Principal 2-bundles and quotient 2-stacks
Elena Caviglia
School of Computing and Mathematical Sciences, University of Leicester, United Kingdom

[2020]18F20, 18F10, 18N10, 14A20

We generalize principal bundles and quotient stacks to the two-categorical context of bisites. We introduce a notion of principal 2-bundle that makes sense for a 2-category with finite flexible limits, endowed with a bitopology. We then use principal 2-bundles to explicitly construct quotient-pre-2-stacks, which are the analogues of quotient stacks one dimension higher. In order to perform this construction, we prove that principal 2-bundles are closed under iso-comma objects and we restrict ourselves to $(2,1)$-categories. Finally, we prove that, if the bisite is subcanonical and the underlying $(2,1)$-category satisfies some mild conditions, quotient pre-2-stacks are 2-stacks.

§ INTRODUCTION

In this paper, we generalize principal bundles and quotient stacks to the two-categorical context of bisites. This is the generalization to one dimension higher of the theory of generalized principal bundles and quotient stacks that we developed in <cit.> in the categorical context of sites. Both these theories will also appear in our PhD thesis <cit.>.

Principal bundles and quotient stacks are key concepts in geometry and topology. Their importance is due, above all, to their fruitful connections with cohomology groups. Given a nice topological space $X$ and a group $G$, the first cohomology group $H^1(X,G)$ of $X$ with coefficients in $G$ corresponds to the set of isomorphism classes of principal $G$-bundles over $X$. And, as also pointed out by Lurie in <cit.>, the approach of principal $G$-bundles turns out to be very useful and more effective than the standard one involving cocycles, especially when considering an arbitrary topological space. The collections of principal $G$-bundles over different topological spaces can then be captured and studied together thanks to the classifying stack $\clst{G}$, which is the archetypal quotient stack.

One of the main motivations that brought us to study categorical generalizations of principal bundles and quotient stacks is the possibility to use such new generalized concepts as tools for the study of cohomology groups in new and broader contexts. We believe that this will have many interesting applications both in mathematics and in mathematical physics.

Principal bundles over topological spaces and over smooth manifolds have been introduced and studied in geometry and topology (see <cit.>, <cit.> and <cit.>).  In  <cit.>, we produced a categorical generalization of  principal bundles that makes sense in any site, i.e. in any category endowed with a Grothendieck topology. The topological group involved in the classical notion of principal bundle is generalized to a group object in the category. And the notion of locally trivial morphism is then internalized by considering pullbacks along the morphisms of a covering family for the Grothendieck topology. This generalized notion of principal bundle recovers the classical notions of principal bundles over topological spaces and smooth manifolds as particular cases.

In recent years, the need of a 2-dimensional notion of principal bundle has appeared in mathematical physics. The particular case of principal bundle over manifolds has been generalized to dimension 2 by Bartels, in their PhD thesis <cit.>. They introduced a notion of principal $G$-2-bundle over a smooth manifold, where $G$ is a Lie 2-group and the smooth manifold is seen as a smooth 2-manifold with only identities. The main motivation arose from applications in higher gauge theory; instances of principal 2-bundles of this form can be found in topological quantum field theory and string theory.

In this paper, we introduce a notion of principal 2-bundle that makes sense in any nice enough 2-category equipped with a two-dimensional Grothendieck topology. More precisely, we use the notion of bitopology introduced by Street in <cit.> for bicategories. The internal groups used in the context of sites are here replaced by (coherent) 2-groups and pullbacks are replaced by iso-comma objects. Our notion of principal 2-bundle recovers Bartels' one when considering the 2-category of differentiable categories, i.e. categories internal to the category of smooth manifolds, with the covering sieves generated by surjective local diffeomorphisms.  We believe that our general theory of principal 2-bundles could have many applications both in higher gauge theory and in other mathematical contexts. It is also worth noticing that an analogous theory can be developed for bicategories using the same ideas.

Principal bundles over topological spaces, schemes or differentiable manifolds have also been used to perform an explicit construction of quotient stacks. Instances of this construction can be found in <cit.>, for the algebraic case, and in <cit.>, for the differentiable case. And in <cit.>, we performed an analogous construction in the categorical context of sites.  In this paper, we generalize this construction to one dimension higher, producing particular trihomomorphisms from a $(2,1)$-category endowed with a bitopology into the tricategory $\twocat$ of 2-categories. These trihomomorphisms are the analogues of quotient prestacks one dimension higher and will be thus called quotient pre-2-stacks.

Since the classical quotient prestacks are always stacks and the generalized quotient stacks of <cit.> are stacks if the site satisfies some mild conditions, it is natural to wonder whether generalized quotient pre-2-stacks satisfy stacky gluing conditions.
As there was no notion of higher dimensional stack suitable for quotient pre-2-stacks, in our paper <cit.>, we introduced a notion of 2-stack suitable for a trihomomorphism from a bisite into $\Bicat$. Our definition naturally generalizes the definition of stack given by Street in <cit.> for pseudofunctors from a 2-category into $\Cat$. But we also proved a useful characterization (Theorem 3.18 of <cit.>) in terms of explicit gluing conditions that can be checked more easily in practice. The obtained gluing conditions generalize one dimension higher the usual gluing conditions satisfied by a stack.

Analogously to what we showed in the one-dimensional case in <cit.>, we prove that, if the bitopology is subcanonical and the underlying $(2,1)$-category is nice enough, quotient pre-2-stacks are 2-stacks. We thus obtain a notion of quotient 2-stack. This is the main theorem of the paper.

Let $\K$ be a bicocomplete $(2,1)$-category with finite flexible limits and such that iso-comma objects preserve bicolimits. Let then $\tau$ be a subcanonical bitopology on $\K$ and $\cX,\cG\in \K$  with $\cG$ an internal 2-group. Then the quotient pre-2-stack $\qst{\cX}{\cG}$ is a 2-stack.

To prove this important result, we use the explicit characterization of 2-stack of our paper <cit.>, as well as the calculus of colimits in two-dimensional slices developed by Mesiti in <cit.> and <cit.>.

A key result needed for the proof of Theorem <ref> is that every object of a subcanonical bisite can be expressed as some kind of two-dimensional colimit of each covering bisieve over it. We proved this result in <cit.>.

Quotient stacks are a fundamental concept in algebraic geometry because many stacks that naturally arise in geometric contexts either are quotient stacks or can be constructed using quotient stacks as building blocks. For this reason, we believe that both the generalized quotient stacks of <cit.> and the quotient 2-stacks studied in this paper could have many useful and interesting applications.

Furthermore, our categorical generalizations of principal bundles and quotient stacks could give a substantial contribution towards the development of a cohomology theory of schemes, and more in general of stacks, with coefficients in stacks of abelian 2-groups. This theory would produce new and refined 2-categorical invariants associated to schemes and algebraic stacks, that could solve numerous open problems in algebraic geometry.

§.§ Outline of the paper

In section <ref>, we introduce and study principal 2-bundles. We start by recalling the notions of equivariant morphisms and 2-cells with respect to actions of internal (coherent) 2-groups. We then construct an action on the iso-comma object of two equivariant morphisms. Moreover, we introduce the notion of 2-locally trivial morphism in a bisite. We use this notion to define principal 2-bundles and cells between them (Definition <ref>). We conclude the section by proving the key result that principal 2-bundles are stable under iso-comma objects (Proposition <ref>).

In section <ref>, we use principal 2-bundles to explicitly construct quotient pre-2-stacks, restricting ourselves to the case of a $(2,1)$-category. We then prove that quotient pre-2-stacks are trihomomorphism into $\twocat$. Finally, we prove the main result of the paper, which is that, if the bisite is subcanonical and the underlying $(2,1)$-category satisfies mild assumptions, our quotient pre-2-stacks are 2-stacks (Theorem <ref>).

§ PRINCIPAL 2-BUNDLES

In this section we introduce a notion of principal 2-bundle in the context of bisites. This notion generalizes the notion of 2-space studied by Bartels in their PhD thesis <cit.>.

Throughout this section $\K$ will be a 2-category with all finite flexible limits and $\tau$ will be a Grothendieck bitopology on it. The notion of bitopology used here is the one introduced by Street in <cit.> (and recalled also in our paper <cit.>).

Throughout this chapter the objects of $\K$ will be denoted by calligraphic letters instead of the usual capital block letters. This is coherent with the choice of other authors using the same concepts. And it is also helpful when comparing the notions presented here to the one-dimensional ones of <cit.>.

In Section <ref>, we will restrict to the case in which $\K$ is a [4] $(2,1)$-category as the invertibility of 2-cells is required to construct the higher dimensional analogues of quotient stacks.

[iso-comma objects]
Since $\K$ has all flexible limits (in the sense of <cit.>), it has in particular all iso-comma objects. If the iso-comma objects exists, it is a special representative of the bi-iso comma objects. For this reason, we will formulate some definitions that are bicategorical in nature and naturally involve bi-iso comma objects using iso-comma objects instead. This will offer substantial simplifications in the calculations. The same applies to biproducts and products.

We aim at introducing a notion of principal 2-bundle that makes sense in $\K$. In this two-dimensional setting instead of an internal group we will consider an internal 2-group. This concept has been introduced by Baez and Lauda in <cit.>, but the idea had already been presented by Joyal and Street in <cit.>.

An internal 2-group in the 2-category $\K$ is an object of $\K$ endowed with a multiplication, a neutral element and an inverse morphism that satisfy associativity and unitality only up to fixed coherent 2-isomorphisms. We refer the reader to <cit.> for a complete definition.

We know recall the notion of action of an internal 2-group $\cG\in \K$ on an object $\cX\in \K$. This definition can be found in <cit.>.

Let $\cG$ be an internal $2$-group in $\K$ and let $\cX$ be an object of $\K$. An action of $\cG$ on $\cX$ is given by a morphism

$$x\: \cG \x \cX \to \cX$$

together with invertible 2-cells

*[l][7][7][μ_x]   𝕀xm𝕀xx
[r,"e𝕀"] [rd,"",equal,""name=A]&  [to=A,"ν_x",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex] [d,"x"]

such that the following equalities of 2-cells hold:

[rd,"𝕀m𝕀"] [d,"𝕀m𝕀"'] [rr,"𝕀𝕀x"] &[-4ex]  &[-4ex]  [rd,"𝕀x"] [ld,"𝕀μ_x", Rightarrow ,shorten <=2.5ex, shorten >= 2.5ex]&

[rd,"m𝕀̧"'] &  [l,"α_m𝕀",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex] [d,"m𝕀"] [rr,"𝕀x"] &  &  [d,"x"] [lld,"μ_x",Rightarrow ,shorten <=8.5ex, shorten >= 8.5ex]

&  [rr,"x"'] &  &

[d,"𝕀m𝕀"'] [rr,"𝕀𝕀x"] &  &[-4ex] [rd,"𝕀x"] [d,"m𝕀"]&

[rr,"𝕀x"][rd,"m𝕀̧"'] &  &  [ld,"μ_x",Rightarrow ,shorten <=2.5ex, shorten >= 2.5ex] [rd,"x"] &  [l,"μ_x",Rightarrow ,shorten <=2.5ex, shorten >= 2.5ex]  [d,"x"]

&  [rr,"x"'] &  &

[r,"𝕀𝕀 x"] [rd,"",equal,""name=A] &  [to=A,"α_e𝕀"inner sep=0.001ex,Rightarrow ,shorten <=1ex, shorten >= 1ex,shift left=-0.51ex][d,"m𝕀"] [r,"𝕀x"] &  [ld,"μ_x",Rightarrow ,shorten <=4.5ex, shorten >= 4.5ex] [d,"x"]

&  [r,"x"']&

[r,"𝕀𝕀 x"] [rd,"",equal] &  [d,"𝕀ν_x",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex] [rd,"",equal]  [r,"𝕀x"] &  [d,"x"]

&  [r,"x"']&

where $\alpha_m$ and $\alpha_e$ are coherence 2-cells associated to the multiplication and the neutral element of $\cG$ (see <cit.>).

For the action of an internal 2-group the usual axioms of action are true only up to coherent isomorphisms. These isomorphisms are $\mu_x$ and $\nu_x$ and their coherences are expressed by the equalities (<ref>) and (<ref>).

We now introduce the notions of $\cG$-equivariant morphism and
$\cG$-equivariant 2-cell in $\K$. These notions are well-known, but we could not find a reference for them in the literature.

[$\cG$-equivariant morphism]
Let $\cG$ be an internal 2-group in $\K$ that acts on  the objects $\cX$ and $\cY$ of $\K$ with actions $x\: \cG\x \cX \to \cX$ and $y\: \cG\x \cY \to Y$ respectively. A morphism $f\: \cX \to \cY$ is said $\cG$-equivariant if there exists an invertible 2-cell
[l][7][7][λ_f]   x𝕀ffy
such that the following equalities hold:

&   [d,"m𝕀"'][ld,"𝕀 𝕀 f"',bend right=30]  [r,"𝕀x"] &  [d,"x"]  [ld,"μ_x",Rightarrow ,shorten <=4.5ex, shorten >= 4.5ex]

[rd,"m𝕀"', bend right=35] &  [r,"x"'] [d,"𝕀f"']&  [d,"f"] [ld,"λ_f",Rightarrow ,shorten <=4.5ex, shorten >= 4.5ex]

&  [r,"y"']&

[20] = [3]

[d,"𝕀 𝕀 f"']  [r,"𝕀x"] &  [d,"𝕀f"] [rd,"x", bend left=30] [ld,"𝕀 λ_f",Rightarrow ,shorten <=4.5ex, shorten >= 4.5ex]&

[r,"𝕀y"'] [d,"m𝕀"'] &  [ld,"μ_y",Rightarrow ,shorten <=4.5ex, shorten >= 4.5ex] [d,"y"]&  [l,"λ_f",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex] [ld,"f",bend left=35]

[r,"y"'] &  &

[d,"𝕀f"'] [r,"e𝕀"] &  [d,"𝕀f"'][r,"x"] &  [ld,"λ_f",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex] [d,"f"]

[rr,"2"', bend right=40,""name=A,pos=0.55] [r,"e 𝕀"'] &  [to=A,"ν_y",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex][r,"y"']&

[rr,"2"', bend right=40,""name=B,pos=0.55] [d,"𝕀f"']  [r,"e𝕀"] &  [to=B,"ν_x",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex] [r,"x"]&   [d,"f"]

[rr,"2"', bend right=40,""name=C,pos=0.5] &  &

Notice that every morphism of $\K$ is $\cG$-equivariant when the source and the target are equipped with trivial actions of $\cG$. Having this in mind, we will always think of the objects of $\K$ as equipped with trivial actions of $\cG$ when the action is not specified.

[$\cG$-equivariant 2-cell]
Let $\cG$ be an internal 2-group in $\K$ that acts on  the objects $\cX$ and $\cY$ of $\K$ with actions $x\: \cG\x \cX \to \cX$ and $y\: \cG\x \cY \to Y$ respectively. Let $f,g\: \cX \to \cY$ be $\cG$-equivariant morphisms. A 2-cell $\gamma\: f \Rightarrow g$ is said $\cG$-equivariant if the following equality holds:

[d,"𝕀g"'] [r,"x"] &  [ld,"λ_g",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex] [d,"g"',""name=B,pos=0.53] [d,"f", bend left=70 ,""name=A]

[from=A,to =B,"γ",Rightarrow ,shorten <=0.5ex, shorten >= 0.001ex] [r,"y"] &

[10] = [2]

[r,"x"]  [d,"𝕀f",""name=C] [d,"𝕀g"',""name=D, bend right=70] [from=C,to=D,"𝕀γ"',Rightarrow ,shorten <=0.9ex, shorten >= 0.7ex]&  [d,"f"] [ld,"λ_f",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex]

[r,"y"] &

The following result might be well-known, but we could not find a reference for it. It generalizes  a well known fact about group objects to one dimension higher.

