentableCategories}

The following refines the [above](#MonoidalCompleteCateories) by demanding the underlying category of a 2-ring to be not just cocomplete but even a [[presentable category]]. This was motivated in ([CJF, remark 2.1.10](#CJF)).

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

Write

$$
  2 Ab \in 2Cat
$$

for the [[2-category]] of [[presentable categories]] and [[colimit]]-preserving [[functors]] between them.

=--

([CJF, def. 2.1.8](#CJF))

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

By the [[adjoint functor theorem]] this is equivalently the 2-category of presentable categories and [[left adjoint]] functors between them.

=--

+-- {: .num_example #CategoryOfModulesAs2AbelianGroup}
###### Example

Given an ordinary [[ring]] $R$, its [[category of modules]] $Mod_R$ is presentable, hence may be regarded as a 2-abelian group.

=--

([CJF, example 2.1.5](#CJF))

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

The 2-category $2Ab$ is a [[closed monoidal 2-category|closed]] [[symmetric monoidal 2-category]] with respect to the [[tensor product]] $\boxtimes \colon 2Ab \times 2Ab \to 2Ab$ such that for $A,B, C \in 2Ab$, $Hom_{2Ab}(A \boxtimes B, C)$ is equivalently the full [[subcategory]] of [[functor category]] $Hom_{Cat}(A \times B, C)$ on those that are [[bilinear function|bilinear]] in that they preserve [[colimits]] in each argument separately.

=--

See also at [[Pr(∞,1)Cat]] for more on this.

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

For $\mathcal{C}$ a [[small category]], the [[category of presheaves]] $Set^{\mathcal{C}}$ is [[presentable category|presentable]] and

$$
  Set^{\mathcal{C}_1} \boxtimes
  Set^{\mathcal{C}_2}
  \simeq
  Set^{\mathcal{C}_1 \times \mathcal{C}_2}
  \,.
$$

=--

+-- {: .num_example #TensorProductModuleCategoriesAsOf2AbelianGroups}
###### Example

For $R$ a [[ring]] the [[category of modules]] $Mod_R$ is presentable and

$$
  Mod_{R_1} \boxtimes Mod_{R_2}
  \simeq
  Mod_{R_1 \otimes R_2}
  \,,
$$

=--

([CJF, example 2.2.7](#CJF))

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

For $R_1, R_2$ two rings,
the category  of 2-abelian group homomorphisms between the [[categories of modules]] is [[natural equivalence|naturally equivalent]] to that of $R_1$-$R_2$-[[bimodules]] and their [[intertwiners]]:

$$
  (-)\otimes (-)
  \;\colon\;
  {}_{R_1}Mod_{R_2}
  \stackrel{\simeq}{\to}
  Hom_{2Ab}(Mod_{R_1}, Mod_{R_2})
  \,.
$$

The equivalence sends a bimodule $N$ to the functor given by the [[tensor product]] over $R_1$:

$$
  (-) \otimes N \;\colon\;
  Mod_{R_1} \to Mod_{R_2}
  \,.
$$

=--

This is the [[Eilenberg-Watts theorem]].

+-- {: .num_defn #2RingAsCompatiblyMonoidalPresentableCategory}
###### Definition

Write

$$
  2Ring \in 2Cat
$$

for the [[2-category]] of [[monoid objects]] [[internalization|internal]] to $2 Ab$. An [[object]] of this 2-category we call a **2-ring**.

Equivalently, a 2-ring in this sense is a [[presentable category]] equipped with the structure of a [[monoidal category]] where the [[tensor product]] preserves [[colimits]].

=--

([CJF, def. 2.1.8](#CJF))

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

The category [[Set]] with its [[cartesian product]] is a 2-ring and it is the [[initial object]] in $2Ring$.

=--

([CJF, example 2.3.4](#CJF))

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

The category [[Ab]] of [[abelian groups]] with its standard [[tensor product of abelian groups]] is a 2-ring.

=--

+-- {: .num_example #CommutativeRingGivesCommutative2Ring}
###### Example

For $R$ an ordinary [[commutative ring]], $Mod_R$ equipped with its usual [[tensor product of modules]] is a commutative 2-ring.

=--

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

For $R$ an ordinary [[ring]] and $Mod_R$ its ordinary [[category of modules]], regarded as a 2-abelian group by example \ref{CategoryOfModulesAs2AbelianGroup}, the structure of a 2-ring on $Mod_R$ is equivalently the structure of a [[sesquiunital sesquialgebra]] on $R$.

If $R$ is in addition a [[commutative ring]] that $Mod_R$ is a commutative 2-ring and is canonically an $Ab$-[[2-algebra]] in that

$$
  Ab \simeq Mod_{\mathbb{Z}} \to Mod_R
  \,.
$$

=--

([CJF, example 2.3.7](#CJF))

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

For $A$ a 2-ring, def. \ref{2RingAsCompatiblyMonoidalPresentableCategory}, write

$$
  2Mod_A \in 2Cat
$$

for the [[2-category]] of [[module objects]] over $A$ in $2Ab$.

This means that a 2-module over $A$ is a [[presentable category]] $N$ equipped with a functor

$$
  A \boxtimes N \to N
$$

which satisfies the evident action property.

=--

([CJF, def. 2.3.3](#CJF))

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

Let $R$ be an ordinary [[commutative ring]] and $A$ an ordinary $R$-[[associative algebra|algebra]]. Then by example \ref{CategoryOfModulesAs2AbelianGroup} $Mod_A$ is a 2-abelian group and by example
\ref{CommutativeRingGivesCommutative2Ring} $Mod_R$ is a commutative ring. By example \ref{TensorProductModuleCategoriesAsOf2AbelianGroups}
$Mod_R$-[[2-module]] structures on $Mod_A$

$$
  Mod_R \boxtimes \Mod_A \to Mod_A
$$

correspond to colimit-preserving functors

$$
  Mod_{R \otimes_{\mathbb{Z}} A} \to Mod_{A}
$$

that satisfy the action property. Such as presented under the [[Eilenberg-Watts theorem]], prop. \ref{EilenbergWattsTheorem}, by $R \otimes_{\mathbb{Z}} A$-$A$ [[bimodules]]. $A$ itself is canonically such a bimodule and it exhibits a $Mod_R$-[[2-module]] structure on $Mod_A$.

=--

## Properties

### Initial object
 {#PropertiesInitialObject}

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

The analog role in 2-rigs to the role played by the [[natural numbers]] among ordinary [[rigs]] should be played by the standard [[categorification]] of the natural numbers: the [[category of finite sets]]. One is therefore inclined to demand that a reasonable definition of 2-rigs should be such that $FinSet$ is the [[initial object]] (in the suitably higher categorical sense) in the [[2-category]] of 2-rigs.

