Proof The proof is straightforward. The first square is given by the $2$ -dimensional part of the pseudo-naturality of applied to the $2$ -cell , whereas the second square is given by the naturality in z of the isomorphisms of Equation (3.1).

Proof Let be a $2$ -diagram which factors through $\mathcal {Y} $ and has a weighted $2$ -colimit F in , with weight . Call the universal cocylinder of such colimit.

In order to prove that is faithful, take two arbitrary $2$ -cells in

such that . We prove that .

As with universal cocylinder $\Lambda $ , it suffices to show that the two modifications

and are equal, as we then conclude by the ( $2$ -dimensional) universal property of the weighted $2$ -colimit F.

It then suffices to prove that, given arbitrary $i\in \mathcal {I} $ and $X\in W(i)$ ,

But as $K(i)\in \mathcal {Y} $ and is faithful by assumption, it then suffices to show that

(3.2)

By Proposition 3.1, we can write as the composite

and analogously for . Since such composites for and for are equal, we conclude that equation (3.2) holds.

Proof (Definition of the assignment) Given $z{\colon }F\to B$ in , we define $p^{\ast } z$ as the upper morphism of the chosen pullback square in on the left below.

Given then a morphism in as in the middle below, we can lift the $2$ -cell in on the right below

along the discrete opfibration p, producing the unique $2$ -cell such that and . By the universal property of the pullback $\operatorname {G}_{}^{z'}$ of p and $z'$ , we factorize v through $p^{\ast } z'$ , obtaining a morphism . We define to be the upper triangle in the following commutative solid:

(3.3)

Given a $2$ -cell in , we define $p^{\ast } \delta $ to be the unique lifting of the $2$ -cell along the Grothendieck opfibration .

Construction 3.6 Let be a fully faithful dense generator of , and let . By Theorem 2.26, there exist a $2$ -diagram which factors through $\mathcal {Y} $ and a weight such that

in and this colimit is I-absolute. By Theorem 2.40, the $2$ -diagram factors through $\mathcal {Y} $ and is such that

Moreover this colimit is still I-absolute by Proposition 2.41. Call

the universal cartesian-marked oplax cocone that presents such colimit.

Consider now a discrete opfibration $p{\colon }E\to B$ in , a morphism $z{\colon }F\to B$ and the chosen pullback in

We want to exhibit $\operatorname {G}_{}^{z}$ as a cartesian-marked oplax conical colimit of a diagram constructed from K and $\Lambda $ .

By [Reference Mesiti11], we can construct from K and $\Lambda $ a $2$ -diagram and a universal cartesian-marked oplax cocone $\Lambda ^{\prime }_z=\Lambda '$ which exhibits

in the lax slice . Explicitly, $K'$ is the 2-diagram that corresponds to the cartesian-marked oplax cocone

associated with z. That is,

Considering the canonical -category structure on , with the tight part given by the morphisms of type $(f,\operatorname {id}_{})$ , we have that $K'$ is an -diagram.

The universal cartesian-marked oplax cocone $\Lambda '$ has component on $(C,X)$ given by the identity filled triangle (which is thus a tight morphism in )

Then $\Lambda ^{\prime }_{f,\nu }=\Lambda _{f,\nu }$ for every morphism $(f,\nu )$ in .

By [Reference Mesiti11], the $2$ -functor

preserves the colimit , since $K'$ is an -diagram, $\Lambda '$ has tight components and the domain of such colimit is , which is I-absolute. Then again by [Reference Mesiti11] the domain $2$ -functor preserves the latter colimit $p^{\ast }(z)$ , since $p^{\ast }$ is an -functor. We obtain that $\operatorname {dom}\circ {p}^{\ast }\circ \Lambda '$ is a universal cartesian-marked oplax cocone that presents

Explicitly, given in , we have that $\operatorname {dom}\big (\mkern 1mu{p^{\ast }\big (\mkern 1mu{\Lambda ^{\prime }_{(C,X)}}\mkern 1mu\big )}\mkern 1mu\big )$ is the unique morphism into the pullback $\operatorname {G}_{}^{z}$ induced by

Then $\operatorname {dom}\big (\mkern 1mu{{p}^{\ast }\big (\mkern 1mu{\Lambda ^{\prime }_{f,\nu }}\mkern 1mu\big )}\mkern 1mu\big )$ is the unique lifting along the discrete opfibration of the $2$ -cell to $\operatorname {dom}\big (\mkern 1mu{p^{\ast }\big (\mkern 1mu{\Lambda ^{\prime }_{(C,X)}}\mkern 1mu\big )}\mkern 1mu\big )$ .