Let $\cG$ in $\K$ be an internal 2-group. The pseudofunctor

$$\cG \x -\: \K \to \K$$

is a pseudomonad with multiplication given by $m\x \id{-}$ and identity given by $e\x \id{-}$.

Moreover, the pseudoalgebras for this pseudomonad are exactly the objects of $\K$ equipped by an action of $\cG$;  the pseudomorphisms between pseudoalgebras are exactly the $\cG$-equivariant morphisms and the pseudotransformations between them are exactly the $\cG$-equivariant 2-cells.

The fact that $\cG \x -\: \K \to \K$ is a pseudomonad easily follows from the definition of internal 2-group. It is,then, straightforward to prove that the pseudoalgebras for this pseudomonad are exactly the objects of $\K$ equipped by an action of $\cG$ using the definition of action. Finally, the facts that pseudomorphisms between pseudoalgebras are exactly the $\cG$-equivariant morphisms and that the pseudotransformations between them are exactly the $\cG$-equivariant 2-cells simply follow by the definitions of $\cG$-equivariant morphisms and $\cG$-equivariant 2-cell respectively.

We now describe how to define an action of $\cG$ on the iso-comma object of two morphisms in $\K$, given actions of $\cG$ on the sources of the morphisms. This action will be largely used throughout the chapter.

Let $\cG$ be an internal 2-group of $\K$ that acts on $\cP\in \K$ with action $p\: \cG\x \cP\to \cP$, on $\cY\in \K$ with action $y\: \cG\x \cY \to \cY$ and on $\cZ\in\K$ with action $z\: \cG\x \cZ \to \cZ$ and let $f:\cP\to \cY$ and $g\: \cZ\to \cY$ be $\cG$-equivariant morphisms.

Consider, then, the iso-comma object square
We want to define an action of $\cG$ on the iso-comma object $\cP\x[\cY]\cZ$ and we can do this using the morphism $\psi\: \cG\x (\cP\x[\cY]\cZ) \to \cP\x[\cY]\cZ$ induced by the universal property of $\cP\x[\cY]\cZ$ as in the following diagram

([])p(̧𝕀fg)z (̧𝕀gf)ψ[]fggffgα^f,g
([]) [d,"z (̧𝕀gf)"'] [r,"p(̧𝕀fg)"] &[6ex]  [d,"f"] [ld,"γ",twoiso ,shorten <=8.5ex, shorten >= 8.5ex]

[r,"g"'] &

where the isomorphic 2-cell $\gamma$  is given by the following pasting diagram

([]) [r,"𝕀fg"]  [d,"𝕀gf"'] & [ld,"𝕀α^f,g",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex] [r,"p"] [d,"𝕀f"] & [d,"f"] [ld,"λ_f",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex]

[r,"𝕀 g"] [d,"z"'] & [r,"y"] & [d,equal,""] [lld,"λ_g^-1",Rightarrow ,shorten <=12.5ex, shorten >= 12.5ex]

[rr,"g"'] & &

The morphism $\psi\: \cG\x (\cP\x[\cY]\cZ) \to \cP\x[\cY]\cZ $ defined in Construction <ref> is an action of $\cG$ on $\cP\x[\cY]\cZ$.

We need to prove that there exist coherent invertible 2-cells

*[l][7][7][μ_ψ]([] )([] ) ([] ) []  𝕀ψm𝕀([] )ψψ
[]  [r,"e𝕀([] )"] [rd,"",equal,""name=A]& ([] ) [to=A,"ν_ψ",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex] [d,"ψ"]

& []

We induce $\mu_\psi$ and $\nu_\psi$ using the two-dimensional universal property of the iso-comma object $\cP\x[\cY] \cZ$.

To induce $\mu_\psi$ we consider the following isomorphic 2-cells:

([]) [rr,"𝕀ψ"] [ddd,"m𝕀[]"'] [rd,"𝕀(𝕀fg)"]& & ([]) [r,"ψ"] [d,"𝕀fg"]& [] [ddd,"fg"]

&  [r,"𝕀p"] [d,"m𝕀"]&  [rdd,"p"] [ld,"μ_p",Rightarrow ,shorten <=4.5ex, shorten >= 4.5ex, shift left=2ex] &

& [rrd,"p"]& &

([]) [ru,"𝕀fg"] [r,"ψ"'] &[] [rr,"fg"'] & &

([]) [rr,"𝕀ψ"] [ddd,"m𝕀[]"'] [rd,"𝕀(𝕀g f)"]& & ([]) [r,"ψ"] [d,"𝕀g f"]& [] [ddd,"g f"]

&  [r,"𝕀z"] [d,"m𝕀"]&  [ld,"μ_z",Rightarrow ,shorten <=4.5ex, shorten >= 4.5ex, shift left=2ex] [rdd,"z"] &

& [rrd,"z"]& &

([]) [ru,"𝕀g f"] [r,"ψ"'] &[] [rr,"g f"'] & &

Using the definition of $\psi$ and the fact that $f$ and $g$ are $\cG$-equivariant, it is straightforward to prove that these 2-cells are compatible and therefore they induce the desired isomorphic 2-cell $\mu_\psi$.

To induce $\nu_\psi$ we consider the following isomorphic 2-cells

([])  [rd,"𝕀fg"][dd,"2"'][rr,"e𝕀[]"]&[-3ex]  &[-3ex] ([]) [d,"𝕀fg"] [r,"ψ"] & [] [dd,"fg"]

&  [rrd,"2"', bend right=10 ,""name=A,pos=0.4][r,"e𝕀"] &  [to=A,"ν_p",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex] [rd,"p"] &

[] [rrr,"fg"'] &  &   &

([]) [rd,"𝕀gf"] [dd,"2"'][rr,"e𝕀[]"]&[-3ex]  &[-3ex] ([]) [d,"𝕀gf"] [r,"ψ"] & [] [dd,"gf"]

&   [rrd,"2"', bend right=10 ,""name=A,pos=0.4] [r,"e𝕀"] &  [to=A,"ν_z",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex] [rd,"z"] &

[] [rrr,"gf"'] &  &   &

Using the definition of $\psi$ and the fact that $f$ and $g$ are $\cG$-equivariant, it is straightforward to prove that these 2-cells are compatible and therefore they induce the desired isomorphic 2-cell $\nu_\psi$.

It remains to prove that $\mu_{\psi}$ and $\nu_{\psi}$ satisfy the coherence conditions (<ref>) and (<ref>) of Definition <ref>. This is straightforward using the fact that the corresponding coherences hold for the actions $p$ and $z$.

To define principal $\cG$-2-bundles, we will need to use the action induced by the internal multiplication of $\cG$ on a product of the form $\cG \x \cX$.

The morphism

$$m\x \id{\cX}\: \cG \x \cG \x \cX \to \cG \x \cX$$

is an action of $\cG$ on $\cG \x \cX$.

To simplify the notation we denote by $\rho$ the morphism $m\x \id{\cX}$. We need to prove that there exist coherent isomorphic 2-cells

*[l][7][7][μ_ρ]   𝕀ρm𝕀ρρ
[r,"e𝕀"] [rd,"",equal,""name=A]&  [to=A,"ν_ρ",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex] [d,"ρ"]

We induce $\mu_\rho$ and $\nu_\rho$ using the two-dimensional universal property of the product $\cG\x\cX$. To induce $\mu_\rho$ we use the following isomorphic compatible 2-cells

[dd,"m𝕀"'] [r,"𝕀ρ"] &  [ldd,"α_m 𝕀",Rightarrow ,shorten <=8.5ex, shorten >= 8.5ex,shift left=-3ex] [d,"m𝕀"] [r,"ρ"] &   [dd,"1"]

&   [rd,"1"] &

[ru,"m𝕀"]  [r,"ρ"']  &  [r,"1"'] &

[dd,"m𝕀"'] [r,"𝕀ρ"]  &  [d,"",aiso] [r,"ρ"] &   [dd,"2"]

& ()  [rd,"2"] &

[ru,"",aiso] [r,"ρ"'] &  [r,"2"'] &

To induce $\nu_\rho$ we use the following isomorphic compatible 2-cells

[rd,"",equal,bend right=30,""name=A][r,"e 𝕀𝕀"',bend right=20]  [dd,equal] [r,"e 𝕀"] &[2ex]  [to=A,"λ_m𝕀"inner sep=0.01ex,Rightarrow ,shorten <=2.5ex, shorten >= 2.5ex,shift left=1.6ex][d,"m𝕀"] [r,"ρ"] &    [dd,"1"]

&   [rd,"1"]  &

[rr,"1"'] & &

[dd,equal] [r,"e 𝕀"] &[2ex]  [d,"",aiso]  [r,"ρ"] &   [dd,"2"]

&  ()  [rd,"2"] &

[rr,"2"'] & &

It remains to prove that $\mu_{\rho}$ and $\nu_{\rho}$ satisfy the coherence conditions (<ref>) and (<ref>) of Definition <ref>. This can be easily done using the coherence diagrams satisfied by $m$ and $e$ respectively.

In order to introduce principal 2-bundles, we need to define local triviality of a morphism with respect to the bitopology on $\K$. The idea is to use iso-comma objects in place of the pullbacks used in the one-dimensional context.

Let $\cX$ and $\cY$ be objects of $\K$ with fixed actions of $\cG$ on them. We say that the morphism $g\: \cY\to \cX$ is 2-locally trivial if there exists a bisieve $S\: R\Rightarrow \K(-, \cX)$ in $\tau(\cX)$ such that, for every $f\: \cU \to \cX$ in the bisieve $S$, the iso-comma object

[] gffggfλ^f,g33

is equivalent to the product $\cG\x \cU$ over $\cU$ via a $\cG$-equivariant equivalence.

[iso-comma objects vs bi-iso-comma objects]
The previous definition could equivalently be given using bi-iso-comma objects instead of iso-comma objects. But since $\K$ has all iso-comma objects and they are representatives of the corresponding bi-iso-comma objects, we can more conveniently ask the required property for iso-comma objects. Notice that we only ask equivalences between the iso-comma objects and the corresponding products. This shows the bicategorical nature of the previous definition.

[reformulation of Definition <ref>]
We can equivalently reformulate Definition <ref> by asking the existence of a bisieve $S\: R\Rightarrow \K(-, \cX)$ in $\tau(\cX)$ such that, for every $f\: \cU \to \cX$ in the bisieve $S$, there exists a bi-iso-comma object of the form

This reformulation will be useful for the proof of Proposition <ref>.

We are now ready to introduce principal 2-bundles and cells between them.

Let $\cY$ be an object of $\K$. A principal $\cG$-2-bundle over $\cY$ is an object $\cP\in\K$ equipped with an action $p\: \cG\x \cP\to \cP$ and a $\cG$-equivariant  2-locally trivial morphism $\pi_{\cP}\: \cP\to \cY$.

A morphism of principal $\cG$-2-bundles over $\cY$ from $\pi_{\cP}\: \cP\to \cY$ to $\pi_{\cQ}\: \cQ\to \cY$ is a $\cG$-equivariant morphism $\phi\: \cP \to \cQ$ in $\K$ together with an isomorphic $\cG$-equivariant 2-cell

[rd,"π_"',""name=A] [rr,"ϕ"] &[-3ex]  &[-3ex]  [ld,"π_"]  [to=A,"γ_ϕ",twoiso ,shorten <=2.5ex, shorten >= 2.5ex,shift left=1ex,start anchor=[xshift=-1.5ex],end anchor=[xshift=-1.5ex]]

&  &

A 2-cell of principal $\cG$-2-bundles over $\cY$ from $\phi\: \cP \to \cQ$ to $\psi\: \cP \to \cQ$ is a $\cG$-equivariant 2-cell $\Gamma\: \phi \Rightarrow \psi$ such that

[rr,"ϕ", bend left=40,""'name=G] [rd,"π_"',""name=A] [rr,"ψ"',""'name=H] [from=G,to=H,"Γ"pos=0.4,Rightarrow,shorten <=-0.3ex, shorten >= 0.3ex] &[-3ex]  &[-3ex]  [ld,"π_"]  [to=A,"γ_ψ",twoiso ,shorten <=2.5ex, shorten >= 2.5ex,shift left=1ex,start anchor=[xshift=-1.5ex],end anchor=[xshift=-1.5ex]]

&  &

[rd,"π_"',""name=A] [rr,"ϕ"] &[-3ex]  &[-3ex]  [ld,"π_"]  [to=A,"γ_ϕ",twoiso ,shorten <=2.5ex, shorten >= 2.5ex,shift left=1ex,start anchor=[xshift=-1.5ex],end anchor=[xshift=-1.5ex]]

&  &

Principal $\cG$-2-bundles over $\cY$ together with morphisms and 2-cells between them form a 2-category that will be denoted $\twoBun{\cG}{\cY}$.

We now aim at proving that principal $\cG$-2-bundles are closed under iso-comma objects. This property is the higher dimensional analogue of the closure of principal bundles under pullbacks and it will be very important in order to construct quotient pre-2-stacks in Section <ref>. To prove this crucial property, we will need to use the following result.

The forgetful functor

$$U\: \opn{Ps}\mbox{-}(\cG\x -)\mbox{-}\opn{Alg} \to \K$$

from the 2-category of objects of $\K$ equipped with actions of $\cG$ (that are the pseudoalgebras of the pseudomonad $\cG\x -$ by Proposition <ref>) to $\K$ lifts bi-iso-comma objects.

Let $(\cP,p),(\cZ,z)$ and $(\cY,y)$ be in $\opn{Ps}\mbox{-}(\cG\x -)\mbox{-}\opn{Alg}$ and let $f\: \cZ \to \cY$ be $\cG$ equivariant morphisms. Consider the bi-iso-comma object

[] gffggfλ^f,g33

If $\cP\x[\cY] \cZ$ is endowed with an action of $\cG$ exactly as the one of Construction <ref> (that is still an action in the case of the bi-iso-comma object), then $f\st g$ and $g\st f$ are $\cG$-equivariant by definition of the action and so is the 2-cell $\lambda^{f,g}$.
Let now $(\cQ,q)$ be in $\opn{Ps}\mbox{-}(\cG\x -)\mbox{-}\opn{Alg}$ and consider the morphism $v\: \cQ \to \cP\x[\cY] \cZ$ induced by te universal property of the bi-iso-comma object $\cP\x[\cY] \cZ$ as in the following diagram

=     *[l][7][7][Ω]rsgf

If $r,s$ and $\Omega$ are $\cG$-equivariant, then so is the morphism $v$. Indeed, the invertible 2-cell

*[l][7][7][α_v]([]) []q𝕀vvψ

can be induced using the universal property of the bi-iso-comma $\cP \x[\cY]\cZ$ starting from the following compatible 2-cells

[rd,"𝕀r",""name=B] [r,"q"] [dd,"𝕀v"'] &  [d,"α_r",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex] [rdd,"r",""name=A] [r,"v"]& [] [dd,"gf"] [to=A,"Ω^-1_1",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex]

&  [d,"",twoiso,shorten <=1.5ex, shorten >= 1.5ex] [rd,"p"] &

([]) [from=B,"𝕀Ω_1",Rightarrow ,shorten <=6.5ex, shorten >= 6.5ex] [ru,"𝕀gf"'][r,"ψ"']& [] [r,"gf"'] &

[rd,"𝕀s",""name=B] [r,"q"] [dd,"𝕀v"'] &  [d,"α_s",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex]  [rdd,"s",""name=A] [r,"v"]& [] [dd,"fg"] [to=A,"Ω_2",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex]

&  [d,"",twoiso,shorten <=1.5ex, shorten >= 1.5ex]  [rd,"z"] &

([]) [from=B,"𝕀Ω^-1_2",Rightarrow ,shorten <=6.5ex, shorten >= 6.5ex][ru,"𝕀fg"'][r,"ψ"']& [] [r,"fg"'] &

It is then easy to prove that $\alpha_v$ satisfies the required coherence conditions.
Analogously, one can easily prove that if the starting data are in $\opn{Ps}\mbox{-}(\cG\x -)\mbox{-}\opn{Alg}$, every 2-cell induced by the universal property of the bi-iso-comma object $\cP \x[\cY] \cZ$ is $\cG$-equivariant. This conclude the proof that $U$ lifts bi-iso-comma objects.