For the notion in def. \ref{BD2Rig} this was conjectured by [[John Baez]], for the notion in def. \ref{2RingAsCompatiblyMonoidalPresentableCategory} this is asserted in ([Chirvasitu & Johnson-Freyd, example 2.3.4](#CJF)).

=--

### Tannaka duality
 {#TannakaDuality}

[[!include structure on algebras and their module categories - table]]

## Related concepts

* [[2-algebraic geometry]]

* A further slight variant of compatibly monoidal cocomplete categories is that of _monoidal [[vectoids]]_.

* [[distributivity for monoidal structures]]

* [[prime spectrum of a monoidal stable (∞,1)-category]]

## References

The proposal that a 2-ring should be a compatibly monoidal cocomplete category is due to

* {#BaezDolan} [[John Baez]], [[James Dolan]], _Higher-dimensional algebra III: $n$-categories and the algebra of opetopes_, _Adv. Math._ **135** (1998), 145-206.  ([arXiv](http://arxiv.org/abs/q-alg/9702014))

The proposal that a 2-ring should be a compatibly monoidal presentable category is due to

* {#CJF} [[Alexandru Chirvasitu]], [[Theo Johnson-Freyd]], _The fundamental pro-groupoid of an affine 2-scheme_, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. ([DOI](http://dx.doi.org/10.1007/s10485-011-9275-y), [arXiv:1105.3104](http://arxiv.org/abs/1105.3104))

see also

* {#Brandenburg14} [[Martin Brandenburg]], _Tensor categorical foundations of algebraic geometry_ ([arXiv:1410.1716](http://arxiv.org/abs/1410.1716))

This is related to

* [[Jacob Lurie]], _[[Tannaka duality for geometric stacks]]_.

A similar notion is that of "monoidal [[vectoid]]" due to

* [[Nikolai Durov]], _Classifying vectoids and generalisations of operads_, Proc. of Steklov Inst. of Math. __273__:1, 48-63 (2011) [arxiv/1105.3114](http://arxiv.org/abs/1105.3114)), the translation of "Классифицирующие вектоиды и классы операд", Trudy MIAN, vol. 273

The role of presentable categories as higher analogs abelian groups in the context of [[(infinity,1)-categories]] have been made by [[Jacob Lurie]],  see at _[[Pr(infinity,1)Cat]]_.

Another, more algebraic, notion of a categorical ring is introduced in

* M. Jibladze , T. Pirashvili, _Third Mac Lane cohomology via categorical rings_, 	J. of homotopy and related structures, __2__(2), 2007, 187–221 [pdf](http://www.emis.de/journals/JHRS/volumes/2007/n2a10/v2n2a10.pdf) [math.KT/0608519](http://arxiv.org/abs/math/0608519)

[[!redirects 2-rig]]
[[!redirects 2-rigs]]
[[!redirects 2-ring]]
[[!redirects 2-rings]]

[[!redirects 2Rig]]
[[!redirects 2Ring]]


2-sheaf

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _2-sheaf_ is the generalization of the notion of [[sheaf]] to the [[higher category theory]] of [[2-categories]]/[[bicategories]].
A 2-category of 2-sheaves forms a [[2-topos]].

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

A _2-sheaf_ is a higher sheaf of [[categories]]. More restrictive than this is a higher sheaf with values in [[groupoids]], which would be a _[[(2,1)-sheaf]]_. Both these notions are often referred to as **[[stack]]**, or sometimes "stack of groupoids" and "stack of categories" for definiteness. But moreover, traditionally a [[stack]] (in either flavor) is considered only over a [[1-site]], whereas it makes sense to consider [[(2,1)-sheaves]] more generally over [[(2,1)-sites]] and 2-sheaves over [[2-sites]].

Therefore, saying "2-sheaf" serves to indicate the full generality of the notion of higher sheaves in [[2-category theory]], as opposed to various special cases of this general notion which have traditionally been considered.

=--

## Definition

Let $C$ be a [[2-site]] having finite [[2-limit]]s (for convenience).  For a covering family $(f_i:U_i\to U)_i$ we have the comma objects
+--{: style="text-align:center"}
<svg xmlns="http://www.w3.org/2000/svg" width="10em" height="10em" viewBox="-30 -20 180 150">
 <desc>Comma Square</desc>
 <defs>
  <marker id="svg295arrowhead" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="8" markerHeight="5" orient="auto">
   <path d="M 0 0 L 10 5 L 0 10 z"/>
  </marker>
 </defs>
 <g font-size="16" markdown="1">
  <foreignObject x="-10"  y="0" width="45" height="25">$(f_i/f_j)$</foreignObject>
  <foreignObject x="100"  y="0" width="20" height="20">$U_i$</foreignObject>
  <foreignObject x="0"  y="100" width="20" height="20">$U_j$</foreignObject>
  <foreignObject x="100"  y="100" width="20" height="20">$U$</foreignObject>
  <foreignObject x="110"  y="50" width="20" height="25">$f_i$</foreignObject>
  <foreignObject x="50"  y="110" width="20" height="25">$f_j$</foreignObject>
  <foreignObject x="-20"  y="50" width="20" height="30">$q_{i j}$</foreignObject>
  <foreignObject x="50"  y="-20" width="20" height="30">$p_{i j}$</foreignObject>
  <foreignObject x="60"  y="60" width="20" height="25">$\mu_{i j}$</foreignObject>
 </g>
 <g fill="none" stroke="#000" stroke-width="1.5" marker-end="url(#svg295arrowhead)">
   <line x1="40" y1="10" x2="90" y2="10"/>
   <line x1="30" y1="110" x2="90" y2="110"/>
   <line y1="30" x1="10" y2="90" x2="10"/>
   <line y1="30" x1="110" y2="90" x2="110"/>
   <line x1="65" y1="55" x2="55" y2="65"/>
 </g>
</svg>
=--

We also have the [[nlab:double comma object|double comma objects]] $(f_i/f_j/f_k) = (f_i/f_j)\times_{U_j} (f_j/f_k)$ with projections $r_{i j   k}:(f_i/f_j/f_k)\to (f_i/f_j)$, $s_{i j k}:(f_i/f_j/f_k)\to (f_j/f_k)$, and $t_{i j k}:(f_i/f_j/f_k)\to (f_i/f_k)$.