Notice that in particular this construction can be applied to $z=\operatorname {id}_{F}$ , exhibiting the domain of any discrete opfibration over F as a cartesian-marked oplax conical colimit (starting from K and $\Lambda $ ). Remember that we chose pullbacks in such that the change of base of an identity is always an identity. Notice also that, given $z{\colon }F\to B$ ,

$$ \begin{align*}\lambda^z=(z\circ -)\circ \lambda^{\operatorname{id}_{}}\end{align*} $$

Whence we can express the -functor $K^{\prime }_z$ constructed from z in terms of the one constructed from $\operatorname {id}_{}$ :

$$ \begin{align*}K^{\prime}_z=(z\circ -)\circ K^{\prime}_{\operatorname{id}_{}}\end{align*} $$

Then the components and the structure 2-cells of $\Lambda ^{\prime }_z$ and $\Lambda ^{\prime }_{\operatorname {id}_{}}$ have the same domains, which determine them.

Proof Let ; we prove that is full. Consider then two morphisms $z,z'{\colon } F\to \Omega $ in and a morphism in . We search for a $2$ -cell such that . The idea is to write F as a colimit and define by giving its “components”, which we can produce by the fullness of the functors with the Y’s in $\mathcal {Y} $ that generate F.

By Theorem 2.26, Theorem 2.40 and Proposition 2.41, there exists a $2$ -diagram which factors through $\mathcal {Y} $ and a universal cartesian-marked oplax cocone

that exhibits F as the I-absolute cartesian-marked oplax conical colimit of K.

Then, in order to produce , it suffices to give a modification

Given , we will have that

So, we will need to have that

and, by Proposition 3.1 and the request that , the right hand side of such equation is equal to the composite

Call $h^{C,X}$ such composite morphism in . Since $K(C,X)\in \mathcal {Y} $ , the functor is fully faithful by assumption, and hence there exists a unique $2$ -cell

in such that . We define the component $\theta _{(C,X)}$ of $\theta $ on $(C,X)$ to be such $2$ -cell $\gamma ^{C,X}$ . The faithfulness of the functors guarantees that $\theta $ becomes a modification. Indeed, given a morphism $(f,\nu ){\colon }(D,X')\to (C,X)$ in , we need to prove that

But since is faithful, it suffices to prove that

and hence that

At this point, it is straightforward to see that such equality is given by the naturality square of (see Proposition 2.11) obtained considering the morphism in . For this, one needs the pseudo-naturality of (Proposition 2.12), the equation and the analogous one for $(D,X')$ , together with both the squares of Proposition 3.1. The square on the right of Proposition 3.1 allows us to calculate, whereas the square on the left allows us to compare the components of on and with and the analogous one with $z'$ instead of z.

Thus we conclude that $\theta $ is a modification, and by the universal property of the colimit F we obtain an induced $2$ -cell in . It remains to prove that .

The idea is to conclude such equality by using the uniqueness part of the universal property of a colimit. As and h are morphisms in , both are totally determined by a morphism in from $\operatorname {G}_{}^{z}$ to $\operatorname {G}_{}^{z'}$ ; we call respectively and h such morphisms in . By Construction 3.6, applied to $z=\operatorname {id}_{F}$ ,

presented by the universal cartesian-marked oplax cocone , with $\Lambda '$ and $K'$ constructed from $\Lambda $ and K as in Construction 3.6.

Then, in order to conclude that and h are equal, it suffices to show that

(3.4)

as cartesian-marked oplax natural transformations. And the last part of Construction 3.6 gives us an explicit calculation of in terms of $\Lambda $ . Precisely, given an arbitrary morphism in ,

and is the unique lifting of the $2$ -cell to along . We now notice that the following two squares are commutative:

Then, to prove that Equation (3.4) holds on component $(C,X)$ , it suffices to show that

as morphisms in . Using Proposition 3.1, we see that is equal to

and thus is equal to ${\Lambda _{(C,X)}}^{\ast }\left (h\right )$ since , by construction of $\theta _{(C,X)}$ .