The previous result of lifting requires an explicit proof as it does not easily follows from the pseudomonadicity of $\cG \x -$ (Proposition <ref>). A result of lifting of PIE limits for pseudomonads is proved in <cit.>, but only for the forgetful functor from 2-category of strict algebras (with pseudomorphisms between them). This result is not strong enough for us as we cannot restrict to strict algebras. In fact, the strict algebras correspond with the actions that satisfy the usual properties of action on the nose.

We will also need to use the following well-known basic property of the bi-iso-comma object, which is a higher dimensional analogue of a well-known property of pullbacks.

Given the following diagram in $\K$

[r,"r"]  [d,"s"']&  [ld,"β",twoiso ,shorten <=2.5ex, shorten >= 2.5ex] [d,"q"][r,"l"] &  [ld,"λ",twoiso, ,shorten <=2.5ex, shorten >= 2.5ex] [d,"f"]

[r,"h"'] &  [r,"g"'] &

where the isomorphic 2-cell $\lambda$ exhibits a bi-iso-comma object, the isomorphic 2-cell $\beta$ exhibits a bi-iso comma object if and only if the pasting diagram exhibits a bi-iso-comma object.

Straightforward using the definition of bi-iso-comma object.

We are  now ready to prove that principal $\cG$-2-bundles are closed under iso-comma-objects.

Let $\pi_\cP\: \cP\to \cY$ be a principal $\cG$-2-bundle  over $\cY$. Then the iso-comma object $\cP\x[\cY]\cZ$ given by

is a principal $\cG$-2-bundle over $\cZ$ with morphism $\pi_{\cP\x[\cY]\cZ}:=f\st \pi_{\cP}$.

By Proposition <ref>, the morphism $f\st \pi_{\cP}$ is $\cG$-equivariant. So it remains to prove that it is 2-locally trivial.
Since $\pi_{\cP}$ is 2-locally trivial, by Remark <ref>, there exists a covering bisieve $S\: R \Rightarrow \K(-,\cY)$ over $\cY$ such that, for every $g\: \cT \to \cY$ in the bisieve $S$, there exists a bi-iso-comma object of the form

We now consider the  covering bisieve $f\st S$.  It is easy to see that every morphism in $f\st S$ is isomorphic a composite

$$\cW \ar{v} \cT\x[\cY]\cZ \ar{f\st g} \cZ$$

with $g\: \cT \to \cY$ in $S$. And so it suffices to prove that there exists a bi-iso-comma object of the form

[rr,"k"] [d,"2"']&  & []  [lld,"", twoiso ,shorten <=9.5ex, shorten >= 9.5ex] [d,"fπ_"]

[r,"v"'] & []  [r,"fg"']&

We start observing that the identity 2-cell inside a commutative square of the form
𝕀u2 2 u
exhibits a bi-iso-comma object. Indeed, it is straightforward to prove that $\cG \x \cU$ satisfies the universal properties of the bi-iso-comma object inducing the required morphisms and 2-cells via the universal properties of the product.
Then, thanks to Lemma <ref>, it is enough to prove that there exists a bi-iso-comma object of the form

([] ) [r,"l"] [d,"2"']& []  [ld,"",twoiso ,shorten <=3.5ex, shorten >= 3.5ex] [d,"fπ_"]

[]  [r,"fg"']&

To prove this, we first look at the following pasting of two bi-iso-comma objects

([] ) [r,"𝕀gf"] [d,"2"']&[3ex]   [r,"h"][d,"2"']&  [ld,"δ",twoiso,Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex] [d,"π_"]

[]  [r,"gf"'] &  [r,"g"'] &

By Lemma <ref>, it
exhibits a bi-iso comma object. Since the following 2-cell is invertible

the following pasting exhibits a bi-iso-comma object as well

([] ) [rr,"l"] [d,"2"']&[3ex]  &  [lld,"β",twoiso,Rightarrow ,shorten <=9.5ex, shorten >= 9.5ex] [d,"π_"]

[]  [r,"fg"'] &  [r,"f"'] &

where $\beta$ is given by the pasting of $\hat{\delta}$ and $\lambda^{f,g}$.

Moreover, by Lemma <ref>, the following pasting of bi-iso-comma object squares

[r,"q_1"]  [d,"q_2"']& []  [ld,"",twoiso ,shorten <=2.5ex, shorten >= 2.5ex] [d,"fπ_"'][r,"π_f"] &  [ld,"λ_^f",twoiso, ,shorten <=2.5ex, shorten >= 2.5ex] [d,"π_"]

[] [r,"fg"'] &  [r,"f"'] &

exhibits a bi-iso-comma object. Since both this bi-iso-comma object and that of diagram <ref> are bi-iso-comma objects of  $\pi_{\cP}$ and $f\c f\st g$, there exists an equivalence

$$\phi\: \cG \x (\cT \x[\cY] \cZ) \aequi{} \cM$$

And it is straightforward to prove that also the following pasting diagram exhibits a bi-iso-comma object

([] )  [rd,"2"', bend right=20 ,""name=A] [r,"ϕ"',aeq] &  [to=A,"",twoiso ,shorten <=2.5ex, shorten >= 2.5ex] [r,"q_1"]  [d,"q_2"']& []  [ld,"",twoiso ,shorten <=2.5ex, shorten >= 2.5ex] [d,"fπ_"'][r,"π_f"] &  [ld,"λ_^f",twoiso, ,shorten <=2.5ex, shorten >= 2.5ex] [d,"π_"]

& [] [r,"fg"'] &  [r,"f"'] &

where the isomorphic two-cell inside the left triangle is the one induced by the universal property of the bi-iso-comma object $\cM$ that induces the equivalence $\phi$.

Finally, applying again Lemma <ref> to diagram <ref> (considering together the two 2-cells on the left), we conclude that the following diagram exhibits a bi-iso-comma object

([] )  [rd,"2"', bend right=20 ,""name=A] [r,"ϕ"',aeq] &  [to=A,"",twoiso ,shorten <=2.5ex, shorten >= 2.5ex] [r,"q_1"]  [d,"q_2"']& []  [ld,"",twoiso ,shorten <=2.5ex, shorten >= 2.5ex] [d,"fπ_"]

& [] [r,"fg"']  &

This concludes the proof that $f\st \pi_{\cP}$ is a principal $\cG$-2-bundle.

§ QUOTIENT 2-STACKS

In this section we will explicitly construct analogues of quotient stacks one dimension higher. We will then prove that they are well-defined trihomomorphisms. We will conclude the section proving that, under mild assumptions, these objects are 2-stacks. This will be the main result of the paper.

Throughout this section $\K$ will be a $(2,1)$-category with all finite flexible limits and $\tau$ will be a bitopology on it. As we will explain in detail later on, we need to restrict ourselves to the case of a $(2,1)$ category to be able to generalize the quotient prestacks of <cit.> to one dimension higher.

We start giving the explicit construction of quotient pre-2-stacks. It generalizes the definition of quotient prestacks of <cit.> to the two-dimensional context.

Let $\cX$ be an object of the $(2,1)$-category $\K$ and let $\cG$ be an internal 2-group in $\K$ that acts on it. The quotient pre-2-stack

$$\qst{\cX}{\cG}\: \K\op \to \twocat$$

is defined as follows:

$\bullet$ for every object $\cY\in \K$ we define $\qst{\cX}{\cG}(\cY)$ as the 2-category that has:

-as objects the pairs $(\cP,\alpha)$ where $\pi_{\cP}\: \cP\to \cY$ is a principal $\cG$-2-bundle over $\cY$ and $\alpha\: \cP\to \cX$ is a $\cG$-equivariant morphism;

- as morphisms from $(\cP,\alpha_{\cP})$  to $(\cQ,\alpha_{\cQ})$ the morphisms of principal $\cG$-2-bundles $\phi\: \cP \to \cQ$ such that there exists an invertible 2-cell

[rd,"α_"',""name=A] [rr,"ϕ"] &[-3ex]  &[-3ex]  [ld,"α_"]  [to=A,"β_ϕ",twoiso ,shorten <=2.5ex, shorten >= 2.5ex,shift left=1ex,start anchor=[xshift=-1.5ex],end anchor=[xshift=-1.5ex]]

&  &

- as 2-cells from $\phi\: (\cP , \alpha_{\cP}) \to  (\cQ , \alpha_{\cQ})$ to $\psi\: (\cP , \alpha_{\cP}) \to  (\cQ , \alpha_{\cQ})$ the 2-cells of principal $\cG$-2-bundles $\Gamma\: \phi \Rightarrow \psi$ such that

[rr,"ϕ", bend left=40,""'name=G] [rd,"α_"',""name=A] [rr,"ψ"',""'name=H] [from=G,to=H,"Γ"pos=0.4,Rightarrow,shorten <=-0.3ex, shorten >= 0.3ex] &[-3ex]  &[-3ex]  [ld,"α_"]  [to=A,"β_ψ",twoiso ,shorten <=2.5ex, shorten >= 2.5ex,shift left=1ex,start anchor=[xshift=-1.5ex],end anchor=[xshift=-1.5ex]]

&  &

[rd,"α_"',""name=A] [rr,"ϕ"] &[-3ex]  &[-3ex]  [ld,"α_"]  [to=A,"β_ϕ",twoiso ,shorten <=2.5ex, shorten >= 2.5ex,shift left=1ex,start anchor=[xshift=-1.5ex],end anchor=[xshift=-1.5ex]]

&  &

$\bullet$ for every morphism $f\: \cZ\to \cY$ in $\K$, we define

$$\qst{\cX}{\cG}(f)\: \qst{\cX}{\cG}(\cY) \to \qst{\cX}{\cG}(\cZ)\vspace{-3.5mm}$$

via iso-comma object along $f$ as the 2-functor that sends:

-an object $(\cP,\alpha_{\cP})$ to the pair $(\cP \x[\cY] \cZ,\alpha_{\cP}\c \pi_{\cP}\st f)$, where $\cP \x[\cY] \cZ$ is given by the iso-comma object

and it is equipped with the morphism $f\st \pi_{\cP}$;

- a morphism $\phi\: (\cP , \alpha_{\cP}) \to  (\cQ , \alpha_{\cQ})$ to the morphism $\qst{\cX}{\cG}(f)(\phi)$ induced by the universal property of the iso-comma object as in the following diagram

[] [d,"fπ_"'] [r,"π_f"] & [r,"ϕ"] [rd,"π_"',""name=A]  [ld,"α_^f",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex]&  [d,"π_"] [to=A,"γ_ϕ",Rightarrow ,shorten <=0.5ex, shorten >= 0.5ex]

[rr,"f"']&  &

- a 2-cell $\Gamma\: \phi \Rightarrow \psi$  from $\phi\: \cP \to \cQ$ to $\psi\: \cP \to \cQ$ to the 2-cell

$$\qst{\cX}{\cG}(f)(\Gamma)\: \qst{\cX}{\cG}(f)(\phi) \Rightarrow\qst{\cX}{\cG}(f)(\psi)$$

induced by the two-dimensional universal property of the iso comma object $\cP\x[\cY]\cZ$ using the following compatible 2-cells

[]  [rd,"π_f"] [dd,"(f)(ψ)"'] [rr,"(f)(ϕ)"]&[-2ex] &[-2ex]  []  [dd,"π_f"]

&  [rd,"ϕ",bend left=30,""'name=A] [rd,"ψ"',bend right=30,""name=B] [from=A,to=B,"Γ",Rightarrow ] &

[]  [rr,"π_f"']&  &

[]  [rd,"fπ_"][d,"(f)(ψ)"'] [r,"(f)(ϕ)"]&[7ex] []  [d,"fπ_"]

[]  [r,"fπ_"'] &

$\bullet$ for every 2-cell $\Lambda\: f \Rightarrow g\: \cZ \to \cY$ in $\K$, we define

$$\qst{\cX}{\cG}(\Lambda)\: \qst{\cX}{\cG}(f) \Rightarrow \qst{\cX}{\cG}(g) $$

as the 2-natural transformation that has component $\qst{\cX}{\cG}(\Lambda)_{\cP}$ relative to $(\cP, \alpha_{\cP})\in \qst{\cX}{\cG}(\cY)$ induced by the universal property of the iso-comma object $\cP \xp{\cY}{g}\cZ$ as follows

f [d,"fπ_"'] [r,"π_f"] &[5ex]  [d,"π_"] [ld,"α_^f"',shift left=-1.3ex,Rightarrow ,shorten <=5.5ex, shorten >= 5.5ex]

[r,"g"',""name=A] [r,"f", bend left=40 ,""'name=B]  [from=B,to=A,Rightarrow,"Λ"]&

Notice that the hypothesis that all the 2-cells of $\K$ are isomorphic is needed to define both the action $\qst{\cX}{\cG}$ on morphisms and 2-cells. Indeed, in order to use the universal property of iso-comma objects to induce the desired morphisms we need to start from isomorphic 2-cells.

We now aim at proving that the quotient pre-2-stacks of Definition <ref> are well-defined and that they are trihomomorphisms from $\K\op$ into $\twocat$. We will split the proof in various intermediate results.

Let $\cY\in \K$. Then $\qst{\cX}{\cG}(\cY)$ (defined in <ref>) is a 2-category.

Straightforward using the fact that $\twoBun{\cG}{\cY}$ is a 2-category and the additional data given by the morphisms into the fixed object $\cX$ behave well with respect to identity and composition.

Let $f:\cZ \to \cY$ be a morphism in $\K$. Then $\qst{\cX}{\cG}(f)$ (defined in <ref>) is a 2-functor.

The fact that $\qst{\cX}{\cG}(f)$ is well-defined follows by the closure of principal-$\cG$-2-bundles under iso-comma-objects (Proposition <ref>). We now prove the [4] 2-functoriality of $\qst{\cX}{\cG}(f)$.

Firstly, we observe that, using the two-dimensional universal property of the iso-comma object, it is straightforward to prove that the assignment on 2-cells preserves identities and composition.

Let now $(\cP,\alpha_{\cP})$ be an object of $\qst{\cX}{\cG}(\cY)$. Since both $\id{(\cP \x[\cY]\cZ,\alpha_{\cP} \x \pi_{\cP}\st f)}$ and [4] $\qst{\cX}{\cG}(f)(\id{(\cP,\alpha_{\cP})})$ can be induced by the universal property of $\cP \x[\cY]\cZ$ using the same 2-cell, they are equal. So $\qst{\cX}{\cG}(f)$ preserves identities.

Let now $\rho\: (\cR,\alpha_{\cR})\to (\cP,\alpha_{\cP})$ and $\phi\: (\cP,\alpha_{\cP})\to (\cQ,\alpha_{\cQ})$ be morphisms in $\qst{\cX}{\cG}(\cY)$. Since both $\qst{\cX}{\cG}(f)(\phi) \c \qst{\cX}{\cG}(f)(\rho)$ and $\qst{\cX}{\cG}(f)(\phi\c \rho)$ can be induced by the universal property of $\cQ \x[\cY]\cZ$ using the same 2-cell, they are equal. So $\qst{\cX}{\cG}(f)$ preserves composition.

Let $\Lambda\: f \Rightarrow g$ be a 2-cell in $\K$ from $f\: \cZ \to \cY$ to $g\: \cZ \to \cY.$ Then $\qst{\cX}{\cG}(\Lambda)$ (defined in <ref>) is a 2-natural transformation.