Now, a functor $X:C^{op} \to Cat$ is called a **2-presheaf**.  It is **1-separated** if

* For any covering family $(f_i:U_i\to U)_i$ and any $x,y\in X(U)$ and $a,b: x\to y$, if $X(f_i)(a) = X(f_i)(b)$ for all $i$, then $a=b$.

It is **2-separated** if it is 1-separated and

* For any covering family $(f_i:U_i\to U)_i$ and any $x,y\in X(U)$, given $b_i:X(f_i)(x) \to X(f_i)(y)$ such that $\mu_{i j}(y) \circ X(p_{i j})(b_i) = X(q_{i j})(b_i) \circ \mu_{i j}(x)$, there exists a (necessarily unique) $b:x\to y$ such that $b_i = X(f_i)(b)$.

It is a **2-sheaf** if it is 2-separated and

* For any covering family $(f_i:U_i\to U)_i$ and any $x_i\in X(U_i)$ together with morphisms $\zeta_{i j}:X(p_{i j})(x_i) \to X(q_{i j})(x_j)$ such that the following diagram commutes:
$$\array{X(r_{i j k})X(p_{i j})(x_i) &
  \overset{X(r_{i j k})(\zeta_{i j})}{\to} &
  X(r_{i j k})X(q_{i j})(x_j) & \overset{\cong}{\to} &
  X(s_{i j k})X(p_{j k})(x_j)\\
  ^\cong \downarrow && && \downarrow ^{X(s_{i j k})(\zeta_{j k})}\\
  X(t_{i j k}) X(p_{i k})(x_i) &
  \underset{X(t_{i j k})(\zeta_{i k})}{\to} &
  X(t_{i j k}) X(q_{i k})(x_k) & \underset{\cong}{\to} &
  X(s_{i j k}) X(q_{j k})(x_k)}$$
there exists an object $x\in X(U)$ and isomorphisms $X(f_i)(x)\cong x_i$ such that for all $i,j$ the following square
commutes:
$$\array{
  X(p_{i j})X(f_i)(X) & \overset{\cong}{\to}
  &     X(p_{i j})(x_i)\\
  ^{X(\mu_{i j})}\downarrow && \downarrow^{\zeta_{i j}}\\
  X(q_{i j})X(f_j)(x)  & \underset{\cong}{\to} &
  X(q_{i j})(x_j).}$$

A 2-sheaf, especially on a 1-site, is frequently called a **[[stack]]**. However, this has the unfortunate consequence that a 3-sheaf is then called a 2-stack, and so on with the numbering all offset by one. Also, it can be helpful to use a new term because of the notable differences between 2-sheaves on 2-sites and 2-sheaves on 1-sites. The main novelty is that $\mu_{i j}$ and $\zeta_{i j}$ _need not be invertible_.

Note, though, they must be invertible as soon as $C$ is (2,1)-site: $\mu_{i j}$ by definition and $\zeta_{i j}$ since an inverse is provided by $\iota_{i j}^*(\zeta_{i j})$, where $\iota_{i j}\mapsto (f_i/f_j) \to (f_j/f_i)$ is the symmetry equivalence.

If $C$ lacks finite limits, then in the definitions of "2-separated" and "2-sheaf" instead of the comma objects $(f_i/f_j)$, we need to use arbitrary objects $V$ equipped with maps $p:V\to U_i$, $q:V\to U_j$, and a 2-cell $f_i p \to f_j q$.  We leave the precise definition to the reader.

A 2-site is said to be **subcanonical** if for any $U\in C$, the representable functor $C(-,U)$ is a 2-sheaf.  When $C$ has finite limits, it is easy to verify that this is true precisely when every covering family is a (necessarily pullback-stable) quotient of its kernel [[2-polycongruence]].  In particular, the regular coverage on a regular 2-category is subcanonical, as is the coherent coverage on a coherent 2-category.

The 2-category $2Sh(C)$ of 2-sheaves on a small 2-site $C$ is, by definition, a [[Grothendieck 2-topos]].

## Properties

### Characterization of over $(n,r)$-sites

If the underlying [[2-site]] happens to be an [[(n,r)-site]] for $n$ and/or $r$ lower than 2, there may be other equivalent ways to think of 2-sheaves.

A [[2-topos]] with a [[2-site]] of definition that happens to be just a 1-site or [[(2,1)-site]] is _1-localic_ or _(2,1)-localic_.

#### Over a 1-site

Over a 1-site, the [[Grothendieck construction]] says that [[2-functors]] on the site are equivalent to [[fibered categories]] over the site. Hence in this case the theory of 2-sheaves can be entirely formulated in terms of fibered categories. See _[References -- In terms of fibered categories](#InTermsOfFiberedCategories)_.

Also, over a 1-site a 2-sheaf is essentially a _[[indexed category]]_. Therefore stacks over 1-sites can also be discussed in this language, see notably the work ([Bunge-Pare](#BungePare)).

In particular, if the 1-site $C$ is a [[topos]], then every topos _over_ $C$ as its [[base topos]] (a $C$-topos) induces an [[indexed category]].

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

If $C$ is a topos and $E$ is a $C$-topos, then (the [[indexed category]] corresponding to) $E$ is a 2-sheaf on $C$ with respect to the [[canonical topology]].

=--

This appears as ([Bunge-Pare, corollary 2.6](#BungePare)).

Moreover, over a [[1-site]] the [[2-topos]] of 2-sheaves ought to be equivalent to the (suitably defined) [[2-category]] of [[internal categories]] in the underlying [[1-topos]]. See _[References -- In terms of internal categories](#ReferencesInTermsOfInternalCategories)_.

#### Over a $(2,1)$-site -- As internal categories

Over a [[(2,1)-site]] the [[2-topos]] of 2-sheaves ought to be equivalent to the [[2-category]] of [[internal (infinity,1)-categories]] in the corresponding [[(2,1)-topos]].

This is discussed at _[2-Topos -- In terms of internal categories](2-topos#InTermsOfInternalCategories)_.

## Examples

### Codomain fibrations / sheaves of modules

A classical class of examples for 2-sheaves are [[codomain fibrations]]
over suitable sites, or rather their [[tangent categories]]. As discussed there, this includes the case of sheaves of categories of [[modules]] over sites of [[algebra over an algebraic theory|algebras]].

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

For $C$ an [[exact category]] with [[finite limits]], the [[codomain fibration]] $Cod : C^I \to C$ or equivalently (under the [[Grothendieck construction]]), the self-[[indexed category|indexing]] of $C$ is a 2-sheaf with respect to the [[canonical topology]].

=--

This is for instance ([Bunge-Pare, corollary 2.4](#BungePare)).