It only remains to prove that Equation (3.4) holds on morphism component $(f,\nu )$ . But this is straightforward using the uniqueness of the liftings through a discrete opfibration, that Equation (3.4) holds on object components and the fact that both and h are morphisms of discrete opfibrations.

Construction 3.9 Let be a fully faithful dense generator of . Assume that $\tau $ satisfies a satisfies a pullback-stable property $\operatorname {P}$ and that for every $Y\in \mathcal {Y} $

is an equivalence of categories. Let then $\varphi {\colon }G\to F$ be a discrete opfibration in that satisfies the property $\operatorname {P}$ . We would like to construct a characteristic morphism $z{\colon }F\to \Omega $ for $\varphi $ . That is, a z such that is isomorphic to $\varphi $ , so that $\varphi $ gets classified by $\tau $ .

There exist a $2$ -diagram which factors through $\mathcal {Y} $ and a universal cartesian-marked oplax cocone $\Lambda $ that exhibits F as the I-absolute

We would like to induce z via the universal property of the colimit F. The idea is condensed in the following diagram:

For every $(C,X)$ in , the change of base of $\varphi $ along $\Lambda _{C,X}$ satisfies the property P and is thus in the essential image of . We can then consider the oplax natural transformation $\chi $ given by the composite

where every is a right adjoint quasi-inverse of giving an adjoint equivalence. The action of such on morphisms $h{\colon }\psi \to \psi '$ is exhibited by the naturality squares of the counit $\varepsilon $

We are also using that for every morphism in the functor restricts to a functor between the essential image of and the essential image of . Moreover, using Proposition 2.12, we have that extends to a pseudo-natural transformation. Its structure $2$ -cell on any in is given by the pasting

where $\eta $ is the unit of the adjoint equivalence, and we denoted $K(C,X)$ as $K^{C,X}$ .

The composite $\chi $ above is readily seen to be a sigma natural transformation (of Descotte, Dubuc, and Szyld’s [Reference Descotte, Dubuc and Szyld4], w.r.t. the cartesian marking). That is, $\chi $ is an oplax natural transformation with the structure 2-cells $\chi _{f,\operatorname {id}_{}}$ being isomorphisms for every morphism of type $(f,\operatorname {id}_{})$ in . If we were in a bicategorical context, this would then be enough to induce a morphism $z{\colon }F\to \Omega $ , as we will explore in detail in future work. In our strict $2$ -categorical context, we further need to be able to “normalize” such sigma natural transformation, ensuring that the structure 2-cells on the morphisms $(f,\operatorname {id}_{})$ are identities. So that the morphisms yield a cartesian-marked oplax cocone. Essentially, this means that we can choose good quasi-inverses of the ’s. Such an extra hypothesis is satisfied by $2$ -dimensional presheaves (i.e., prestacks), see Theorem 4.14. By Theorem 3.24, it is then satisfied by nice sub-2-categories of prestacks, such as the 2-category of stacks (Theorem 5.11).

Proof Let $\varphi {\colon }G\to F$ be a discrete opfibration in that satisfies the property $\operatorname {P}$ . Consider the associated $\chi $ and as in the statement. Let $z{\colon }F\to \Omega $ be the unique morphism induced by via the universal property of the colimit . We prove that z is a characteristic morphism for $\varphi $ with respect to $\tau $ .

Consider the pullback

We want to prove that there is an isomorphism $j{\colon }G\cong V$ such that . Applying Construction 3.6 to $\varphi $ and $\operatorname {id}_{F}$ , we construct $K'=K^{\prime }_{\operatorname {id}_{}}$ and $\Lambda '=\Lambda ^{\prime }_{\operatorname {id}_{}}$ that exhibits

Applying again Construction 3.6 to $\tau $ and $z{\colon }F\to \Omega $ we obtain

We show that

Notice that $(z\circ -)\circ K'$ is the $2$ -functor associated with the oplax natural transformation , as described in Construction 3.6. Since , we obtain that $(z\circ -)\circ K'\cong U$ where U is the $2$ -functor associated with the sigma natural transformation $\chi $ . Moreover the general component $t_{C,X}$ on of such isomorphism has first component equal to the identity. This produces an isomorphism

whose general component on $(C,X)$ is over $K(C,X)$ . Indeed by Remark 3.5

and is thus an isomorphism that makes the following triangle commute:

In this way, we have handled the isomorphisms given by the normalization process. We now show that

Given , by Remark 3.5 and by construction of $\chi $ we have

So the counits of the adjoint equivalences give isomorphisms

over $K(C,X)$ , where the map to $K(C,X)$ from the right hand side is . We prove that these isomorphisms form a $2$ -natural transformation $\xi $ . Take then $(f,\nu ){\colon }(C,X)\to (D,X')$ in . By Remark 3.5, we can express the action of $\operatorname {dom}\circ \tau ^{\ast }$ on the morphism

in terms of . Analogously, we express $\operatorname {dom}\left (\varphi ^{\ast } \left (K(f,\nu ),\Lambda _{f,\nu }\right )\right )$ in terms of . Consider the composite pullbacks

and call $R^H$ and $R^Q$ řespectively the pullbacks of $\varphi $ and of $\tau $ along the composites. By construction of $\chi $ and by the action of on morphisms, we calculate that is precisely the composite

Notice that we also need one triangular equality of the adjoint equivalence to handle the $\eta ^{-1}$ part of . In order to obtain $\operatorname {dom} \left (\tau ^{\ast }\left (K(f,\nu ),\chi _{f,\nu }\right )\right )$ , by Remark 3.5, we compose with

We conclude the naturality of $\xi $ by definition of . To prove that $\xi $ is $2$ -natural, consider a $2$ -cell in . Both

give the unique lifting of to along . Thus $\xi $ is $2$ -natural. We then obtain a $2$ -natural isomorphism

whose general component on is over $K(C,X)$ .

As a consequence, there is an isomorphism $j{\colon }G\cong V$ , respecting the universal cartesian-marked oplax cocones $\Theta ^G$ and $\Theta ^V$ that exhibit the two as colimits:

We want to show that the following triangle is commutative:

(3.5)

Since G is a cartesian-marked oplax conical colimit, it suffices to show that

Whence it suffices to show that

Given , the two have equal components on $(C,X)$ since by Construction 3.6

using that $\zeta _{(C,X)}$ is over $K(C,X)$ . Given in , also the structure $2$ -cells of the two cartesian-marked oplax natural transformations on $(f,\nu )$ are equal, since by Construction 3.6

Therefore the triangle of Equation (3.5) is commutative and $\varphi $ is in the essential image of .

Proof The proof of is exactly as the proof of Theorem 3.10, using the essential image of in place of .

We prove . By assumption, there exists a characteristic morphism z for $\varphi $ . For every $(C,X)$ , we then have that $z\circ \Lambda _{(C,X)}$ is a characteristic morphism for . It remains to prove that the operation of normalization described in Theorem 3.10 starting from $\varphi $ is possible. We can choose the quasi-inverse of (restricted to its essential image) so that for every $b{\colon }K(C,X)\to F$ in

and the component of the counit on is given by the pseudofunctoriality of the pullback. Then for every morphism in

$$ \begin{align*}\chi_{(C,X)}=z\circ \Lambda_{(C,X)}=z\circ \Lambda_{(D,F(f)(X))}\circ K(f,\operatorname{id}_{})=\chi_{(D,F(f)(X))}\circ K(f,\operatorname{id}_{}).\end{align*} $$

In order to prove that $\chi _{f,\operatorname {id}_{}}=\operatorname {id}_{}$ , it suffices to prove that . It is straightforward to see that this holds, using the recipe described in the proof of Theorem 3.10. Indeed it is just given by the compatibilities of a pullback along a composite of three morphisms with the composite pullbacks.

Proof By Remark 2.16, taking $\tau $ to be the lax limit of the arrow $\omega $ , we have that for every . We conclude by Theorem 3.10.

Notation 3.16 Throughout the rest of this section, we fix $\operatorname {P}$ an arbitrary pullback-stable property $\operatorname {P}$ for discrete opfibrations in . We assume of course that $\operatorname {P}$ only depends on the isomorphism classes of discrete opfibrations.

Proof We prove that, for every , there is an isomorphism

(3.6)

Given , consider the comma objects

in . It is straightforward to see that, since $\ell $ is a fully faithful morphism, exhibits the comma object on the right hand side. Then over $F'$ , and such isomorphism is natural in z by the universal property of the comma object. $\ell \circ -$ is fully faithful by definition of fully faithful morphism in . Hence is fully faithful.