Let $\phi\: (\cP, \alpha_{\cP}) \to (\cQ, \alpha_{\cQ})$ be a morphism in $\qst{\cX}{\cG}(\cY)$. We need to prove that the following diagram is commutative:
[][12][12]fgfg(Λ)_(, α_)(f)(ϕ) (g)(ϕ)(Λ)_(, α_)
Thanks to the universal property of the iso-comma object $\cQ \xp{\cY}{g}\cZ$, this is implied by the fact that the following diagrams are commutative by construction:

f [dd,"(f)(ϕ)"'] [rd,"π_f"'] [r,"(Λ)_(, α_)"] &[5ex] g [d,"π_g"] [r,"(g)(ϕ)"] &[5ex] g [dd,"π_g"]

&  [rd,"ϕ"] &

f [rr,"π_f", bend left=20] [r,"(g)(ϕ)"'] & g [r,"π_g"'] &

f  [rrdd,"fπ_"][dd,"(f)(ϕ)"'] [r,"(Λ)_(, α_)"] &[5ex] g [rdd,"gπ_"] [r,"(g)(ϕ)"] &[5ex] g [dd,"gπ_"]

&  &

f [rr,"fπ_", bend left=20] [r,"(g)(ϕ)"'] & g [r,"gπ_"']&

The previous three propositions (<ref>, <ref> and <ref>) show that $\qst{\cX}{\cG}$ takes values in $\twocat$.

Before proving that $\qst{\cX}{\cG}$ is a trihomomorphism, we recall the definition of trihomomorphism. We take <cit.> as main reference.

Let $T$ and $T'$ be tricategories. A trihomomorphism $F\: T \to T'$ is given by the following data:

* a function $F:\opn{Ob}(T) \to \opn{Ob}(T')$;

* given $a,b\in T$ a pseudofunctor
$F_{a,b}\: T(a,b) \to T'(F(a),F(b));$

* given $a,b,c,d\in T$ an adjoint equivalence $\chi\: \otimes' \c (F\x F) \Rightarrow F \c \otimes $ with left adjoint

T(b,c) T(a,b) [r,"FF"] [d,"⊗"'] & T'(F(b),F(c))T'(F(a), F(b))[d," ⊗'"] [ld,"χ", Rightarrow, shorten <= 6ex, shorten >= 7ex ]

T(a,c) [r,"F"', ]& T'(F(a),F(c));

* given $a\in T$ an adjoint equivalence $\iota\: I'_{F(a)} \Rightarrow F\c I_a$ with left adjoint

1 [rd,"I'_F(a)"',""name=G] [r,"I_a"]& T(a,a)[to=G,"ι", Leftarrow] [d,"F"]

& T'(F(a),F(a));

* given $a,b,c,d\in T$  an invertible modification

& T^3 [rr,"F FF"] [ld,"1 ⊗"'] [rd," ⊗1 "]&  & T'^3 [ld,"χ1", Rightarrow, shorten <=1.2ex,shorten >=1.2ex ][rd,"1 ⊗'"] &

T^2 [rd,"1 ⊗"'] &  & T^2 [ll," α",Rightarrow, shorten <=3.2ex,shorten >=3.2ex] [rr,"FF"'] [ld," ⊗"]&  & T'^2 [llld,"χ", Rightarrow, shorten <=6.7ex,shorten >=6.7ex] [ld,"⊗'"]

& T [rr,"F"'] &  & T' &

& T^3 [rr,"FF F"] [ld,"1⊗"'] &  & T'^3 [llld,"1 χ", shorten <=6.5ex, shorten >= 6.5 ex, Rightarrow][ld,"1 ⊗'"][rd,"⊗' 1"] &

T^2 [rr,"FF"'] [rd,"⊗"'] &  & T'^2 [ld,"χ", Rightarrow, shorten <= 1ex, shorten >= 1ex] [rd,"⊗'"] &  & T'^2 [ll,"α'", Rightarrow, shorten <=2.5 ex, shorten >= 2.5 ex] [ld,"⊗'"]

& T [rr,"F"'] &  & T'; &

* given $a,b\in T$ invertible modifications

& T'^2  [rd,"⊗'"] [rddd,"χ", Rightarrow, shorten <=10ex, shorten >= 10ex]   &

T' [rd,"ι1", Rightarrow, shorten <=3ex, shorten >= 3ex]    [ru,"I'1"] &  & T'

& T^2 [uu,"FF"]  [rd,"⊗"”] [d,Rightarrow,"l",shorten <=1.5ex, shorten >= 1.5ex] &

T [uu,"F"] [rr,"1"']  [ru,"1I"']&  & T [uu,"F"”]

& T'^2 [rd,"⊗'"] [d,"l", Rightarrow,shorten <=1.5ex,shorten >=1.5ex] &

T'  [rr,"1"', ""name=O]  [ru,"I' 1"] &  & T' [lldd,"", equal, shorten <= 11.3ex, shorten >=11.3 ex]

&  &

T [uu,"F"] [rr,"1"'] &  & T [uu,"F"]

& T^2 [rd,"⊗"] [d,"r̂", Leftarrow,shorten <=1.5ex,shorten >=1.5ex] &

T  [rr,"1"', ""name=O]  [ru,"1I"] &  & T [lldd,"", equal, shorten <= 11.3ex, shorten >=11.3 ex]

&  &

T' [uu,"F", leftarrow] [rr,"1"'] &  & T' [uu,"F", leftarrow]

& T^2  [rd,"⊗"]   &

T   [ru,"1 I"] &  & T

& T'^2 [ru,"χ", Rightarrow, shorten <=3ex, shorten >= 3ex]  [uu,"FF", leftarrow]  [rd,"⊗'"”] [d,Leftarrow,"l",shorten <=1.5ex, shorten >= 1.5ex] &

T' [ruuu,"1 ι", Rightarrow, shorten <=10ex, shorten >= 10ex]   [uu,"F", leftarrow] [rr,"1"']  [ru,"1I'"']&  & T' [uu,"F"”, leftarrow]

See Definition 3.3.1 of <cit.> for the axioms that these data are required to satisfy.

Let $F,G\: T \to T'$ be trihomomorphisms. A tritransformation $\theta\: F \Rightarrow G$ is given by the following data:

* given $a\in T$ a morphism $\theta_a\: F(a) \to G(a)$ in $T'$;

* given $a,b\in T$ an adjoint equivalence
[l][7][7][θ]T(a,b)T'(F(a),F(b))T'(G(a),G(b))T'(F(a),G(b));FGT'(1, θ_b)T'(θ_a,1)

* given $a,b,c\in T$ invertible modifications

&  & T(b,c)T(a,b) [ddl,"GG"'] [ddll,"⊗"', bend right=15] [r,"FF"] [d,"GF"]& T'(F(b),F(c))T'(F(a),F(b)) [ld,"θ1", Rightarrow, ,shorten <=9.5ex, shorten >= 11.5ex] [d,"T'(1,θ_c)1"]

&  & T'(G(b),G(c))T'(F(a),F(b)) [ld,"1θ", Rightarrow,shorten <=9.5ex, shorten >= 11ex][r,"T'(θ_b)1"] [d,"1 T'(1,θ_b)"]& T'(F(b),G(c))T'(F(a),F(b)) [ld,"α",shorten <=10.5ex, shorten >= 10.5ex, Rightarrow] [d,"⊗"]

T(a,c) [r,"χ", Leftarrow,shorten <=1.5ex, shorten >= 1.5ex] [rrd,"G"', bend right=15] & T'(G(b),G(c))T'(G(a),G(b)) [r,"1T'(θ_a,1)"] [rd,"⊗"] & T'(G(b),G(c))T'(F(a),G(b)) [d,"α̂", Rightarrow,shorten <=1.5ex, shorten >= 1.5ex] [r,"⊗"] & T'(F(a), G(c))

&  & T'(G(a), G(c)) [ru,"T'(θ_a,1)"'] &

[d,"Π", triple]

&  & T(b,c)T(a,b)  [ddll,"⊗"', bend right=15] [r,"FF"] [d,"FF"]& T'(F(b),F(c))T'(F(a),F(b)) [ldd,"α", Rightarrow, ,shorten <=15.5ex, shorten >= 15.5ex] [d,"T'(1,θ_c)1"]

&  & T'(F(b),F(c))T'(F(a),F(b)) [lld,"χ", Rightarrow,shorten <=9.5ex, shorten >= 9.5ex]  [d,"⊗"]& T'(F(b),G(c))T'(F(a),F(b))  [d,"⊗"]

T(a,c) [rr,"F"] [rrd,"G"', bend right=15] &    & T'(F(a),F(c))  [d,"α̂", Rightarrow,shorten <=1.5ex, shorten >= 1.5ex] [r,"T'(1,θ_c)"] & T'(F(a), G(c))

&  & T'(G(a), G(c)) [ru,"T'(θ_a,1)"'] &

& 1[ld,"ι", Rightarrow ,shorten <=7.5ex, shorten >= 7.5ex] [d,"I_F(a)"] [ld,"I_a"',""name=B, bend right= 25] [rdd,"θ_a",""name=A, bend left=35] &

T(a,a) [r,"F"] [d,"G"']& T'(F(a), F(a)) [ld,"θ", Rightarrow ,shorten <=5.5ex, shorten >= 5.5ex]  [from=A,"r̂", Rightarrow ,shorten <=2.5ex, shorten >= 2.5ex] [rd,"T'(1,θ_a)"] &

T'(G(a), G(a)) [rr,"T'(θ_a,1)"'] &  & T'(F(a), G(a))

[r,"M", triple]&

& 1 [ld,"I_a"',""name=B, bend right= 25] [rdd,"θ_a",""name=A, bend left=35] [ddl,"I_G(a)",""name=C,bend left=35] &

T(a,a) [d,"G"]&  [l,"ι", Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex]&

T'(G(a), G(a)) [from=A,"l̂", Rightarrow ,shorten <=10.5ex, shorten >= 10.5ex, shift left= 3ex] [rr,"T'(θ_a,1)"']&  & T'(F(a), G(a));

See Definition 3.3.6 of <cit.> for the axioms that these data are required to satisfy.

The assignment $\qst{\cX}{\cG}$ of Definition <ref> is a trihomomorphism.

Let $\cY,\cZ$ in $\K$. We prove that the assignment

$$\qst{\cX}{\cG}_{\cY,\cZ}\: \K\op (\cY,\cZ) \to\twocat(\qst{\cX}{\cG}(\cY), \qst{\cX}{\cG}(\cZ))$$

that sends a morphism $f\: \cZ \to \cY$ into $\qst{\cX}{\cG}(f)$ and a 2-cell $\Gamma\: f \Rightarrow g$ into $\qst{\cX}{\cG}(\Gamma)$ is a 2-functor.

Let $f\: \cZ \to \cY$ be a morphism in $\K$. The 2-natural transformation $\qst{\cX}{\cG}(\id{f})$ has component relative to $(\cP, \alpha_{\cP})\in \qst{\cX}{\cG}(\cY)$ induced by the universal property of the iso-comma object $\cP\xp{\cY}{f}\cZ$ as in the following diagram

f [d,"fπ_"'] [r,"π_f"] &[5ex]  [d,"π_"] [ld,"α_^f"',shift left=-1.3ex,Rightarrow ,shorten <=5.5ex, shorten >= 5.5ex]

[r,"f"',""name=A] &

and so $\qst{\cX}{\cG}(\id{f})_{(\cP,\alpha_{\cP})}$ is equal to $(\id{\qst{\cX}{\cG}(f)})_{(\cP,\alpha_{\cP})}$. This shows that $\qst{\cX}{\cG}_{\cY,\cZ}$ preserves identities.

Let now $f,g,h\: \cZ \to \cY$ be morphisms in $\K$ and $\Gamma\: f \Rightarrow g$ and $\Lambda \: g \Rightarrow h$ be 2-cells in $\K$. The 2-natural transformation $\qst{\cX}{\cG}(\Lambda \c \Gamma)$ has component relative to $(\cP, \alpha_{\cP})\in \qst{\cX}{\cG}(\cY)$ induced by the universal property of the iso-comma object $\cP\xp{\cY}{h}\cZ$ as in the following diagram

f [d,"fπ_"'] [r,"π_f"] &[5ex]  [d,"π_"] [ld,"α_^f"',shift left=-1.3ex,Rightarrow ,shorten <=5.5ex, shorten >= 5.5ex]

[r,"f",bend left=40 ,""'name=B] [r,"h"',""name=A] [from=B,to=A,Rightarrow,"ΛΓ̧"]&

and so $\qst{\cX}{\cG}(\Lambda \c \Gamma)_{(\cP,\alpha_{\cP})}$ is equal to $\qst{\cX}{\cG}(\Lambda)_{(\cP,\alpha_{\cP})} \c \qst{\cX}{\cG}(\Gamma)_{(\cP,\alpha_{\cP})}$. This shows that $\qst{\cX}{\cG}_{\cY,\cZ}$ preserves composition.

Let now $\cY\in \K$. We need to prove that there exists an adjoint equivalence

$$\iota\: \id{ \qst{\cX}{\cG}(\cY)}\Rightarrow \qst{\cX}{\cG}( \id{\cY})$$

Given $(\cP,\alpha_{\cP})\in \qst{\cX}{\cG}(\cY)$, we define

$$\iota_{(\cP,\alpha_{\cP})}\: \id{ \qst{\cX}{\cG}(\cY)}((\cP,\alpha_{\cP})) \to \qst{\cX}{\cG}(\id{\cY})((\cP,\alpha_{\cP}))$$

as an adjoint equivalence between $(\cP, \alpha_{\cP})$ and $(\cP\xp{\cY}{\id{\cY}}\cY, \alpha_{\cP}\c \pi_{\cP}\st \id{\cY})$ given by the fact that they are both bi-iso-comma objects of  $\pi_{\cP}$ and $\id{\cY}$.
Then, given a morphism $\phi\: (\cP,\alpha_{\cP}) \to (\cQ\alpha_{\cQ})$ in $\qst{\cX}{\cG}(\cY)$, we can induce the invertible 2-cell
[l][7][7][ι_ϕ]𝕀 𝕀 ι_(,α_) ϕ (𝕀)(ϕ) ι_(,α_)
using the universal property of the iso-comma object $\cQ \xp{\cY}{\id{\cY}}\cY$ thanks to the following compatible isomorphic 2-cells:

[dd,"ϕ"'] [rd,equal] [r,"ι_(,α_)"] &[5ex] 𝕀 [d,"π_𝕀"] [r,"(𝕀)(ϕ)"] &[5ex] 𝕀 [dd,"π_𝕀"]

&  [rd,"ϕ"] &

[rr,"",equal, bend left=20] [r,"(𝕀)(ϕ)"'] & 𝕀 [r,"π_𝕀"'] &

[rrdd," π_"][dd,"ϕ"'] [r,"ι_(,α_)" ,""name=F] &[5ex] 𝕀 [rdd,"𝕀π_"] [r,"(𝕀)(ϕ)"] &[5ex] 𝕀 [dd,"𝕀π_"]

&  &

[rr,"π_", bend left=20] [r,"(𝕀)(ϕ)"',""name=G,pos=0.2] [from=F,to=G,"γ_ϕ^-1",Rightarrow ,shorten <=9.5ex, shorten >= 6.5ex]& 𝕀 [r,"𝕀π_"']&

Moreover, it is straightforward to prove that the assignment of the structure 2-cells $\iota_\phi$ is pseudofunctorial and that $\iota$ is an adjoint equivalence.

Let now $\cY,\cZ,\cT\in \K$. We need to prove that there exists an adjoint equivalence

$$\chi\: \otimes' \c (\qst{\cX}{\cG} \x \qst{\cX}{\cG}) \Rightarrow \qst{\cX}{\cG} \c \otimes$$

Given $g\: \cT \to \cZ$ and $f\: \cZ \to \cY$, we need to define a 2-natural transformation

$$\chi_{f,g}\:  \qst{\cX}{\cG}(g) \c \qst{\cX}{\cG}(f) \Rightarrow \qst{\cX}{\cG}(f\c g)$$

Given $(\cP, \alpha_{\cP})\in \qst{\cX}{\cG}(\cY)$, we define

$$(\chi_{f,g})_{(\cP, \alpha_{\cP})}\:  (\qst{\cX}{\cG}(g) \c \qst{\cX}{\cG}(f))((\cP, \alpha_{\cP})) \Rightarrow \qst{\cX}{\cG}(f\c g) ((\cP, \alpha_{\cP}))$$

as an adjoint equivalence between $(\cP \x[\cY]\cZ)\x[\cZ] \cT$ and $\cP\x[\cY]\cT$ given by the fact that they are both iso-comma objects of $\pi_{\cP}$ and $f\c g$.