## Related concepts

* [[presheaf]] /  [[sheaf]] / [[cosheaf]]

* **2-sheaf** / [[stack]]

* [[(∞,1)-sheaf]] / [[∞-stack]]

* [[(∞,2)-sheaf]]

* [[(∞,n)-sheaf]]

* [[descent]]

## References

Historically, the original definition of _[[stack]]_
included the case of category-valued functors, hence of 2-sheaves, in:

* [[Jean Giraud]], *Cohomologie non abélienne* Grundlehren **179**, Springer (1971) [[doi:10.1007/978-3-662-62103-5](https://www.springer.com/gp/book/9783540053071)]

* [[Jean Giraud]], _Classifying topos_, in: [[William Lawvere]] (ed.) _Toposes, Algebraic Geometry and Logic_, Lecture Notes in Mathematics **274**, Springer (1972) [[doi:10.1007/BFb0073964](https://doi.org/10.1007/BFb0073964)]

### In terms of categories internal to sheaf toposes
 {#ReferencesInTermsOfInternalCategories}

Category-valued stacks as [[internal categories]] in the underlying [[sheaf topos]]:

* {#BungePare} [[Marta Bunge]], [[Robert Paré]], _Stacks and equivalence of indexed categories_, [[Cahiers|Cahiers de Top. et Géom. Diff. Catég]] **20** 4 (1979) 373-399 [[numdam:CTGDC_1979__20_4_373_0](http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0)]

* {#Bunge} [[Marta Bunge]], *Stack completions and Morita equivalence for categories in a topos*, [[Cahiers|Cahiers de Top. et Géom. Diff. Catég]] **20** 4, (1979) 401-436 [[numdam](http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0), [MR558106](http://www.ams.org/mathscinet-getitem?mr=558106)]

* {#JoyalTierney} [[André Joyal]], [[Myles Tierney]], section 3 of: *Strong stacks and classifying spaces*, in: *Category Theory* ([[Como]], 1990), Lecture Notes in Mathematics **1488**, Springer (1991) 213-236 [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]

  > (establishing the [[canonical model structure on Cat]] in the internal generality)

### In terms of fibered categories
 {#InTermsOfFiberedCategories}

A discussion of stacks over [[1-sites]] in terms of their [[Grothendieck construction|associated]] [[fibered categories]] is in

* [[Angelo Vistoli]], _Notes on Grothendieck topologies, fibered categories and descent theory_ ([pdf](http://homepage.sns.it/vistoli/descent.pdf))

### 2-Sites

The above text involves content transferred from

* [[Michael Shulman]], _[[michaelshulman:2-site]]_

2-sites were earlier considered in

* [[Ross Street]], _[[StreetCBS]]_

[[!redirects 2-sheaves]]


2-site

This entry is about the notion of [[site]] in [[2-category]] theory. For the notion "bisite" of a 1-categorical site equipped with two coverages see instead [[separated presheaf]].

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

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

## Idea

The notion of [[2-site]] is the generalization of the notion of [[site]] to the [[higher category theory]] of [[2-categories]] ([[bicategories]]).

Over a 2-site one has a [[2-topos]] of [[2-sheaves]].

## Definition

A **[[coverage]]** on a [[2-category]] $C$ consists of, for each object $U\in C$, a collection of families $(f_i: U_i\to U)_i$ of morphisms with codomain $U$, called _covering families_, such that

* If $(f_i:U_i\to U)_i$ is a covering family and $g:V\to U$ is a morphism, then there exists a covering family $(h_j:V_j\to V)_j$ such that each composite $g h_j$ factors through some $f_i$, up to isomorphism.

This is the 2-categorical analogue of the 1-categorical notion of [[nlab:coverage|coverage]] introduced in the [[nlab:Elephant|Elephant]].

A 2-category equipped with a coverage is called a **2-site**.

## Examples

* If $C$ is a [[michaelshulman:regular 2-category]], then the collection of all singleton families $(f:V\to U)$, where $f$ is eso, forms a coverage called the _regular coverage_.

* Likewise, if $C$ is a [[michaelshulman:coherent 2-category]], the collection of all finite jointly-eso families forms a coverage called the _coherent coverage_.

* On $Cat$, the _canonical coverage_ consists of all families that are jointly essentially surjective on objects.

## Saturation conditions

A **pre-Grothendieck coverage** on a 2-category is a coverage satisfying the following additional conditions:

* If $f:V\to U$ is an equivalence, then the one-element family $(f:V\to U)$ is a covering family.

* If $(f_i:U_i\to U)_{i\in I}$ is a covering family and for each $i$, so is $(h_{i j}:U_{i j} \to U_i)_{j\in J_i}$, then $(f_i h_{i j}:U_{i j}\to U)_{i\in I, j\in U_i}$ is also a covering family.

This is the 2-categorical version of a [[Grothendieck pretopology]] (minus the common condition of having actual [[pullbacks]]).

Now, a **sieve** on an object $U\in C$ is defined to be a functor $R:C^{op}\to Cat$ with a transformation $R\to C(-,U)$ which is objectwise fully faithful (equivalently, it is a [[fully faithful morphism]] in $[C^{op},Cat]$).  Equivalently, it may be defined as a subcategory of the [[slice 2-category]] $C/U$ which is closed under precomposition with all morphisms of $C$.

Every family $(f_i\colon U_i\to U)_i$ generates a sieve by defining $R(V)$ to be the full subcategory of $C(V,U)$ on those $g:V\to U$ such that $g \cong f_i h$ for some $i$ and some $h:V\to U_i$.  The following observation is due to [StreetCBS](#StreetCBS).

+--{: .num_lemma}
###### Lemma
A 2-presheaf $X:C^{op}\to Cat$ is a 2-sheaf for a covering family $(f_i:U_i\to U)_i$ if and only if
$$X(U) \simeq[C^{op},Cat](C(-,U),X) \to [C^{op},Cat](R,X)$$
is an equivalence, where $R$ is the sieve on $U$ generated by $(f_i:U_i\to U)_i$.
=--

Therefore, just as in the 1-categorical case, it is natural to restrict attention to covering _sieves_.  We define a **Grothendieck coverage** on a 2-category $C$ to consist of, for each object $U$, a collection of sieves on $U$ called covering sieves, such that

* If $R$ is a covering sieve on $U$ and $g:V\to U$ is any morphism, then $g^*(R)$ is a covering sieve on $V$.

* For each $U$ the sieve $M_U$ consisting of _all_ morphisms into $U$ (the sieve generated by the singleton family $(1_U)$) is a covering sieve.