Moreover, we obtain that if $\tau $ satisfies $\operatorname {P}$ then the lax limit $\xi $ of the arrow satisfies $\operatorname {P}$ . Since i lifts comma objects and the terminal objects, the lax limit of the arrow can be calculated in . So satisfies $\operatorname {P}$ as well.

Let now . Since i preserves discrete opfibrations, i induces a fully faithful functor

Notice then that the following square is commutative:

(3.7)

Indeed by assumption i lifts comma objects and the terminal object. Whence is fully faithful and is a 2-classifier.

Consider now a discrete opfibration $\varphi $ in . Following the diagrams of Equations (3.6) and (3.7), we obtain that if $i(\varphi )$ is classified by $\tau $ via a characteristic morphism z that factors through $\ell $ then $\varphi $ is classified by . This can also be seen from the diagram on the right in the statement, using the pullbacks lemma, after Remark 3.22.

Construction 3.23 Let be a fully faithful 2-functor. Consider then a fully faithful 2-functor such that is a dense generator of . By Kelly’s [Reference Kelly8, Theorem 5.13], then I is a fully faithful dense generator of .

Moreover, let . We want to exhibit F as a nice colimit of the objects that form the dense generator I. By Construction 3.6, there exist a 2-diagram which factors through $i\circ I$ and a weight such that, calling ,

and this colimit is $(i\circ I)$ -absolute. Call $\Lambda $ the universal cartesian-marked oplax cocone that presents such colimit. Notice that, as K factors through $i\circ I$ , it also factors through i. Call the resulting diagram; so that . It is clear that factors through I. Take to be the unique cartesian-marked oplax cocone such that . Then, since a fully faithful 2-functor reflects colimits (see also Proposition 2.41),

exhibited by . Moreover, this colimit is I-absolute, as $\widetilde {I}\cong \widetilde {(i\circ I)}\circ i$ .

Proof By Construction 3.23, is a fully faithful dense generator of . By Proposition 3.21, the lax limit of the arrow satisfies $\operatorname {P}$ and for every $Y\in \mathcal {Y} $

is an equivalence of categories. Indeed is fully faithful and, given $\psi {\colon }H\to I(Y)$ a discrete opfibration in that satisfies $\operatorname {P}$ , any characteristic morphism of $i(\psi )$ in factors through . In order to prove that is a good 2-classifier in , by Corollary 3.13, it only remains to prove that the operation of normalization described in the proof of Theorem 3.10 is possible.

So let $\varphi {\colon }G\to F$ be a discrete opfibration in that satisfies $\operatorname {P}$ . By Construction 3.23, we express

exhibited by . Looking at the proof of Corollary 3.13 (and Construction 3.9), we consider the sigma natural transformation given by the composite

We can visualize it as follows:

For this, we need to choose an adjoint quasi-inverse of for every . Let then be a discrete opfibration in that satisfies $\operatorname {P}$ . By assumption, the “normalized” characteristic morphism defined as in the proof of Corollary 3.12 starting from $i(\varphi )$ and K and $\Lambda $ , factors through . We define to be the morphism in corresponding to the resulting morphism in given by the factorization. So that . We then extend to a right adjoint quasi-inverse of , choosing the components of the counit on the objects $\psi $ to be the isomorphism corresponding to the one in from the comma of and to the comma of and composed with the counit of .

We prove that is cartesian-marked oplax natural. It suffices to prove that

where $\chi $ is the cartesian-marked (i.e., “normal”) oplax natural transformation produced as in the proof of Corollary 3.12, starting from $i(\varphi )$ and K and $\Lambda $ . Indeed $\ell $ is a chronic arrow and i is injective on objects and fully faithful. So we show that the following diagram of oplax natural transformations is commutative:

The square on the left is commutative by construction of . The square in the middle is commutative because pullbacks along opfibrations in are calculated in , by assumption. We also use that the structure of a discrete opfibration in is just given by the structure of the underlying discrete opfibration in . We prove that the square on the right is commutative as well. Let . Given a discrete opfibration in that satisfies $\operatorname {P}$ ,

by construction of . Given $\theta {\colon }\psi \to \psi '$ in ,

because they are equal after applying the fully faithful , by construction of the counit of (together with the proof of Proposition 3.21). Finally, let in . The two composite oplax natural transformations of the square on the right also have the same structure 2-cells on $(f,\nu )$ . Indeed it suffices to show it after applying the fully faithful . And this is straightforward to prove, following (part of) the recipe for described in the proof of Theorem 3.10 (together with the construction of the counit of and the proof of Proposition 3.21). We conclude that is cartesian-marked oplax natural.