Given a morphism $\phi\: (\cP,\alpha_{\cP}) \to (\cQ\alpha_{\cQ})$ in $\qst{\cX}{\cG}(\cY)$, the following square is then commutative
([])[] ([])[] (χ_f,g)_(,α_) ((g) (f))(ϕ) (fģ)(ϕ) (χ_f,g)_(,α_)
thanks to the following commutative diagrams:

([])[] [rd,"(fπ_)g"] [ddd,"((f) (g))(ϕ)"'][rr,"(χ_f,g)_"]&  & [] [d,"π_(fģ)"]  [r," (fģ)(ϕ)"]&[4ex] [] [ddd,"π_(fģ)"]

& [] [r,"π_"]  [d,"(f)(ϕ)"description]&  [rdd,"ϕ"] &

& [] [rrd,"π_"] &  &

([])[] [ru,"(fπ_)g"] [rr,"(χ_f,g)_"'] &  & [] [r,"π_(fģ)"']  &

([])[] [rrrd,"g(fπ_)"]  [d,"((f) (g))(ϕ)"'][rr,"(χ_f,g)_"]&  & [] [rd,"(fģ)π_"] [r," (fģ)(ϕ)"]&[4ex] [] [d,"(fģ)π_"]

([])[] [rrr,"g(fπ_)", bend left=13] [rr,"(χ_f,g)_"'] &  & [] [r,"(fģ)π_"']  &

This shows that $\chi_{f,g}$ is 2-natural. Moreover, when $f$ and $g$ vary the $\chi_{f,g}$'s form a 2-natural transformation. Indeed, given morphisms $g,g'\: \cT\to \cZ$ and $f,f'\: \cZ\to \cY$ and 2-cells $\Gamma\: g \Rightarrow g'$ and $\Lambda\: f \Rightarrow f'$ the following equality holds

$$\qst{\cX}{\cG}(\Lambda \star \Gamma) \c \chi_{f,g} = \chi_{f',g'} \c (\qst{\cX}{\cG}(\Lambda) \star \qst{\cX}{\cG}(\Gamma).$$

This is because, for every $(\cP, \alpha_{\cP})\in \qst{\cX}{\cG}(\cY)$, the following diagram is commutative

(f) g [rr,"(χ_f,g)_"] [d,"(g)(((Λ))_)"']&[6ex]  &[3ex] fģ [d,"((Λ⋆Γ))_"]

(f') g [r,"((Γ))_f'"']& (f') g' [r,"χ_f',g'"'] & f'ģ'

thanks to the commutativity of

(f) g [rd,"(fπ_)g"description] [r,"(χ_f,g)_"] [d,"(g)(((Λ))_)"']&[4ex] fģ [rddd,"π_(fģ)", bend left=15] [r,"((Λ⋆Γ))_"] &[4ex] f'ģ' [ddd,"π_(f'ģ')"]

(f') g [rd,"((f')π_)g"description] [dd,"((Γ))_f'"'] & f [rdd,"π_f", bend left=15] [d,"(Λ)_"description] &

& f' [rd,"π_f'"]&

(f') g' [ru,"((f')π_)g'"] [r,"χ_f',g'"'] & f'ģ' [r,"π_(f'ģ')"'] &

(f) g [rrdd,"g(fπ_)"description, bend left=10] [r,"(χ_f,g)_"] [d,"(g)(((Λ))_)"']&[4ex] fģ [rdd,"(fģ)π_"description, bend left=10] [r,"((Λ⋆Γ))_"] &[4ex] f'ģ' [dd,"π_(f'ģ')"]

(f') g [rrd,"g((f')π_"description, bend left=10] [d,"((Γ))_f'"'] &  &

(f') g' [rr,"(g')((f')π_)", bend left=10][r,"χ_f',g'"'] & f'ģ' [r,"π_(f'ģ')"'] &

This shows that $\chi$ is an adjoint equivalence.

Consider now a chain of morphisms

$$\cW \ar{h} \cT \ar{g} \cZ \ar{f} \cY$$

in $\K$. The following equality of 2-natural transformation holds

$$\chi_{f,g\c h} \c (\chi_{g,h} \star \qst{\cX}{\cG} (f)) = \chi_{f\c g,h} \c (\qst{\cX}{\cG}(h) \star \chi_{f,g})$$

because it holds on components thanks to the compatibility between the iso-comma objects along the composite of three morphisms and the pasting of the three iso-comma objects. This shows that the invertible modification $\omega$ required in Definition <ref> is the identity for $\qst{\cX}{\cG}$.

Moreover, the invertible modification $\gamma$ of Definition <ref>, is the identity as well. Indeed, given a morphism $\cZ\ar{f}\cY$ in $\K$, the pseudonatural transformations $\chi_{f,\id{\cZ}} \c (\iota \star  \qst{\cX}{\cG}(f))$ and $\id{ \qst{\cX}{\cG}(f)}$ have the same components by the universal properties of the iso-comma object and the same structure 2-cells thanks to the fact that the structure 2-cells of $\iota$ of the form $\iota_{ \qst{\cX}{\cG}(f)(\phi)}$ with $\phi$ a morphism in $\qst{\cX}{\cG}(\cY)$ are identities by definition of $\qst{\cX}{\cG}(f)$. Analogously, the invertible modification $\delta$ of Definition <ref> is the identity as well.

Finally, the axioms of trihomomorphisms are trivially satisfied by $\qst{\cX}{\cG}$ since all the modifications involved are identities.

Analogously to the one-dimensional case, we can consider the particular case of $[\cT/\cG]$ where $\cT$ is the terminal object of $\K$ and so the  morphisms with target $\cT$ are uniquely determined and the  2-category $\qst{\cT}{\cG}(\cY)$ is isomorphic to $\twoBun{\cG}{\cY}$.

The quotient pre-2-stack $\qst{\cT}{\cG}$ is called classifying pre-2-stack and will be denoted $2\clst{G}$.

In the remaining part of this section we will prove that, if the $(2,1)$-category $\K$ is bicocomplete and such that iso-comma objects preserve bicolimits and the bitopology $\tau$ is subcanonical, the quotient pre-2-stacks  of Definition <ref> are 2-stacks in the sense of <cit.>. To prove this important result, we will use the following explicit characterization of 2-stack that we prove in <cit.>.

[<cit.>, Theorem 3.18 ]
Let $(\K,\tau)$ be a bisite. A trihomomorphism $F\: \K\op \to \Bicat$ is a 2-stack if and only if for every $C\in \K$ and every covering bisieve $S\in \tau(C)$ the following conditions are satisfied:

(O) every weak descent datum for $S$ of elements of $F$ is weakly effective;

(M) every descent datum for $S$ of morphisms of $F$ is effective;

(2C) every matching family for $S$ of 2-cells of $F$ has a unique amalgamation.

Moreover, we will need the crucial result that every object of a subcanonical bisite is the sigma-bicolimit of each covering bisieve over it. We prove this result in <cit.>.

[<cit.>,Theorem 2.24]
Let $\tau$ be a subcanonical bitopology and let $S\: R \Rightarrow y(C)$ be a covering bisieve over $C$. Then $S$ is a sigma-bicolim bisieve, that is

where $F\: \Groth{R}\to \K$ is the 2-functor of  projection to the first component.

Another important result proven in <cit.> that will be helpful for the proof is the following.

Let $\K$ be a small 2-category with iso-comma objects and let $\tau$ be a subcanonical bitopology on it. Let then $S\: R \Rightarrow \K(-,Y)$ be a covering bisieve over $Y\in\K$ and $f\: X\to Y$ be a morphism in $\K$. Then $X$ is the sigma-bicolimit of the 2-functor

$$F\:  \Grothdiag{R} \ar{\opn{inc}} \laxslice{\K}{Y} \ar{f\st} \laxslice{\K}{X}  \ar{\opn{dom}} \K$$

where $\opn{inc}=\Groth{S}\: \Groth{R} \to \Groth{\K(-,Y)}$ is the inclusion 2-functor  and $f\st$ is the 2-functor of iso-comma object along the morphism $f$.

Finally, we will need to apply the results about colimits in lax slices proved by Mesiti in <cit.> and more in detail in the case of sigma-bicolimits in his PhD thesis <cit.>.

Let $\K$ be a bicocomplete $(2,1)$-category with finite flexible limits and such that iso-comma objects preserve bicolimits. Let then $\tau$ be a subcanonical bitopology on $\K$ and $\cX,\cG\in \K$  with $\cG$ an internal 2-group. Then the quotient pre-2-stack $\qst{\cX}{\cG}$ is a 2-stack.

The hypothesis that iso-comma objects preserve bicolimits in $\K$ means that for every morphism $a\: \cT \to \cW$ in $\K$ the 2-functor

$$a\st\: \laxslice{\K}{\cW} \to \laxslice{\K}{\cT}$$

of iso-comma along  the morphism $a$ preserves bicolimits. This hypothesis will be crucial in our proof of Theorem <ref>.

Let $\cY\in \K$ and let $S\: R \Rightarrow \K(-,\cY)$ be a covering bisieve over $\cY$. Thanks to Theorem <ref>, it suffices to prove that the following conditions hold:

(O) every weak descent datum for $S$ of elements of $F$ is weakly effective;

(M) every descent datum for $S$ of morphisms of $F$ is effective;

(2C) every matching family for $S$ of 2-cells of $F$ has a unique amalgamation.

We start proving (2C).

Let $\sigma,\rho\: (\cP, \alpha_{\cP}) \to (\cQ, \alpha_{\cQ})$ be morphisms in $\qst{\cX}{\cG}(\cY)$. Consider a matching family for $S$ of 2-cells of $\qst{\cX}{\cG}$ that assigns to each morphism $f\: \cZ\to \cY\in S$ a 2-cell

$$w_f\: \qst{\cX}{\cG}(f)(\sigma) \Rightarrow \qst{\cX}{\cG}(f)(\rho)$$

We need to prove that there exists a unique 2-cell

$$w\: \sigma \Rightarrow \rho$$

such that $\qst{\cX}{\cG}(f)(w)=w_f$ for every  $f\: \cZ\to \cY\in S$.

By Proposition <ref>, $\cP$ is the sigma-bicolimit of the 2-functor

$$F\:  \Grothdiag{R} \ar{\opn{inc}} \laxslice{\K}{\cY} \ar{\pi\st_{\cP}} \laxslice{\K}{\cP}  \ar{\opn{dom}} \K$$

with universal sigma-bicocone given by the identical 2-cells of the form

h [rd,"π_ h",""name=A] [d,"F((𝕀,α))"',] &[10ex]

lģ   [r,"π_(lģ)"'pos=0.4] [d,"F((g,𝕀))"',""name=C]&

l [ru,"π_l"',""name=D] &

for every morphism $(g,\alpha)\: (\cW, \cW \ar{h} \cY) \to (\cT, \cT \ar{l} \cY)$ in $\Groth{R}$.

For every $f\: \cZ\to \cY\in S$, we consider the 2-cell

&[5ex]   [rd,"σ", bend left=35] &[5ex]

f [r,"(f)(σ)", ,""name=D, bend left=35] [r,"(f)(ρ)"', ,""name=E, bend right=35] [ru,"π_ f", bend left=35] [rd,"π_ f"', bend right=35] & f [r,"π_ f"'] &  [from=D,to=E,"w_f",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex]

&   [ru,"ρ"', bend right=35]&

Using the fact that the $w_f$'s are the assignment of a matching family, it is straightforward to prove that the 2-cells of the form (<ref>) are compatible. In particular, one needs to use the compatibility of the matching family with respect to composition to deal with the morphisms of  $\Groth{R}$ of type $(g,\id{})$ and the compatibility of the matching family with respect to the 2-cells in $S$ to deal with the morphisms of  $\Groth{R}$ of type $(\id{},\alpha)$.
So, by the  two-dimensional universal property of the sigma-bicolimit of $F$, there exists a unique 2-cell $w\: \sigma \Rightarrow \rho$ such that for every $f\: \cZ \to \cY\in S$, the whiskering

f [r,"π_ f"]&  [r,"σ", bend left=30 ,""'name=A] [r,"ρ"', bend right=30 ,""name=B] [from=A,to=B,"w", Rightarrow ] &

is equal to the 2-cell of diagram (<ref>). We now show that $w$ is the desired amalgamation. By definition of $\qst{\cX}{\cG}$, the 2-cell $\qst{\cX}{\cG}(f)(w)$ is the unique that has whiskering with $f\st \pi_{\cQ}$ equal to the identical 2-cell and whiskering with $\pi\st_{\cQ} f$ equal to the 2-cell

f [rd,"π_"] [rr,"(f)(σ)"] [dd,"(f)(ρ)"'] &   & f [dd,"π_"]

&  [rd,"σ" ,""'name=A,bend left=30] [rd,"ρ"' ,""name=B,bend right=30] [from=A,to=B,"w",Rightarrow ] &

f [rr,"π_"']&  &

But the whiskering of $w_f$ with $f\st \pi_{\cQ}$ is equal to the identical 2-cell because $w_f$ is a 2-cell in $\qst{\cX}{\cG}(\cZ)$ and the 2-cells $\gamma_{ \qst{\cX}{\cG}(f)(\sigma)}$ and $\gamma_{ \qst{\cX}{\cG}(f)(\rho)}$ are identities by definition of $\qst{\cX}{\cG}$. Moreover, the whiskering of $w_f$ with $\pi\st_{\cQ}f$, is equal to the 2-cell of diagram (<ref>) by construction of $w$. This shows that $\qst{\cX}{\cG}(f)(w)=w_f$ and concludes the proof of the condition (2C).

We now prove the condition $(M)$.

Let $(\cP, \alpha_{\cP}), (\cQ, \alpha_{\cQ})\in \qst{\cX}{\cG}(\cY)$. Consider a descent datum for $S$ of morphisms of $\qst{\cX}{\cG}$ that assigns to every $\cZ \ar{f} \cY\in S$ a morphism

$$w_f\: \cP\xp{\cY}{f}{\cZ}\to \cQ\xp{\cY}{f}{\cZ}$$

and to every 2-cell $\Gamma\: f \Rightarrow f'$ with $f,f'\: \cZ \to \cY\in S$  an (invertible) 2-cell
[l][7][7][η_Γ]f ff' f'w_f(Γ)_(Γ)_w_f'
Moreover, for every chain $\cZ' \ar{g} \cZ \ar{f} \cY$ with $f\in S$, we have an (invertible) 2-cell

(f)g' [r,"gw_f"]  [d,"",aeq] &[7ex] (f)g' [d,"",aeq]  [ldd,"ϕ^f,g",Rightarrow ,shorten <=6.5ex, shorten >= 6.5ex]

fģ' [d,"",aiso]& fģ' [d,"",aiso]

fģ' [r,"w_fģ"'] &  fģ'

We need to prove that there exist a morphism

$$w\: \cP \to \cQ$$

in $\qst{\cX}{\cG}(\cY)$ and for every morphism $\cZ \ar{f} \cY\in S$ an (invertible) 2-cell

$$\psi^f\: w_f \Rightarrow \qst{\cX}{\cG}(f)(w)$$

that satisfy the conditions required by the definition of effective descent datum on morphisms (see Definition 3.9 of <cit.>).
The idea is to induce the morphism $w$ using the universal property of $\cP$ seen as a sigma-bicolimit of a diagram of shape $\Groth{R}$. This is similar to what we did to prove the condition (2C).

As already observed when proving (2C), by Proposition <ref>, $\cP$ is the sigma-bicolimit of the 2-functor

$$F\:  \Grothdiag{R} \ar{\opn{inc}} \laxslice{\K}{\cY} \ar{\pi\st_{\cP}} \laxslice{\K}{\cP}  \ar{\opn{dom}} \K$$

with universal sigma-bicocone as shown in diagram (<ref>).