* If $R$ is a covering sieve on $U$ and $S$ is an arbitrary sieve on $U$ such that for each $f:V\to U$ in $R$, $f^*(S)$  is a covering sieve on $V$, then $S$ is also a covering sieve on $U$.

Here if $R$ is a sieve on $U$ and $g:V\to U$ is a morphism, $g^*(R)$ denotes the sieve on $V$ consisting of all morphisms $h$ into $V$ such that $g h$ factors, up to isomorphism, through some morphism in $R$.

As in the 1-categorical case, one can then show that every coverage generates a unique Grothendieck coverage having the same 2-sheaves.

## Properties

* The [[2-category]] of [[2-sheaf|2-sheaves]] on a 2-site is a [[Grothendieck 2-topos]].

* If $C$ is a [[1-category]] regarded as a 2-category with only [[identity morphism|identity]] [[2-morphisms]], then a coverage (pretopology, topology) on $C$ reduces to the usual notion of [[coverage]], [[Grothendieck pretopology]], or [[Grothendieck topology]].

## Related concepts

* [[(n,r)-site]]

* [[1-site]]

* **2-site**, [[(2,1)-site]]

* [[(∞,1)-site]]

  * [[model site]], [[simplicial site]]

## References

Strict 2-sites were considered in

* [[Ross Street]], _Two-dimensional sheaf theory_ J. Pure Appl. Algebra __23__ (1982), no. 3, 251-270, [MR83d:18014](http://www.ams.org/mathscinet-getitem?mr=644277), doi

Bicategorical 2-sites in

* [[Ross Street]], [[zoranskoda:Characterization of Bicategories of Stacks]], [MR84d:18006](http://www.ams.org/mathscinet-getitem?mr=682967), p. 282-291 in: Category theory (Gummersbach 1981) Springer LNM __962__, 1982
{: #StreetCBS }

See also _[[StreetCBS]]_.

More discussion is in

* [[Michael Shulman]], _[[michaelshulman:2-site]]_

[[!redirects 2-sites]]

[[!redirects (2,2)-site]]
[[!redirects (2,2)-sites]]

[[!redirects bisite]]
[[!redirects bisites]]
[[!redirects Grothendieck 2-topology]]
[[!redirects 2-Grothendieck topology]]
[[!redirects Grothendieck 2-topologies]]
[[!redirects 2-Grothendieck topologies]]
[[!redirects Grothendieck 2-pretopology]]
[[!redirects 2-Grothendieck pretopology]]
[[!redirects Grothendieck 2-pretopologies]]
[[!redirects 2-Grothendieck pretopologies]]
[[!redirects 2-coverage]]
[[!redirects 2-coverages]]
[[!redirects topology on a 2-category]]
[[!redirects topologies on 2-categories]]
[[!redirects Grothendieck topology on a 2-category]]
[[!redirects Grothendieck topologies on 2-categories]]
[[!redirects coverage on a 2-category]]
[[!redirects coverages on 2-categories]]
[[!redirects topology on a bicategory]]
[[!redirects topologies on bicategories]]
[[!redirects Grothendieck topology on a bicategory]]
[[!redirects Grothendieck topologies on bicategories]]
[[!redirects coverage on a bicategory]]
[[!redirects coverages on bicategories]]


2-spectral triple

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#### Noncommutative geometry
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[[!include noncommutative geometry - contents]]
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#### Functorial quantum field theory
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[[!include functorial quantum field theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

An ordinary [[spectral triple]] is, as discussed there, the abstract algebraic data characterizing [[supersymmetric quantum mechanics]] on a [[worldline]] and thereby spectrally encoding an effective (possibly [[non-commutative geometry|non-commutative]], hence "[[non-geometric background|non-geometric]]") [[target space]] [[geometry]].
Ordinary [[Riemannian geometry]] with [[spin structure]] is the special case of this where the [[Hilbert space]] in the spectral triple is that of [[square integrable function|square integrable]] [[sections]] of the [[spinor bundle]] on a [[spin structure|spin]] [[Riemannian manifold]] and the operator $D$ acting on that is the standard [[Dirac operator]], hence the "supercharge" of the worldline supersymmetry of the [[spinning particle]].

In generalization of this, a "2-spectral triple" should be the analogous algebraic data that encodes the [[worldsheet]] theory of a [[superstring]] propagating on a [[target space]] geometry which is a generalization of [[Riemannian geometry]] with ([[twisted string structure|twisted]]) [[string structure]].

Of course such data is just that of a [[2d superconformal field theory]], realized locally by,  for instance, a [[vertex operator algebra]] or by a [[conformal net]] of [[local net of observables|local observables]]. But for emphasis it may be useful to speak of such data as constituting a "2-spectral triple", for emphasizing more the important and intricate relation to the concept of [[spectral triples]], which in much of the literature seems to be unduly ignored.

| [[quantum system]]    | [[supercharge]] | formalization | algebra |
|----------------------|---------|------------------------|-----|
| quantum [[spinning particle]] | [[Dirac operator]] | [[spectral triple]]    | [[operator algebra]]
| quantum [[spinning string]] |  [[Dirac-Ramond operator]] | [[2d SCFT]] | [[vertex operator algebra]] |

That the 0-mode sector of a [[2d SCFT]] -- hence the quantum point [[particle]] limit of a quantum [[superstring]] dynamics -- yields a [[spectral triple]] was maybe first highlighted in ([Fröhlich-Gawędzki 93](#FroehlichGawedzki93)) by way of a series of concrete examples, such as the [[WZW model]].

Here the role of the [[Dirac operator]] of the spectral triple is played by the [[Dirac-Ramond operator]] of the [[superstring]], hence the operator whose [[index]] (in the [[large volume limit]]) is the [[Witten genus]].

That hence the superstring quantum theory should be regarded as a kind of higher spectral triple was maybe first suggested in ([Chamseddine 97](#Chamseddine97)), together with arguments that the associated [[spectral action]] indeed reproduces the [[action functional]] of the string's [[target space]] [[effective quantum field theory|effective]] [[supergravity]] theory. An exposition of this perspective is in ([Fröhlich-Grandjean-Recknagel 97, section 7.2](#FroehlichGrandjeanRecknagel97)).