Notation 4.1 Given a discrete opfibration in with small fibres, we denote as $(p)_B$ the fibre of p on .

Construction 4.2 We search for a good 2-classifier $\omega {\colon }1\to \Omega $ in . Looking at the archetypal example of , we expect such a good 2-classifier to classify all discrete opfibrations in with small fibres. By Example 2.28, representables form a dense generator

Then, by our theorems of reduction to dense generators (see Corollary 3.13), we will be able to look just at the functors

with . Notice that the left hand side is isomorphic to $\Omega (C)$ , by the Yoneda lemma. As we want all the ’s with to be equivalences of categories (forming then a pseudo-natural equivalence by Proposition 2.12), the assignment of $\Omega $ is forced up to equivalence to be

This is a nice generalization of what happens in dimension 1. Indeed recall that the subobject classifier in 1-dimensional presheaves sends C to the set of sieves on C. And sieves on C are equivalently the subfunctors of . In line with the philosophy to upgrade subobjects to discrete opfibrations (discussed in Section 2.1), discrete opfibrations over generalize the concept of sieve to dimension $2$ . Notice that $\Omega $ coincides with the composite

and is thus a pseudofunctor by Proposition 2.11. We then take as $\omega {\colon }1\to \Omega $ the pseudonatural transformation with component on that picks the identity on . However, $\Omega $ is not a strict 2-functor and, a priori, does not land in ; so $\Omega $ cannot be a 2-classifier in . Thanks to our joint work with Caviglia [Reference Caviglia and Mesiti3], we can produce a nice concrete strictification of $\Omega $ . Although it was already known before [Reference Caviglia and Mesiti3] that any pseudofunctor can be strictified, by the theory developed by Power in [Reference Power13] and later by Lack in [Reference Lack9], the work of [Reference Caviglia and Mesiti3] can be applied to produce an explicit and easy to handle strictification of $\Omega $ , which in addition lands in . Moreover, as we will present in Section 5, such strictification can also be restricted in a natural way to a good 2-classifier in stacks.

Proof By Remark 2.16 any comma object from $\widetilde {\omega }$ can be expressed as a pullback of $\widetilde {\tau }$ , and by Remark 2.8 the property of having small fibres is stable under pullback. So we can just look at $\widetilde {\tau }$ . Since comma objects in are calculated pointwise, for every the component $\widetilde {\tau }_C$ of $\widetilde {\tau }$ on C is given by the comma object in from $\widetilde {\omega }_C=\Delta 1$ to $\operatorname {id}_{\widetilde {\Omega }(C)}$ . Given ,

and thus $\widetilde {\tau }_C$ has small fibres. Indeed a natural transformation from $\Delta 1$ to Z is the same thing as an element in $Z(\operatorname {id}_{C})$ , by the naturality condition. This is similar to the proof of the Yoneda lemma; see also Remark 4.9.

Proof Taking comma objects from the morphism $\omega {\colon }1\to \Omega $ in certainly extends to a functor

by Remark 2.16. But, given $z{\colon }F\to \Omega $ , the comma object

in is calculated pointwise. Since $1$ and F are both strict 2-functors, the universal property of the comma object induces a strict 2-functor L and a strict 2-natural transformation s. The functor also sends modifications between F and $\Omega $ to strict 2-natural transformations over F. Moreover, every component $s_C$ of s on needs to be a discrete opfibration in with small fibres, by Remark 4.9 and the fact that the property of having small fibres is pullback-stable. Since , it follows that $s_C$ is a discrete opfibration in with small fibres. And then s is a discrete opfibration in with small fibres, by Proposition 2.6 (and Definition 2.7).