We construct a sigma-bicocone $\Xi$ of shape $\Groth{R}$ over $\cQ$. Given an object $(\cZ, \cZ\ar{f}\cY)$, we define $\Xi_f$ as the composite

$$\cP\xp{\cY}{f}\cZ \ar{w_f} \cQ\xp{\cY}{f}\cZ \ar{\pi\st_{\cQ}} \cQ$$

Given a morphism $(g,\id{})\: (\cZ', \wt{f\c g}) \to (\cZ,f)$ in $\Groth{R}$, we define $\Xi_{(g,\id{})}$ as the following 2-cell

fģ' [dd,"F((g,𝕀))"'][r,"w_fģ"] & fģ' [dd,"g_"] [rd,"π_ (fģ)"] [ldd,"κ_g",Rightarrow ,shorten <=5.5ex, shorten >= 5.5ex]&[5ex]

&  &

f [r,"w_f"]& f [ru,"π_ f"'] &

where $\widehat{g}_{\cQ}$ is the morphism induced by the universal property of ${\cQ \xp{\cY}{f}\cZ}$ using the 2-cell

fģ [rr,"π_(fģ)"]  [d,"(fģ)π_"']&  &  [lld,"λ^fģ_"'inner sep=0.1ex, Rightarrow ,shorten <=8.5ex, shorten >= 8.5ex, shift left=-2ex] [d,"π_"]

' [r,"g"'] [rr,"fģ", bend left=35,""'name=A] &  [r,"f"'] [from=A,"σ_f,g"pos=0.8, Rightarrow ,]&

and the 2-cell $\kappa_g$ is induced by the two-dimensional universal property of ${\cQ \xp{\cY}{f}\cZ}$ using the compatible 2-cells

fģ '  [rrr,"F((g,𝕀))"] [rd,"",aiso]  [ddddd,"w_fģ"']&[-5ex]  &[-5ex]  &[-3ex] f [r,"w_f"] & f [ddddd,"π_f"]

& fģ ' [rd,"",aeq]  &  	&  &

&  & (f )g '  [llddd,"ϕ^f,g", Rightarrow ,shorten <=8.5ex, shorten >= 8.5ex, shift left=-6ex] [d,"gw_f"]	[ruu,""] &  &

&  & (f )g '  [rruuu,"", bend right=15]	&  &

& fģ ' [ru,"",aeq]  [rrrd,"", bend left=25] &  	&  &

fģ' [ru,"",aiso] [rrrr,"", bend left=15] [rrr,"_g"']&  &  	& f [r,"π_ f"']  &

fģ '  [rrr,"F((g,𝕀))"] [rd,"",aiso]  [ddddd,"w_fģ"']&[-5ex]  &[-5ex]  &[-3ex] f [rddddd,"fπ_",""name=P] [r,"w_f"] [dddd,"λ^g_f", Rightarrow ,shorten <=7.5ex, shorten >= 7.5ex] & f [to=P,"γ_w_f", Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex] [ddddd,"fπ_"]

& fģ ' [rd,"",aeq]  &  	&  &

&  & (f )g '  [rdd,"", bend left=16] [ruu,""] [llddd,"ϕ^f,g", Rightarrow ,shorten <=8.5ex, shorten >= 8.5ex, shift left=-6ex] [d,"gw_f"] &  &

&  & (f )g ' 	 [rd,""]&  &

& fģ '  [rr,""][ru,"",aeq]   &  	& '  [rd,"g"']&

fģ' [rrru,"", bend right=5] [ru,"",aiso] [rrr,"g_"']&  &  	& f [r,"π_ f"']  &

where the unnamed arrows are the projections and the 2-cell $\widehat{\phi^{f,g}}$ is defined by whiskering $\phi^{f,g}$ with the isomorphic 2-cells of pseudoinvertibility of the involved equivalences.

Given  a morphism $(\id{}, \Gamma)\: (\cZ, \cZ \ar{f} \cY) \to (\cZ, \cZ \ar{f'} \cY)$ in $\Groth{R}$, we define $\Xi_{(\id{}, \Gamma)}$ as the following 2-cell

f [dd,"F((𝕀,Γ))"'] [r,"w_f"] & f [dd,"(Γ)_ "] [rd,"π_f"] [ldd,"η_Γ",Rightarrow ,shorten <=5.5ex, shorten >= 5.5ex]&[5ex]

&  &

f' [r,"w_f'"]& f' [ru,"π_ f'"'] &

One can then construct the structure 2-cell of the form $\Xi_{(g,\alpha)}$ using the factorization of $(g,\alpha)$ as a composite of a morphism of type $(g,\id{})$ with one of type $(\id{},\alpha)$ and use the compatibility conditions satisfied by the descent datum to verify that these assignment defines a sigma-bicocone of shape $\Groth{R}$ over $\cQ$. This implies that there exists a unique morphism $w\: \cP \to \cQ$ together (invertible) 2-cells

&   [dd,"θ_f",twoiso ,shorten <=2.5ex, shorten >= 2.5ex] [rd,"w", bend left=30] &

f  [ru,"π_ f", bend left=30] [rd,"w_f"', bend right=30] &  &

& f  [ru,"π_ f"', bend right=30]&

for every $\cZ \ar{f} \cY$ such that, given a morphism $(g,\alpha)\: (\cZ, \cZ \ar{f} \cY) \to (\cZ', \cZ' \ar{h} \cY)$, the following equality of 2-cells holds:

&   [dd,"θ_f",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex] [rd,"w", bend left=20] &

f [dd,"F((g,α))"']  [ru,"π_ f", bend left=20] [rd,"w_f"', bend right=20] &  &

& f [d,"Ξ_(g,α)",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex]  [ru,"π_ f"'inner sep=0.1ex, bend right=20]&

h' [r,"w_h"']& h' [ruu,"π_ h"', bend right=30]&

&  [ddd,"θ_h",Rightarrow ,shorten <=5.5ex, shorten >= 5.5ex]  [rd,"w", bend left=20] &

f [dd,"F((g,α))"']  [ru,"π_ f", bend left=20] &  &

&   &

h' [ruuu,"π_ h"', bend left=15] [r,"w_h"']& h' [ruu,"π_ h"', bend right=30]&

We now need to prove that $w\: \cP \to \cQ$ is the desired morphism. It is straightforward to prove that $w$ is a morphism in $\qst{\cX}{\cG}(\cY)$. Moreover, given $\cZ \ar{f} \cY\in S$, we can induce the desired 2-cell $\psi^{f}\: w_f \Rightarrow f\st w$ applying the two-dimensional universal property of the iso-comma object $\cQ \xp{\cY}{f}\cZ$ to the compatible 2-cells

f  [rd,"π_"'] [rr,"w_f"] [dd,"fw"']&  & f  [ld,"θ_f^-1", Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex] [dd,"π_ f"]

&  [rd,"w"'] &

f  [rr,"π_ f"'] &  &

f   [rrdd,"fπ_"',""name=E][rr,"w_f"] [dd,"fw"']&  & f  [dd,"f π_"]  [to=E,"γ_w_f",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex]

&  &

f  [rr,"f π_"'] &  &

Finally, one can prove that the required conditions are satisfied tanks to equalities of type (<ref>) and the compatibility conditions satisfied by the descent datum. This concludes the proof that (M) holds.

It remains to prove that the condition (O) holds. This will be the trickiest condition to prove.

Consider a weak descent datum for S of elements of $\qst{\cX}{\cG}$ that assigns to every $\cZ \ar{f} \cY\in S$ a principal $\cG$-2-bundle

$$\pi_f\: \cW_f \to \cZ\in \qst{\cX}{\cG}(\cZ)$$

and to every 2-cell $\Gamma\: f \Rightarrow f'$ with $f,f'\: \cZ \to \cY$ a morphism

$$w_{\Gamma}\: \cW_f \Rightarrow \cW_{f'}$$

in $\qst{\cX}{\cG}(\cZ)$. Moreover, for every chain $\cZ' \ar{g} \cZ \ar{f} \cY$ with $f\in S$, we have an equivalence

$$\phi^{f,g}\: \cW_{f\c g} \aequi g\st \cW_f$$

in $\qst{\cX}{\cG}(\cZ)$ and all the other data required by the definition of weak descent datum (see <cit.>) satisfying the required axioms.

We need to prove that there exist a principal $\cG$-2-bundle

$$\pi\: \cW \to \cY\in \qst{\cX}{\cG}(\cY)$$

and for every $\cZ \ar{f} \cY\in S$ an equivalence

$$\psi^f\: W_f \to f\st W$$

in $\qst{\cX}{\cG}(\cY)$ and all the other data required by the definition of weak descent datum satisfying the required conditions.

We use the assignment of the weak descent datum to construct a pseudofunctor
$$\Lambda \: \Groth{R} \to \psslice{\K}{\cY}$$

Given an object $(\cZ,\cZ \ar{f}\cY)\in \Groth{R}$, we define $\Lambda((\cZ,\cZ \ar{f}\cY))$ as the morphism

$$\cW_f \ar{\pi_f} \cZ \ar{f} \cY$$

Given a morphism $(g,\id{})\: (\cZ', \wt{f\c g}) \to (\cZ, f)$ in $\Groth{R}$, we define $\Lambda((g,\id{}))$  as the 2-cell

_fģ [rr,"ϕ^f,g"] [rd,"π_fģ"',""name=A] &[-3ex]  &[-3ex] g_f [to=A,"γ_ϕ^f,g",Rightarrow ,shorten <=2.5ex, shorten >= 2.5ex, shift left=-1ex] [ld,"gπ_f"] [rr,"π_f g"]&[-3ex]  &[-3ex] _f [ld,"π_f"]  [llld,"λ^π_f,g",Rightarrow ,shorten <=9.5ex, shorten >= 9.5ex]

& ' [rr,"g"] [rd,"fģ"' ,""name=F] &  &  [to=F,"σ^-1_f,g"inner sep=0.01ex, Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex] [ld,"f"]&

&  &  &  &

Given a morphism $(\id{}, \Gamma)\: (\cZ, \cZ \ar{f} \cY) \to (\cZ, \cZ \ar{f'} \cY)$ in $\Groth{R}$, we define $\Lambda((\id{},\Gamma))$ as the 2-cell

_f [rrrr,"_Γ"]  [rd,"π_f"']&[-3ex]  &[-3ex]  &[-3ex]  &[-3ex] _f' [ld,"π_f'"] [lllld,"γ__Γ",Rightarrow ,shorten <=11.5ex, shorten >= 11.5ex]

&  [rd,"f"',""name=G] [rr,"",equal] & &   [ld,"f'",""name=H] [ll,"Γ^-1",Rightarrow ,shorten <=3.5ex, shorten >= 3.5ex, shift left=4ex]&

&  &  &  &

One can then construct the image of a generic morphism of $\Groth{R}$ using the factorization in terms of morphisms of type $(g,\id{})$ and $(\id{},\Gamma)$.
Finally, given a 2-cell $\delta\: (g,\alpha) \to (h,\beta)$ in $\Groth{R}$ with $(g,\alpha),(h,\beta)\: (\cZ, \cZ \ar{f} \cY) \to (\cZ',\cZ' \ar{f'} \cY)$, we define $\Lambda(\delta)$ as the 2-cell

& _f'ģ [dd,"_R(δ)_f'"']  [r,"ϕ^f',g"] & g_f'  [rd,"π_f' g"] [dd,"(δ)__f'"] [ldd,"(α_δ)_f'", Rightarrow ,shorten <=4.5ex, shorten >= 4.5ex]&

_f  [ru,"_α"] [rd,"_β"']&  &  & _f'

& _f'ḩ [r,"ϕ^f',g"'] & h_f' [ru,"π_f' h"'] &

where $(\alpha_{\delta})_{f'}$ is the 2-cell assigned by the weak descent datum (see Definition 3.14 of <cit.>).

We now prove that $\Lambda$ is a pseudofunctor.

Given a chain of morphisms

$$(\cZ,\cZ \ar{f} \cY) \ar{(g,\alpha)} (\cZ',\cZ' \ar{f'} \cY) \ar{(g',\alpha')} (\cZ'',\cZ'' \ar{f''} \cY)$$

in $\Groth{R}$, the isomorphic 2-cell

$$\varepsilon_{(g,\alpha), (g', \alpha')}\: \Lambda((g',\alpha'))\c \Lambda((g,\alpha)) \Rightarrow \Lambda((g',\alpha')\c (g,\alpha))$$
is given by the following pasting diagram:

&[-4ex] _f'ģ [rd,"_α' ⋆g"inner sep=0.1ex,""'name=D] [d,"",aiso] [r,"ϕ^f',g"] &[-4ex] g_f' [d,"",twoiso ,shorten <=1.5ex, shorten >= 1.5ex] [rd,"g_α"] [r,"π_f'g"] &[-4ex] _f' [r,"_α"] &[-4ex] _f” ģ' [r,"ϕ^f”,g"] &[-4ex] (g')_f” [rd,"π_f”g"]&[-4ex]

_f [ru,"_α"] [rd,"_ζ"'] & _f'ģ [from=D,"",twoiso] [r,"",aiso]& _(f” ģ')ģ [ld,"",aiso] [r,"ϕ^f”ģ',g"]& g_f”ģ' [r,"gϕ^f”,g'"] [ru,""] & g(g')_f” [ld,"",aeq]  [ru,""] [llld,"β^g',g_f”",Rightarrow ,shorten <=11.5ex, shorten >= 11.5ex]&  & _f”

& _f” (̧g' ģ) [rr,"ϕ^f”,g'ģ"'] &  & (g' ģ)_f” [rrru,"π_f”(g'ģ)"', bend right=20] &  &  &

where $\beta^{g',g}_{f''}$ is the weak cocycle condition on $f'',g',g$ and $\zeta$ is the 2-cell such that $(g,\alpha) \c (g',\alpha) = (g\c g', \zeta)$.
Moreover, it is straightforward to prove that $\Lambda((\id{},\id{}))$ is isomorphic to the identity and that the required conditions on 2-cells are satisfied.

Let then $W$ be the sigma-bicolimit of the pseudofunctor $\opn{dom} \c \Lambda$ (that exists because $\K$ has all sigma-bicolimits by hypothesis). Since $\Lambda$ is pseudofunctorial it corresponds to a sigma-bicocone over $\cY$ of shape $\Groth{R}$ and so, using the universal property of the sigma-bicolimit, we can induce a morphism

$$\pi\: \cW \to \cY$$

Moreover, by the calculus of colimits in lax slices developed by Mesiti in <cit.> (and more in detail in <cit.> in the sigma-bicolimit case), we obtain the following equality

$$\pi= \sigmabicolim{\Lambda}$$

We now prove that the morphism $\pi$ is a principal $\cG$-2-bundle. $\pi$ is trivially $\cG$-equivariant (since we are considering trivial actions of $\cG$) so it remains to prove that it is 2-locally trivial.
Given $\cZ \ar{f} \cY\in S$, since $\pi_f\: \cW_f\to \cZ$ is 2-locally trivial, there exists a covering bisieve $S_f\in \tau(\cZ)$ such that for every $\cT \ar{h} \cZ\in S_f$ the iso-comma object of $\pi_f$ and $h$ is equivalent to the product $\cG \x \cT$. Consider then the collection

$$S'=\{ \cT \ar{h} \cZ \ar{f} \cY | f\in S, h\in S_f\}$$

Using the axiom $(T2)$ of bitopology, it is easy to prove that $S'$ is a covering bisieve over $\cY$.  Given a morphism $\cT\ar{l} \cY \in S'$ that is equal to the composite $\cT \ar{r} \cZ' \ar{g} \cY$, we obtain the following chain of equalities

$$l\st \pi= l\st \sigmabicolim{\Lambda}= \sigmabicolim{l\st \Lambda},$$

where the second equality is given by the hypothesis that iso-comma objects preserve colimits.
And given an object $(\cZ,f)\in \Groth{R}$ the following is an iso-comma object square

[r,"",iso,shift left=-5ex][r,"q_1"] [d,"q_2"']& _f ([] ') [r,"",iso,shift left=-5ex] [r,""] [d,"(fg)π_f"]& _f [d,"π_f"]

([] ')['] [r,"",iso,shift left=-5ex] [r,"(fg)r"] [d,""]& [] ' [r,"",iso,shift left=-5ex] [d,"gf"]  [r,"fg"]&  [d,"f"]

[r,"r"'] & ' [r,"g"'] &

thanks to the fact that the covering bisieve generated by $(f\st g)\st h$ with $h\in S_g$ trivializes $(f\st g)\st \pi_f$ as shown in the proof of Proposition <ref>. This implies that the iso-comma of the morphism $\cW_f \ar{\pi_f} \cZ \ar{f} \cY$ along $l$ is equivalent to the product $\cG \x \cT$. And so the morphism $\pi$ is 2-locally trivial.