From [Fröhlich 92, p. 11](non-perturbative+quantum+field+theory#Froehlich92):

> I still have hopes, perhaps romantic ones, that [[string theory]], or something inspired by it, will come back to life again. I believe it is interesting to attempt to formulate string theory in an "invariant" way, quite like it is useful to formulate geometry in a coordinate-independent way. One might, for example, start with a family $\mathcal{F}$, of [[von Neumann algebra factor|hyperfinite type $III_1$ von Neumann algebras]] -- to be a little technical -- [[conformal net|indexed by]] intervals of the circle with non-empty complement (or of the super-circle). It may pay to formulate the starting point using the language of [[sheaves]]. $[...]$ This structure determines a [[braided monoidal category|braided monoidal]] [[C*-category]] with unit, ...; briefly, a quantum theory. From a combination of such [[tensor categories]] (left and right movers) one would attempt to reconstruct (symmetries of) _physical space-time_. String amplitudes would correspond to [[morphisms|arrows]] ([[intertwiners]]) of the [[tensor category]]. $[...]$ it would provide a general way of thinking about string theory that does not presuppose knowing the target space-time of the theory.

Later it was shown more formally ([Roggenkamp-Wendland 03](#RoggenkampWendland03)), reviewed in ([Roggenkamp-Wendland 08](#RoggenkampWendland08)), that there is a precise algebraically formalization of taking the "point particle limit" of a quantum string, by sending its [[vertex operator algebra]] to a spectral triple obtained by suitably retaining only [[worldsheet]] 0-modes.

In ([Soibelman 11](#Soibelman11)) this was used as a means to systematically study the large volume limit of [[effective quantum field theory|effective]] string [[spacetimes]] (and hence aspects of the [[landscape of string theory vacua]]) by studying the spectral geometries (i.e. the Connes-style noncommutative geometries) of the spectral triples arising from the string's point particle limit this way.

Now, since there is information lost in passing from a stringy "2-spectral triple" (a [[2d SCFT]]) to its underlying point particle [[spectral triple]], not all spectral triples are to be expected to have a lift to a 2-spectral triple (possibly corresponding to a [[UV-completion]] of the corresponding target space [[effective field theories]]).

In view of this, it is noteworthy that the spectral triple of the [[Connes-Lott-Chamseddine model]] shares a few key properties with the 2d SCFTs considered in [[string phenomenology]]:

The [[Connes-Lott-Chamseddine model]] is an encoding in a spectral triple of the [[standard model of particle physics]] coupled to [[gravity]] realized as a kind of spectral [[Kaluza-Klein compactification]] on a non-commutative fiber space down to ordinary 4d [[Minkowski spacetime]] (or possibly its [[Wick rotation|Wick rotated]] Euclidean version). In order for this to work out, it turns out that the compactified non-commutative fiber space needs to have [[KO-dimension]] equal to $6$. (Here the fiber space is classically just a ("non-commutative") point, but it appears as the singular collapsing limit of a space of finite dimension. This actual dimension is the [[KO-dimension]].)

Hence the claim of the [[Connes-Lott-Chamseddine model]] is that if the [[standard model of particle physics]] is encoded as a singular limit of a [[Kaluza-Klein compactification]] modeled via a [[spectral triple]] then the [[dimensions]] of the [[KK-compactification]] are

$$
  4 + 6 \;\;\; (mod\;8)
$$

with 4-dimensional base space and 6-dimensional fiber space, to a total of a 10-dimensional [[spacetime]] at high energy (after uncompactification of the fiber).

This, of course, is precisely the dimensionality of the target spacetime of [[perturbative string theory vacua]] for the critical [[superstring]].

This point was highlighted in [Connes 06, p. 8](#Connes06):

> {#Connes06OnRelationToStringVacua} When one looks at the table (7.2) of Appendix 7 giving the [[KO-dimension]] of the finite space $[$ i.e. the [[noncommutative geometry|noncommutative]] [[KK-compactification]]-[[fiber]] $F$ $]$ one then finds that its [[KO-dimension]] is now equal to 6 [[modulo]] 8 (!). As a result we see that the [[KO-dimension]] of the [[Cartesian product|product space]] $M \times F$ $[$ i.e. of 4d [[spacetime]] $M$ with the [[noncommutative geometry|noncommutative]] [[KK-compactification]]-[[fiber]] $F$$]$ is in fact equal to $10  \sim 2$ [[modulo]] 8. Of course the above 10 is very reminiscent of string theory, in which the finite space $F$ might bea good candidate for an "[[effective field theory|effective]]" [[KK-compactification|compactification]] at least for low energies. But 10 is also 2 [[modulo]] 8 which might be related to the observations of [Lauscher-Reuter 06](#LauscherReuter06) about [[gravity]].

Algebraically, this arises from the fact that the [[BRST complex]] for the [[superstring]] [[worldsheet]] theory is consistent (has BRST differential squaring to 0) precisely if the corresponding [[2d SCFT]] has conformal [[central charge]] 15, and each spacetime dimension contributes $1 \tfrac{1}{2}$ to this central charge (a contribution of 1 from each bosonic direction, and another $\tfrac{1}{2}$ for the corresponding fermionic contribution).

## Examples

### Flop transition

There is at least evidence that there is a continuous path in the space of 2-spectral triples that starts and ends at a point describing the ordinary geometry of a complex 3-dimensional [[Calabi-Yau space]] but passes in between through a 2-spectral triple/2d SCFT (a [[Gepner model]]) which is not the $\sigma$-model of an ordinary geometry, hence which describes "noncommutative 2-geometry" (to borrow that terminology from the situation of ordinary spectral triples). This is called the [[flop transition]] (alluding to the fact that the geometries at the start and end of this path have different [[topology]]). This was further expanded on and used for the mathematical study of the [[large volume limit]] of [[string theory]] [[vacua]] in ([Soibelman 11](#Soibelman11)).

## Related concepts

* [[spectral triple]]

* [[spectral action]]

* [[D-brane geometry]]

* [[automorphism of a 2-spectral triple]]

* [[swampland]]

## References
  {#References}

An early observation that the 0-mode sector of a [[2d SCFT]] is a [[spectral triple]], demonstrated in a series of concrete examples, is

* {#FroehlichGawedzki93} [[Jürg Fröhlich]], [[Krzysztof Gawędzki]], _Conformal Field Theory and Geometry of Strings_,  extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 ([arXiv:hep-th/9310187](http://arxiv.org/abs/hep-th/9310187))

The suggestion to understand, conversely, the [[string theory|string]]'s [[worldvolume]] [[2d SCFT]] as a higher spectral triple is due to

*  {#Chamseddine97} [[Ali Chamseddine]], _An Effective Superstring Spectral Action_, Phys.Rev. D56 (1997) 3555-3567 ([arXiv:hep-th/9705153](http://arxiv.org/abs/hep-th/9705153)),

which claims to show that the corresponding [[spectral action]] reproduces the correct effective background action known in [[string theory]]. A more expository account of this perspective is in

* {#FroehlichGrandjeanRecknagel97} [[Jürg Fröhlich]], Oliver Grandjean, [[Andreas Recknagel]], section 7 of _Supersymmetric quantum theory, non-commutative geometry, and gravitation_,  Lecture Notes Les Houches (1995) ([arXiv:hep-th/9706132](http://arxiv.org/abs/hep-th/9706132)).