Proof Given , call the corresponding element in $\Omega (C)$ via the Yoneda lemma and the morphism on the left of the comma object

in . We show that there is a 2-natural isomorphism $L\cong G$ over . Given , we have that $L(D)\cong G(D)$ over because of the following bijection between the fibres, which is natural in $f{\colon }D\to C$

The first isomorphism is given by the explicit construction of comma objects in . It is natural by construction of the structure of discrete opfibration on s induced by the comma object (see Remark 2.16). The second natural isomorphism is given by pseudonaturality of z. The third one is given by the Yoneda lemma and trivially natural. The fourth one is given by the explicit construction of pullbacks in and is natural by construction of on morphisms. It is straightforward to show that the isomorphism $L(D)\cong G(D)$ is 2-natural over and to conclude the proof.

Proof We prove that there is an isomorphism

(4.1)

Given , consider the comma objects

respectively in and in . Notice that the left hand side comma is also a comma object in because commas are calculated pointwise in both 2-categories. We have that $\widetilde {L}(D)\cong L(D)$ over for every , since $j_D$ is an equivalence of categories. Indeed $(j\ast \widetilde {\lambda })_D$ exhibits the comma object on the right hand side in component D. It is straightforward to show that the isomorphism $\widetilde {L}(D)\cong L(D)$ is 2-natural in over and to conclude the isomorphism of Equation (4.1). Notice that $j\circ \widetilde {\omega }$ is isomorphic to $\omega $ , whence is isomorphic to .

By Proposition 4.12, the functor is an equivalence of categories. By the Yoneda lemma, also $j\circ -$ is an equivalence of categories. Indeed the following square is commutative:

And thus $j\circ -$ is an equivalence of categories by the two out of three property. Therefore also is an equivalence of categories.

Proof Consider the fully faithful dense generator formed by representables. By Proposition 4.6, the lax limit of the arrow $\widetilde {\omega }$ has small fibres. By Proposition 4.13, we have that for every the functor

is an equivalence of categories. In order to prove that $\widetilde {\omega }{\colon }1\to \widetilde {\Omega }$ is a good 2-classifier in with respect to the property of having small fibres, by Corollary 3.13, it only remains to prove that the operation of normalization described in Theorem 3.10 is possible.

So let $\varphi {\colon }G\to F$ be a discrete opfibration in with small fibres. Using the dense generator , we express F as a cartesian-marked oplax colimit of representables. By Example 2.42,

whence , with the universal cartesian-marked oplax cocone $\Lambda $ given by

Looking at the proof of Corollary 3.13 (and Construction 3.9), we consider the sigma natural transformation $\chi $ given by the composite

We can visualize it as follows:

For this, we need to choose an adjoint quasi-inverse of for every . By Proposition 4.13, we can construct such a quasi-inverse by taking quasi-inverses of $j\circ -$ and . Both the latter are given by the Yoneda lemma, respectively by the proof of Proposition 4.13 and by Proposition 4.12. So given , we can take to be the morphism which corresponds to $j_C^{-1}(\psi )$ (see Proposition 4.3). With this choice, $\chi _{(C,X)}$ would be the morphism which corresponds to

Using the explicit construction of pullbacks in to calculate , we do not obtain a cartesian-marked oplax natural transformation $\chi $ . This is due to the unnecessary keeping track of the morphisms $f{\colon }D\to C$ other than the objects of $G(D)$ .

Instead, we choose on the objects so that $\chi _{(C,X)}$ is the morphism which corresponds to

This is the operation of normalization that we need. Notice that

and that such isomorphism is natural in . So that, thanks to the argument above, is indeed isomorphic to . We then extend to a right adjoint quasi-inverse of , choosing the components of the counit on the objects to be the just obtained isomorphisms.

We prove that $\chi $ is cartesian-marked oplax natural. Given a morphism in , it is straightforward to show that

using that F is a strict 2-functor. We still need to show that $\chi _{f,\operatorname {id}_{}}=\operatorname {id}_{}$ . For this, it is straightforward to prove that

following the recipe given in the proof of Theorem 3.10. So we conclude using the fully faithfulness of .

Proof By Theorem 2.36, is a bireflective sub-2-category of . So by Remark 4.8 it has all bilimits, calculated in . Consider then a flexible weight W and a 2-diagram F in . Then the flexible limit of F weighted by W exists in , by Remark 4.8. In particular, by flexibility of W, it satisfies the universal property of a bilimit of a 2-diagram that factors through . And it is then a pseudofunctorial stack. It follows that it is the flexible limit of F weighted by W in , since fully faithful 2-functors reflect 2-limits.