Furthermore, we can induce a morphism $\alpha\: \cW \to \cX$ applying the universal property of the sigma-bicolimit to the sigma-bicocone of shape $\Groth{R}$ whose structure 2-cells are simply given by the fact that the morphisms involved are morphisms in 2-categories of the form $\qst{\cX}{\cG}(\cT)$ with $\cT\in\K$. This conclude the proof of the fact that $(\cW \ar{\pi} \cY, \cW \ar{\alpha} \cX)$ is in $\qst{\cX}{\cG}(\cY)$.

We now need to induce the equivalence $\psi^{f}\: \cW_{f} \aequi f\st \cW$ for every $f\in S$. We equivalently induce the pseudoinverse $\theta^{f}\: f\st \cW \to \cW_f$.
We observe that the following chain of equalities holds

$$f\st \cW = f\st \sigmabicolim{\Lambda}= \sigmabicolim{f\st \Lambda}$$

and so we can use the universal property of the sigma-bicolimit to induce $\theta^{f}$. To do so, we construct a sigma bicocone $\Sigma$ over $\cW_f$. Given $(\cZ', \cZ' \ar{g} \cY)\in \Groth{R}$, we define $\Sigma_g$ as the composite

$$f\st \cW_g \ar{\lambda_{f,g}} (g\st f)\st \cW_{g} \ar{\rho^{g,g\st f}} \cW_{g\c g\st f} \ar{\cW_{\sigma_{f,g}}} \cW_{f\c f\st g} \ar{\phi^{f,f\st g}} (f\st g)\st \cW_{f} \ar{\pi\st_{f} (f\st g)} \cW_f,$$

where $\lambda_{f,g}$ is the isomorphism induced by the universal property of $(g\st f)\st \cW_g$ using the 2-cell of iso-comma object of $f$ and $g\c \pi_g$ and $\rho^{g,g\st f}$ is the pseudoinverse of $\psi^{g,g\st f}$.

Given a morphism $(\id{},\alpha)\: (\cZ',g) \to (\cZ', g')$ in $\Groth{R}$, the structure 2-cell $\Sigma_{(\id{},\alpha)}$ is given by the pasting diagram

f_g [dd,"f_α"'] [r,"λ_f,g"] & (gf)_g [dd,"δ_α"] [r,"ρ^g,gf"] & _gģf [r,"_σ_f,g"]  & _ff̧g [r,"ϕ^f,fg"]  & (fg)_f [rd,"π_f (fg)"] [dd,"π_f(fα)"] [llldd,"ζ",twoiso ,shorten <=20.5ex, shorten >= 20.5ex]&

&  &  &  &  & _f

f_g' [r,"λ_f,g'"'] & ((g')f)_g' [r,"ρ^g',(g')f"'] & _g'(̧g')f [r,"_σ_f,g'"']  & _ff̧g' [r,"ϕ^f,fg'"']  & (fg')_f [ru,"π_f (fg')"']&

where the morphism $\delta_{\alpha}$ is induced by the universal property of the iso-comma object $((g')\st f)\st W_{g'}$ using the 2-cell

(gf)_g [r,""] [d,""]& _g [ld,"λ^π_g,gf"',Rightarrow,shorten <=4.5ex, shorten >= 4.5ex, shift left=-1ex] [r,"_α"] [rd,"π_g"'pos=0.4,""name=D]& _g' [d,"π_g'"]  [from=D,"",iso]

g' [r,"fα"'] [rr,"gf",bend left=24,""name=H] & g'' [r,""]  & '

and the 2-cell $\zeta$ is given by the following pasting diagram

(gf)_g [rd,""] [dddd,"δ_α"'] [rr,"ρ^g,gf"] &[-7ex]  &[-2ex] _gģf  [d,""] [r,"_σ_f,g"] &[-0.5ex] _ff̧g [d,"",aiso] [rr,"ϕ^f,fg"] &[-4ex]  &[-4ex] (fg)_f [dddd,"π_f(fα)"]

& ((g')f)_g [ru,"",iso] [r,"ρ^g',(g')f f̧α"] [dd,"",aiso]& _g'(̧gf f̧α) [ru,"",iso][r,""] [d,"",aiso] & _ff̧g' Af̧α [dd,"ϕ^ff̧g,fα"] [rru,"β^fg',fα_f",iso]& (fα)(fg')_f) [ru,"",aeq] [rddd,""] &

&  & _(g'(̧g')f)f̧α [d,"ϕ^g'(̧g')f,fα",bend left=20,""name=A] &  &  &

& (fα)((g')f)_g [ruu,"β^(g')f,fα_g'"inner sep=0.3ex,iso] [r,"(fα)ρ^g',(g')f"'inner sep=1ex] [ld,""]& (fα)_g (̧g')f [r,"(fα)_σ_f,g'"'inner sep=1ex] [d,""] [u,"ρ^g'(̧g')f,fα",bend left=20,""name=B] [from=B,to=A,"",iso] [ruu,"",iso] & (fα)_f(̧f')g [d,""] [ruu,"(fα)ϕ^f,fg'"'] &  &

((g')f)_g' [ruuuu,"",iso] [rr,"ρ^g',g'f"] &[-1ex]  &[-1ex] _(g')(̧g')f [r,"_σ_f,g'"] &[-0.5ex] _ff̧g' [rr,"ϕ^f,fg'"] &[-3ex]  &[-3ex] (fg')_f

Given a morphism $(h, \id{})\: (\cZ'',\wt{g\c h}) \to (\cZ', g)$ in $\Groth{R}$, the structure 2-cell $\Sigma_{(h,\id{})}$ is given by the pasting diagram

f_gḩ [d,"fϕ^g,h"'] [r,"λ_f,gḩ"] &[-2ex] ((gḩ)f)_gḩ [dd,"ι_g,h"] [r,"ρ^gḩ,(gḩ)f"] &[-2ex] _(gḩ)(̧( gḩ)f) [r,"_σ_f,gḩ"]  &[-2ex] _ff̧( gḩ) [r,"ϕ^f,f(gḩ)"]  &[-2ex] (f(gḩ))_f [llldd,"ξ",twoiso ,shorten <=22.5ex, shorten >= 22.5ex] [rd,"π_f(f(gḩ))"] [dd,"l_g,h"]&[-2ex]

fh_g [d,"fπ_g h"'] &  &  &  &  & _f

f_g [r,"λ_f,g"'] & (gf)_g [r,"ρ^g,gf"'] & _gģf [r,"_σ_f,g"']  & _ff̧g [r,"ϕ^f,fg"']  & (fg)_f [ru,"π_f (fg)"']&

where the morphism $\iota_{g,h}$ is induced by the universal property of the iso-comma object $(g\st f)\st W_g$ using the 2-cell

((gḩ)f)_gḩ [r,""]  [dd,""]& _gḩ [ldd,"λ^gπ̧_g,f",Rightarrow ,shorten <=9.5ex, shorten >= 9.5ex][rd,"",iso,xshift=-5ex, shift left=2ex] [r,"ϕ^g,h"] [d,"π_gḩ"'] & h_g [ld,"hπ_g"inner sep=0.1ex,""name=N] [d,"π_g"]

& ” [r,"h"] [rd,"gḩ"' ,""name=G] & ' [to=G,"σ^-1_g,h",Rightarrow] [d,"g"]

[rr,"f"'] &  &

the morphism $l_{g,h}$ is induced by the universal property of the iso-comma object $(f\st g)\st W_f$ using the 2-cell

(f(gḩ))W_f  [r,""] [d,""]& _f [dd,"π_f"]  [ld,"λ^π_f,f(gḩ)"inner sep=0.1ex,Rightarrow,shorten <=5.5ex, shorten >= 5.5ex]

f”  [rd,"f(gḩ)",""name=D] [d,"fh"']&

f' [from=D,"fσ_g,h",Rightarrow ,shorten <=1.5ex, shorten >= 1.5ex][r,"fg"'] &

the commutativity of the diagram on the left can be checked on components and the 2-cell $\xi$ is obtained as a pasting of appropriate weak cocycle condition 2-cells and cells given by the assignment of the descent datum in a way similar to the one used to construct $\zeta$ in $\Sigma_{(\id{},\alpha)}$.
One can then prove that these data give a sigma-bicocone $\Sigma$. And we can thus induce the desired morphism $\theta^f$. Furthermore, since the morphism

$$f\st \cW_g \ar{\lambda_{f,g}} (g\st f)\st \cW_{g} \ar{\rho^{g,g\st f}} \cW_{g\c g\st f} \ar{\cW_{\sigma_{f,g}}} \cW_{f\c f\st g} \ar{\phi^{f,f\st g}} (f\st g)\st \cW_{f}$$

is an equivalence, it follows that $\theta^f$ is an equivalence as well. And so we have induced the desired equivalence $\psi^f\: \cW_f \aequi f\st \cW$ (pseudoinverse of $\theta^{f}$).

We now need to prove that, given $\cZ \ar{f} \cY\in S$ and a morphism $\cZ' \ar{g} \cZ$, there exists an invertible 2-cell

_fģ [rr,"ϕ^f,g"] [d,"ψ^fģ"']&  & g_f [d,"gψ^f"] [lld,"(ε_g)_f",Rightarrow ,shorten <=10.5ex, shorten >= 12.5ex]

(fģ) [r,"",aiso]& (fģ)[r,"",aeq] & gf

We equivalently prove that there exists an invertible 2-cell

gf [r,"gθ^f"] [d,"",aeq]& g_f [r,"ρ^f,g"]  & _fģ [dd,"",equal] [lldd,"(ι_g)_f",Rightarrow ,shorten <=8.5ex, shorten >= 8.5ex]

(fģ) [d,"",aiso] &  &

(fģ) [rr,"θ^fģ"'] &  & _fģ

We notice that

$$g\st f\st \cW =g\st f\st \sigmabicolim{(\opn{dom}\c\Lambda)}=\sigmabicolim{(g\st \c f\st \c\opn{dom}\c\Lambda)}$$

and so we can induce the 2-cell $(\iota_g)_f$ via the universal property of the sigma-bicolimit. Given $(\cT, \cT \ar{t} \cY)\in \Groth{R}$, we define a 2-cell $\Gamma_t$ as the following pasting diagram

gf_t [rdddd,"",aeq][rrr,"gfσ_t"] [ddd,"gfσ_t"] [rrd,"gλ_f,t"] &[-3ex] &[-3ex]  &[-3ex] gf [rrr,"gθ^f"]&[-2ex]  &[-2ex]  &[-2ex] g_f [ddddddddd,"ρ^f,g"]

&  & g(tf)_t [r,"gρ^t,tf"] [d,"",aiso] & g_tţf [r,"",iso, shift left =6ex] [r,"g_σ_f,t"] [d,"",aiso]& g_ff̧t [r,"gϕ^f,ft"] [d,"",aiso] & g(ft)_f [ru,"g(π_f (ft))"] [d,"",aiso] &

&  & m(tf)_t [r,"β^tf,m_t",iso, shift left=-12ex] [r,"mρ^t,tf"] [dd,"",aiso] & m_tţf [r,""], [r,"",iso, shift left=-4ex] [d,"ρ^tţf,m"]& m_ff̧t [r,"β^ft,m_f",iso, shift left=-12ex] [r,"mϕ^f,ft"] [d,"ρ^ff̧t,m"', bend right=15,""name=A] & m(ft)_f [dd,"",aiso] &

gf [d,"",aiso] &  &  & _tţfm̧ [r,"",iso, shift left=-6ex] [r,""] [d,"",aiso] & _ff̧tm̧ [d,"",aiso] [u,"ϕ^ff̧t,m"', bend right=15,""name=B] [from= B,to =A,"",iso]&  &

(fģ)  [ddddd,"",aiso]  &(fģ)_t  [l,"(fģ) σ_t"] [dddd,"",aiso] [r,"λ_fģ,t"] & (t(fģ))_t [r,"",iso, shift left=-22ex] [dddd,""] [r,"ρ^t,tf m̧"]  & _tţ(fģ) [r,"",iso, shift left=-22ex][dddd,""] [r,""]& _ff̧t m̧ [r,"",iso, shift left=-6ex] [r,"ϕ^f,ft m̧"] [d,""] & (ft m̧)_f [d,""]&

&  &  &  & _fģ (̧fģ) t [r,"β^g, (fģ) t_f",iso, shift left=-10ex] [r,"ϕ^f,g(̧fģ) t"] [dd,"",aiso] & (g(̧fģ)t) _f [d,"",aiso] &

&  &  &  &  & ((fģ) t)(g_f)  [d,"((fģ) t)ρ^f,g"pos=0.4, bend left=20 ,""name=F] &

&  &  &  & _fģ(̧fģ) t [r,"",iso, shift left=-6ex] [r,"ϕ^fģ,(fģ) t"] [d,""] & ((fģ) t)_fģ [d,""] [u,"((fģ) t)ϕ^f,g"pos=0.6, bend left=20 ,""name=G] [from= G,to =F,"",iso]&

& (fģ)_t [r,"λ_fģ,t"] [ld,"(fģ)σ_t"] & (t(fģ))_t [r,"ρ^t,t(fģ)"]& _tţ(fģ) [r,"",iso, shift left=-6ex] [r,"_fģ,t"]& _(fģ)(̧fģ)t [r,"ϕ^fģ,(fģ)t"] & ((fģ)t)_fģ [rd,""] &

(fģ) [rrrrrr,"θ^fģ"'] &  &  &  &  &  & _fģ

where $\varepsilon$ is the canonical equivalence between $(f\c g)\st \cT$ and $g\st f\st \cT$ and $m=(f\st t)\st g \c \varepsilon$. Using the compatibility conditions of the weak descent datum, one can then prove that the $\Gamma_t$'s form a modification and so the 2-cell $(\iota_g)_f$ is induced by the universal property of the sigma-bicolimit.

We now need to prove that, given a 2-cell $\gamma\: f\Rightarrow f'$ with $f,f'\: \cZ\to \cY$ in $R(\cZ)$  there exists an isomorphic 2-cell

_f [r,"ψ^f"] [d,"η_γ"']& f [d,"(γ) _"] [ld,"ψ_γ",twoiso, shorten <= 2ex, shorten >= 2ex]

_f' [r,"ψ^f'"'] & f'

We equivalently induce an isomorphic 2-cell

f [r,"θ^f"] [d,"(γ)_"']& _f [d,"η_γ"] [ld,"κ_γ",twoiso ,shorten <=2.5ex, shorten >= 2.5ex]

f'  [r,"θ^f'"']& _f'

We notice that

$$f\st \cW=f\st \sigmabicolim{\opn{dom}\c \Lambda}= \sigmabicolim{f\st \c \opn{dom}\c \Lambda}$$

and so we can induce the desired 2-cell $\kappa_{\gamma}$ via the universal property of the sigma-bicolimit using a modification $\Delta$. Given $(\cT, \cT \ar{t} \cY)\in \Groth{R}$, we define the 2-cell

f_t  [r,"fσ_t"] [d,"fσ_t"']& f [r,"θ^f"] & _f [d,"η_γ"] [lld,"Δ_t",twoiso ,shorten <=5.5ex, shorten >= 5.5ex]

f [r,"(γ)_"'] & f' [r,"θ^f'"']& _f'

as a pasting diagram analogous to the one used to define $\Gamma_t$. And it is easy to see that we obtain a modification $\Delta$ that induces the desired 2-cell $\kappa_{\gamma}$.