A more formal derivation of how ordinary [[spectral triples]] arise as point particle limits of [[vertex operator algebra]]s for [[2d SCFTs]] then appears in

* {#RoggenkampWendland03} [[Daniel Roggenkamp]], [[Katrin Wendland]], _Limits and Degenerations of Unitary Conformal Field Theories_, Commun.Math.Phys. 251 (2004) 589-643 ([arXiv:hep-th/0308143](http://arxiv.org/abs/hep-th/0308143))

summarized in

* {#RoggenkampWendland08} [[Daniel Roggenkamp]], [[Katrin Wendland]], _Decoding the geometry of conformal field theories_, Proceedings of the 7th International Workshop "Lie Theory and Its Applications in Physics", Varna, Bulgaria ([arXiv:0803.0657](http://arxiv.org/abs/0803.0657))

A brief indication of some ideas of [[Yan Soibelman]] and [[Maxim Kontsevich]] on this matter is at

* [[Urs Schreiber]],  [_Spectral triples and graph field theory_](http://golem.ph.utexas.edu/category/2007/06/had_the_pleasure_of_talking.html)

Further development of this and application to the study of the [[large volume limit]] of [[superstring]] [[vacua]] is in

* {#Soibelman11} [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry_ ([pdf](http://www.math.ksu.edu/~soibel/nc-riem-3.pdf), [pdf](https://ncatlab.org/schreiber/files/SoibelmanCFTandRicciCurvature.pdf)), in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2001)

based on

* [[Maxim Kontsevich]], [[Yan Soibelman]], section 2 of _Homological mirror symmetry and torus fibrations_, Proceedings of KIAS Annual International Conference on Symplectic Geometry and Mirror Symmetry ([arXiv:math/0011041](http://arxiv.org/abs/math/0011041), [spire](http://inspirehep.net/record/536540/))

(discussing aspects of [[homological mirror symmetry]]).

Analogous detailed discussion based not on the [[vertex operator algebra]] description of local [[CFT]] but on the [[AQFT]] description via [[conformal nets]] is in

* {#CHKL09} [[Sebastiano Carpi]], Robin Hillier, [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Spectral triples and the super-Virasoro algebra_, Commun.Math.Phys.295:71-97 (2010) ([arXiv:0811.4128](http://arxiv.org/abs/0811.4128))

where [[2d SCFTs]] are related essentially to [[local nets]] of [[spectral triples]].

Exposition of these results is in

* [[Urs Schreiber]], _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_, PhysicsForums--Insights (2016)

See also the references at [[geometric model for elliptic cohomology]].

That the [[Connes-Lott models]] could be [[effective field theory]]-limits of [[perturbative string theory vacua]] is also mentioned in

* {#Connes06} [[Alain Connes]],  p. 8 of _Noncommutative Geometry and the standard model with neutrino mixing_, JHEP0611:081,2006 ([arXiv:hep-th/0608226](http://arxiv.org/abs/hep-th/0608226))

[[!redirects 2-spectral triples]]


2-topos

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#### $(\infty,2)$-Topos theory
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[[!include (infinity,2)-topos theory - contents]]
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#### 2-Category theory
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[[!include 2-category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The notion of **2-topos** is the generalization of the notion of [[topos]] from [[category theory]] to the [[higher category theory]] of [[2-categories]].

There are multiple conceivable such generalizations, depending in particular on whether one tries to generalize the notion of [[Grothendieck topos]] or of [[elementary topos]], and in the latter case what axioms one chooses to take as the basis for generalization.

In contrast, [[(n,1)-topos|(2,1)-toposes]] are much better understood.

A _Grothendieck 2-topos_ is a [[2-category]] of [[2-sheaves]] over a [[2-site]].

A _Grothendieck (2,1)-topos_ is a [[(2,1)-category]] of [[(2,1)-sheaves]] over a [[(2,1)-site]].

See also [[higher topos theory]].

## Properties

### Characterization of 2-sheaf 2-toposes

The [[2-toposes]] of [[2-sheaves]] over a [[2-site]] are special among all 2-toposes, in direct generalization of how [[sheaf toposes]] ("[[Grothendieck toposes]]") are special among all [[toposes]]. In that case, _[[Giraud's theorem]]_ famously characterizes sheaf toposes. This characterization has a 2-categorical analog: the _[[2-Giraud theorem]]_.

### $(n,r)$-Localic 2-toposes

A [[2-sheaf]] [[2-topos]] is "$(n,r)$-localic" or "$(n,r)$-truncated" if it has an [[(n,r)-site]] of definition.

In particular a $(2,1)$-localic 2-topos is the same as a [[(2,1)-topos]].

### In terms of internal categories
 {#InTermsOfInternalCategories}

Given a 2-topos $\mathcal{X}$, regard it is a [[2-site]] by equipping it with its [[canonical topology]].

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

Write $Cat(\mathcal{X})$ for the 2-category of _[[internal categories]]_ in $\mathcal{X}$, precisely: the 2-category of [[2-congruences]] and internal [[anafunctors]] between them (see [here](http://ncatlab.org/nlab/show/2-congruence#2CategoryOf2Congruences)).

=--

+-- {: .num_theorem #2SheavesAsInternalCategories}
###### Theorem

For $\mathcal{X}$ a 2-topos of [[2-sheaves]] on a [[2-site]], there is an [[equivalence of 2-categories]]

$$
  \mathcal{X} \simeq Cat(\mathcal{X})
  \,.
$$

If $\mathcal{X}$ is $(2,1)$-localic, with a [[(2,1)-site]] of definition $C$, then there is already an equivalence

$$
  \mathcal{X} \simeq Cat(Sh_{(2,1)}(C))
$$

with the 2-category of categories internal to the underlying [[(2,1)-topos]].

If $\mathcal{X}$ is $1$-localic, with 1-site of definition, then there is even already an equivalence

$$
  \mathcal{X} \simeq Cat(Sh(C))
$$

with the internal categories in the underlying [[sheaf topos]].