Finally, it is straightforward to verify that the defined data satisfy the axioms required by the definition of weakly effective weak descent datum. This concludes the proof of the condition (O).


# Quantum Stoner-Wohlfarth model of two-dimensional single domain magnets

Essa M. Ibrahim  Shufeng Zhang  Department of Physics, University of Arizona,
1117 E 4th Street, Tucson, AZ 85721

###### Abstract

The Stoner-Wohlfarth (SW) model is a classical model for magnetic hysteresis
of single domain particles. For two dimensional magnets at finite temperature,
the SW model must be extended to include intrinsic strong spin fluctuations.
We predict several fundamentally different hysteresis properties between 2D
and 3D magnets. The magnetization switching diagram known as the Asteroid
figure in the conventional SW model becomes highly temperature dependent and
asymmetric with respect to the transverse and longitudinal magnetic fields.
Our results provide new insights for 2D magnetic materials based spintronics
applications.

††preprint: APS/123-QED

##  I Introduction

In the last several years, many 2D magnetic materials with novel magnetic and
spin transport phenomena have been discovered [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17]. These new classes of 2D magnetic materials
generate an interesting perspective for their possible applications in
spintronics. To elucidate the fundamental differences between 2D and 3D
magnets in response to an external magnetic field, we start with a single
domain magnet in which the magnetization is spatially uniform across the
sample. The single domain magnet is usually a building block for magnetic
memory devices in which the direction of the magnetization can be well
controlled by either the magnetic field or the electric currents. The most
elementary magnetic property of a single domain is its very simple magnetic
hysteresis described by the classical Stoner-Wohlfarth (SW) model [18, 19],
whose magnetic energy is

$E_{sw}=-K({\bf\hat{z}\cdot M})^{2}-{\bf H}\cdot{\bf M}$ (1)

where ${\bf M}$ is the magnetization vector, $\hat{\bf z}$ is the anisotropy
axis with anisotropy energy $K$, and ${\bf H}$ is the applied magnetic field.
The above simple SW model immediately gives rise to the well-known hysteresis
loops for the different directions of the applied magnetic field, as shown in
Fig. (1) (a-d).

In this paper, we study the magnetic hysteresis of two-dimensional single-
domain magnetic particles with uniaxial anisotropy, i.e., the 2D Stoner-
Wohlfarth (SW) model. Why does the above successful SW model for the
conventional 3D magnet fail for 2D magnets? In 3D, the magnitude of the
magnetization $M_{s}(T)=|{\bf M}|$ is controlled by the exchange interaction
between the neighboring spins and thus it weakly depends on the magnetic field
or the magnetic anisotropy for temperatures sufficiently less than the Curie
temperature. Since the hysteresis is measured with a constant temperature,
$M_{s}$ does not change for the entire range of the field in the hysteresis.
In 2D, however, the magnitude of the magnetization depends on both the
exchange interaction and the total effective field ${\bf H}_{eff}$ (the sum of
the anisotropy and the applied field). If the total effective field is zero,
the magnitude of the magnetization would be zero; this is known as the Wigner
and Mermin theorem [20]. The dependence of the magnetization on the total
effective field is due to fundamentally strong spin fluctuation in low
dimensions in which the number of low-energy excitations (long-wavelength
magnons) diverges, i.e., the long-range ordering disappears. In the hysteresis
loop, when the magnetic field is reversed to the opposite direction of the
magnetization, the total effective field becomes small and thus the
magnetization reduces. To describe the variation of both magnitude and
direction of the magnetization with the applied field, we use the self-
consistent spin-wave method, which is equivalent to the random phase
approximation [21], to model the magnitude of the magnetization. Let us first
show the 2D hysteresis along with the above 3D hysteresis in Fig. (1) (e-h)
followed by our detailed theory and calculation in the next Section.

Figure 1: Magnetic hysteresis and switching Asteroid for the 3D (four left
panels a-d) and 2D (four right panels e-h) Stoner-Wohlfarth model (where
$m\equiv M/M_{s}$ , $h_{3D}\equiv H/2K$ and $h_{2D}\equiv H/2zAM_{s}$). $(a)$,
$(b)$, and $(c)$, The hysteresis loops of a 3D magnet with the angles between
the magnetic fields and the anisotropy axis at $0^{\circ}$, $45^{\circ}$, and
$90^{\circ}$. $(d)$ The 3D Asteroid figure for magnetization switching fields.
$(e)$, $(f)$, and $(g)$, are the hysteresis loops of the 2D magnet with the
angles between the magnetic fields and the anisotropy axis at $0^{\circ}$,
$45^{\circ}$, and $90^{\circ}$ at $T=0.6T_{c}$. $(h)$ The 2D Asteroid figure
at $T=0.6T_{c}$.

##  II Model

The quantum version of the 2D SW model is

${\hat{\cal H}}=-J\sum_{<i,j>}{\hat{\bf S}}_{i}\cdot{\hat{\bf
S}}_{j}-A\sum_{<i,j>}\hat{S}_{i}^{z}\hat{S}_{j}^{z}-\sum_{i}{\bf
H}\cdot\hat{\bf S}_{i}$ (2)

where ${\hat{\bf S}}_{i}$ and $\hat{S}_{i}^{z}$ respectively are the spin and
the $z$-component (taken as perpendicular to the two-dimensional plane) of the
spin operators at lattice site ${\bf R}_{i}$, $J$ is the isotropic exchange
integral, $A$ is the anisotropic exchange integral (it is worth mentioning
here that the anisotropy energy in the 3D classical limit $K$ is equivalent to
$zAM_{s}$ in the quantum model, where z is the number of the nearest-neighbour
sites), $<ij>$ indicates the sum over nearest neighbors, and ${\bf H}$ is the
external field. To determine the magnetization, we have developed a random
phase approximation (RPA) in which the transverse spin fluctuation and the
longitudinal spin fluctuation are decoupled, and we have arrived at the self-
consistent equation for the magnetization [21],

$\displaystyle M=M_{s}-\int\frac{d^{2}k}{(2\pi)^{2}}\frac{2M}{e^{\beta
E_{k}}-1}$ (3)

where $M_{s}$ is the magnetization at zero temperature, and $E_{k}$ is the
magnon energy; in the long wave length limit, $E_{k}=zM(Jk^{2}/2+2A)+H$
(assuming the field is along the direction of the anisotropy field). Eq. (3)
has a straightforward explanation: the magnetization is subtracted by the
number of the magnons which are softened by the factor of $M$ at the finite
temperature. We note that a) Eq. (3) is the RPA approximation for spin-1/2;
the higher spins would lead to a more complicated equation; b) the RPA is
considered an excellent approximation for temperature sufficiently lower than
the Curie temperature, and c) we consider the magnetic anisotropy from the
anisotropic exchange rather than on-site anisotropy in the form of
$-A(S_{i}^{z})^{2}$. By using the quadratic dispersion in the energy, we may
integrate out $d^{2}k$, resulting to a simple analytical expression,

$M=M_{s}-\frac{1}{\pi
zJ}\left(\frac{1}{\beta}\ln\left|\frac{e^{\beta(\Delta+W)}-1}{e^{\beta\Delta}-1}\right|-W\right)$
(4)

where $\Delta=2zAM+H$ and $W=2\pi zJM$ are the effective magnon gap and the
magnon bandwidth, respectively. Clearly, the magnetization depends on the
magnetic field even for temperature significantly lower than the Curie
temperature.

##  III External field in the direction of the anisotropy

Figure 2: The solutions of Eq. (4) for three different magnetic fields. For
the field less than $|H|<H_{c}$ the equation has two solutions $m_{1}$ and
$m_{2}$. At $H=-H_{c}$ (the coercive field) the equation has one solution
which is the critical magnetization at which the magnetization reversal
occurs.  Figure 3: $(a)$ The hysteresis of a single domain 2D magnet at
different temperatures ($2zAM_{s}=1$). $(b)$ The temperature dependence of the
critical magnetization. $(c)$ The temperature dependence of the coercive
field.

The numerical solution of $M$ for the field in the direction of the anisotropy
field is readily solved from Eq. (4). A simple way to obtain a solution for
given parameters (temperature, field, and anisotropy) is to plot two functions
$y=M$ and $y=f(M)$ where $f(M)$ is the right side of Eq. (4). Note that the
function $f(M)$ is only physically meaningful when the magnon energy gap is
positive, i.e., $\Delta>0$. The negative gap is unstable such that the
magnetization reversal takes place. In Fig. (2), we show $f(M)$ for three
different magnetic fields: for a positive or small negative field, $M=f(M)$
has two solutions, representing an energy minimum (the solution with a larger
$M$) and an energy maximum. At a critical negative magnetic field, there is
only one solution, which is also known as the coercive field. Beyond the
critical field, there is no solution for $M>0$, indicating magnetization
reversal occurs.

Compared with the conventional SW model, the hysteresis shown in Fig. (1e) is
no longer square. The reduction of the magnetization near the critical value
of the field is caused by the reduced effective gap and thus the increased
number of magnons. Since the magnon population depends on the temperature, the
magnetization at the critical magnetic field decreases significantly at higher
temperatures as shown in Fig. (3). This contrasts with the 3D magnet, which is
essentially independent of temperature.

Figure 4: The temperature dependence of the Asteroid diagram of 2D single
domain magnet ($2zAM_{s}=1$).

##  IV External field at an arbitrary direction

We now consider the hysteresis loop with the field in an arbitrary direction,
${\bf H}=H_{0}(\hat{\bf z}\cos\theta+\hat{\bf x}\sin\theta)$ where $\theta$ is
the angle between the applied field and the ${\bf z}$-axis . The total
effective field is the sum of the external and anisotropic field, ${\bf
H}_{t}={\bf H}+2zA({\bf M}\cdot{\hat{\bf z}})\hat{\bf z}$ where the direction
and the magntitude of ${\bf M}$ need to be self-consistently determined. At
equilibrium, the magnetization ${\bf M}$ is always parallel to $H_{t}$, i.e.,

$\frac{M_{x}}{M_{z}}=\frac{H_{0}\sin\theta}{H_{0}\cos\theta+2zAM_{z}}$ (5)

Eq. (4) remains valid as long as the magnon gap is replaced by the total
magnetic field in the direction of the magnetization, i.e., $\Delta={\bf
H}_{t}\cdot{\bf M}/M$. Thus, Equations (4) and (5) determine the magnetization
for arbitrary direction of the magnetic field. As an example, we show in Fig.
(1f) the hysteresis for the field direction at $\theta=45^{\circ}$. Comparing
3D, Fig. (1b), and 2D, Fig. (1f), the coercivity is smaller while the
magnetization jumps at a positive value. For $\theta=90^{\circ}$, i.e., the
hard axis loops, both 2D and 3D hysteresis are single valued. However, the 2D
SW model has a non-zero slope even above the anisotropy field while the 3D SW
would be completely saturated above the anisotropy field.

Finally, we construct the critical values of the magnetic field for the
magnetization reversal for all directions of the magnetic field, known as the
Asteroid figure. When the magnetic field increases across the Asteroid line,
the reversal occurs. In the classical SW model, the Asteroid line can be
readily derived from Eq. (1) and the analytic expression of the Asteroid is
$H_{z}^{2/3}+H_{x}^{2/3}=(2K)^{2/3}$. In 2D SW model, the Asteroid is highly
temperature dependent as shown in Fig. (4). At low temperature, the Asteroid
figure resembles that of the 3D SW model. At higher temperature, the
magnetization reversal for the longitudinal field (parallel to the anisotropy
field) is more effective than for the transverse field; this is because for
the same magnitude of the field, the longitudinal direction reduces the magnon
gap more than the transverse direction, leading to the asymmetry of the
Asteroid figure in the direction of the applied field.

In summary, we have formulated the magnetic hysteresis loops of 2D single
domain magnets. Compared to 3D Stoner-Wohlfarth single domain model, the 2D
magnetic hysteresis is more complicated due to the fundamentally stronger spin
fluctuations. We have used the equilibrium magnetization formulation which has
been derived previously by the random phase approximation. As long as the
temperature is not too close to the Curie temperature, the RPA provides an
excellent approximation.

This work was partially supported by the U.S. National Science Foundation
under Grant No. ECCS-2011331.

## References

  * [1] B. Huang, et al, (2017) Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit Nature 546, 270.
  * [2] Gong, C. et al., (2017) Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature 546, 265–269.
  * [3] D. J. O’Hara, et al., (2018) Room Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer Limit, Nano Lett. 18, 3125.
  * [4] D. R. Klein, et al., (2018) Probing magnetism in 2D van der Waals crystalline insulators via electron tunneling, Science 360, 1218.
  * [5] S. Jiang, et al., (2018) Controlling magnetism in 2D $CrI_{3}$ by electrostatic doping, Nature Nanotechnol. 13, 549.
  * [6] J. U. Lee, et al., (2016) Ising-Type Magnetic Ordering in Atomically Thin $FePS_{3}$, Nano Lett. 16, 7433–7438.
  * [7] B. Huang et al., (2020) Emergent phenomena and proximity effects in two-dimensional magnets and heterostructures, Nat. Mater. 19, 1276.
  * [8] C. Gong and X. Zhang, (2019) Two-dimensional magnetic crystals and emergent heterostructure devices, Science 363, 4450.
  * [9] Song, T. et al., (2018) Giant tunneling magnetoresistance in spin-filter van der Waals heterostructures, Science 360, 1214–1218.
  * [10] V. Gupta et al., (2020) Manipulation of the van der Waals Magnet $Cr_{2}Ge_{2}Te_{6}$ by Spin–Orbit Torques, Nano Lett. 20, 7482.
  * [11] D. MacNeill, et al., (2017) Control of spin–orbit torques through crystal symmetry in $WTe_{2}$/ferromagnet bilayers, Nature Phys. 13, 300.
  * [12] M. Alghamdi, et al., (2019) Highly Efficient Spin–Orbit Torque and Switching of Layered Ferromagnet $Fe_{3}GeTe_{2}$, Nano Lett. 10, 4400.
  * [13] X. Wang, et al., (2019) Current-driven magnetization switching in a van der Waals ferromagnet $Fe_{3}GeTe_{2}$, Science Adv. 5, 8904.
  * [14] C. Fang, et al., (2019) Observation of large anomalous Nernst effect in 2D layered materials $Fe_{3}GeTe_{2}$, Appl. Phys. Lett. 115, 212402.
  * [15] T. Liu, et al., (2020) Spin caloritronics in a $CrBr_{3}$ -based magnetic van der Waals heterostructure, Phys. Rev. B 101, 205407.
  * [16] N. Ito, et al., (2019) Spin Seebeck effect in the layered ferromagnetic insulators $CrSiTe_{3}$ and $CrGeTe_{3}$, Phys. Rev. B. 100, 060402(R).
  * [17] J. Xu, et al., (2019) Large Anomalous Nernst Effect in a van der Waals Ferromagnet $Fe_{3}GeTe_{2}$, Nano Lett. 19, 8250.
  * [18] Stoner E. C. and Wohlfarth E. P. (1948) A mechanism of magnetic hysteresis in heterogeneous alloys, Philos. Trans. Royal Soc. A 240 (826): 599–642.
  * [19] C. Tannous and J. Gieraltowski (2008) The Stoner–Wohlfarth model of ferromagnetism, Eur. J. Phys. 29 475
  * [20] N. D. Mermin and H. Wagner, (1966) Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett. 17, 1133.
  * [21] P. Tang, X. F. Han, and S. Zhang, (2021) Quantum theory of spin-torque driven magnetization switching, Phys. Rev. B 103, 094442.