=--

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

By the [[2-Giraud theorem]], $\mathcal{X}$ is an [[exact 2-category]]. With this, the first statement is [this theorem](2-congruence#nCongExidempotent) at _[[2-congruence]]_.

By the discussion at [[n-localic 2-topos]], a 2-sheaf 2-topos has _[[core in a 2-category|enough groupoids]]_ precisely if it has a [[(2,1)-site]] of definition, and has _[[core in a 2-category|enough discretes]]_ precisely if it has a 1-site of definition.
With this the second and third statement is [this theorem](2-congruence#nCongOnGroupoidsAndDiscretes) at _[[2-congruence]]_.

=--

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

The noteworthy point about theorem \ref{2SheavesAsInternalCategories} is that for an ambient context which is a $(2,1)$-localic [[(2,1)-topos]], the straightforward morphisms of [[internal categories]], hence the notion of [[internal functors]], needs no further [[localization]]. This is in stark contrast to the situation for an ambient [[1-category]]. The generalization of this phenomenon is discussed at _[[category object in an (∞,1)-category]]_.

=--

## Examples

### The archetypical 2-topos

The archetypical 2-topos is [[Cat]]. This plays the role for 2-toposes as [[Set]] does for [[1-toposes]].

### Internal categories in a $(2,1)$-topos

Given any [[(2,1)-topos]] $\mathcal{X}$, the [[2-category]] $Cat(\mathcal{X})$ of [[internal (infinity,1)-category|internal categories]] in $\mathcal{X}$ ought to be a 2-topos. But it seems that at the moment there is no proof of this in the literature.

For literature on [[internal categories]] in [[1-toposes]] see at _[[2-sheaf]]_.

## Related concepts

* [[elementary theory of the 2-category of categories]] ([[ETCC]])

* [[formal category theory]]

[[!include flavors of higher toposes -- list]]

## References

An introduction is in

* [[Mike Shulman]], _[[michaelshulman:What is a 2-topos]]?_

Early developments include

* [[Ross Street]], _Two dimensional sheaf theory_, J. Pure and Appl. Algebra 24 (1982) 2Opp.

* [[Dominique Bourn]], _Sur les ditopos_, C. R. Acad. Sci. Paris 279, 911–913 (1974).

A detailed discussion from the point of view of [[internal logic]] is at

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

Discussion of the 2-categorical [[Giraud theorem]] for [[2-sheaf]] 2-toposes is in

* [[Ross Street]], _[[zoranskoda:Characterization of Bicategories of Stacks]]_  Category theory (Gummersbach 1981) LNM 962, 1982, MR0682967 (84d:18006)

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

Discussion of the [[elementary topos]]-analog of 2-toposes is in

* [[Mark Weber]], *Yoneda structures from 2-toposes*, Appl Categor Struct **15** (2007) 259–323 [[doi:10.1007/s10485-007-9079-2](https://doi.org/10.1007/s10485-007-9079-2), [pdf](https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes)]

A notion of "flat 2-functor" (cf [[Diaconescu's theorem]]) perhaps relevant to the "points" of 2-toposes is in

* M.E. Descotte, [[Eduardo Dubuc]], M. Szyld, _On the notion of flat 2-functors_, arXiv:[1610.09429](https://arxiv.org/abs/1610.09429)

Discussion of 2-classifiers for 2-toposes is in

* [[Luca Mesiti]], _2-classifiers via dense generators and Hofmann-Streicher universe in stacks_ [[arXiv:2401.16900](https://arxiv.org/abs/2401.16900)]

[[!redirects 2-toposes]]
[[!redirects 2-topoi]]

[[!redirects (2,1)-topos]]
[[!redirects (2,1)-toposes]]

[[!redirects (2,2)-topos]]
[[!redirects (2,2)-toposes]]

[[!redirects Grothendieck 2-topos]]
[[!redirects Grothendieck 2-toposes]]


2-topos theory

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#### 2-Category theory
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[[!include 2-category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The theory of [[2-topos]]es, generalizing [[topos theory]] from [[category theory]] to [[2-category theory]].

## Related concepts

* [[topos theory]]

* **2-topos theory**

* [[(∞,1)-topos theory]]

* [[higher topos theory]]

## References

The analog of the [[Elephant]] for 2-topos theory still needs to be written.

For some speculations and further references, see [[michaelshulman:2-categorical logic|this page]].


2-trivial model structure

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#Contents#
* table of contents
{:toc}

## Definition

Every [[strict 2-category]] $K$ with finite [[strict 2-limits]] and finite strict 2-colimits becomes a [[model category]] (or, rather, its underlying 1-category does) in a canonical way, where:

* The weak equivalences are the [[equivalences]].

* The fibrations are the morphisms that are representably [[isofibrations]], i.e. the morphisms $e\to b$ such that $K(x,e)\to K(x,b)$ is an isofibration for all $x\in K$.

* The cofibrations are determined.

We call it the **2-trivial model structure**, as it is a 2-categorical analogue of the [[trivial model structure]] on any 1-category.  It can be said to regard $C$ as an [[(∞,1)-category]] with only trivial [[k-morphisms]] for $k \geq 3$.

## Properties

* Every object is fibrant and cofibrant.

* By duality, any such category has another model structure, with the same weak equivalences but where the cofibrations are the iso-cofibrations and the fibrations are determined.  In $Cat$, the two model structures are the same.

## Examples

* In [[Cat]], this produces the [[canonical model structure]].

* If $T$ is an [[accessible functor|accessible]] [[strict 2-monad]] on a [[locally finitely presentable category|locally finitely presentable]] [[strict 2-category]] $K$.  Then the category $T Alg_s$ of strict $T$-[[algebra over a monad|algebras]] admits a [[transferred model structure]] from the 2-trivial model structure on $K$.  The cofibrant objects therein are the [[flexible algebra]]s.

## Related pages

* In model categories built from various kinds of [[topological spaces]], there are often analogous [[model structure on topological spaces#Hurewicz (or Strøm) Model Structure|Hurewicz model structures]].  These are not actually examples of a 2-trivial model structure (for instance, the 2-category of spaces, continuous functions and homotopy classes of homotopies does not have finite limits as a 2-category, or even as a 1-category), but they share a common intuition and can sometimes be obtained as two instances of a more general construction.

## References

* [[Steve Lack]], *Homotopy-theoretic aspects of 2-monads*, [arXiv](http://arxiv.org/abs/math.CT/0607646)

[[!redirects trivial model structure on a 2-category]]
[[!redirects trivial model structure for a 2-category]]


2-type > history

< [[2-type]]
[[!redirects 2-type -- history]]