Consider now a 2-diagram F in and a flexible weight W. The flexible limit of F weighted by W exists in , calculated pointwise. Then it is also the flexible limit in , as the latter is as well calculated pointwise. So it is a stack, as the 2-diagram in factors through . Whence we have produced the flexible limit of F weighted by W in .

Finally, consider $p{\colon } E\to B$ and $z{\colon } F\to B$ in , with p a discrete opfibration. The pullback of p and z exists in , calculated pointwise. Then it is also the bi-iso-comma object of p and z in , by Remark 4.8. We conclude that it is a stack and hence the pullback of p and z in , by the argument above.

Proof The “if” part is clear by the fact that is fully faithful. The “only if” part follows from Proposition 2.6, since J is subcanonical.

Proof The proof is straightforward.

Construction 5.6 We want to produce the object of Theorem 3.24 in our case with . That is, a stack, which we will call $\Omega _J$ , that is a nice restriction of the good 2-classifier $\widetilde {\Omega }$ in prestacks. Recall that, in dimension 1, the subobject classifier in sheaves is given by taking closed sieves. We produce a 2-categorical notion of closed sieve.

We have already said in Construction 4.2 that discrete opfibrations over representables generalize the concept of sieve to dimension 2; we call them 2-sieves. Thanks to Proposition 4.3 (indexed Grothendieck construction, explored in our joint work with Caviglia [Reference Caviglia and Mesiti3]), we can equivalently consider presheaves on slice categories. We now need to generalize closedness of a sieve to dimension 2. The indexed Grothendieck construction can be restricted to a bijection between 1-dimensional sieves on and presheaves . It can be shown that closed 1-sieves correspond with sheaves , and that the closure of a sieve corresponds to the sheafification of the corresponding presheaves. So we define closed 2-sieves to be the sheaves (with respect to the Grothendieck topology induced by J on the slices, that we call again J). And we use them to restrict our good 2-classifier $\omega {\colon }1\to \widetilde {\Omega }$ in prestacks to a good 2-classifier $\omega _J{\colon }1\to \Omega _J$ in stacks (Theorem 5.11).

Proof The second part of the statement is clear after Proposition 5.5 and Remark 5.7.

We prove that $\Omega _J$ is a stack (recall from Definition 2.34 the definition). So let and $S\in J(C)$ a covering sieve on C.

We first prove the uniqueness of gluings of morphisms. Let and let two natural transformations such that for every . We show that . Given ,

Let now in and consider $g^{\ast } S\in J(g)=J(E)$ . Since M is a sheaf,

$$ \begin{align*}M(g)\cong \operatorname{Match}^{M}\mkern-1.3mu\left(g^{\ast} S\right)\end{align*} $$

where the right hand side denotes the set of matching families for M with respect to the covering sieve $g^{\ast } S$ . And this holds analogously for N. We produce a commutative square

and analogously for $\beta $ . We define to send a matching family to the matching family . The latter is indeed a matching family by naturality of . The square above commutes since, for every $m\in \operatorname {Match}^{M}\mkern -1.3mu\left (g^{\ast } S\right )$ , calling X the amalgamation of m, we have that is an amalgamation of . Notice now that , as for every $h\in g^{\ast } S$ we have that $g\circ h\in S$ and hence . Thus .

We prove that we have the gluings of morphisms. Let and consider a matching family

So that for every with $f\in S$ it holds that . We produce a natural transformation such that for every . Given , we would like to define . Let $g{\colon }E\to C$ in . We define $\lambda _g$ to be the composite

where $[\lambda _g]$ sends a matching family to the matching family . The latter is indeed a matching family by naturality of . It is then straightforward to prove that $\lambda $ is natural. Given , we have that , since $\operatorname {id}_{D}\in f^{\ast } S$ and hence the amalgamation of any matching family on $f^{\ast } S$ is just the datum on $\operatorname {id}_{D}$ . So . Indeed for every $l{\colon }D'\to D$ in

It remains to prove that we have the gluing of objects. So consider a descent datum

with $\varphi ^{f,h}{\colon }h^{\ast } M_f\cong M_{f\circ h}$ such that the cocycle condition

holds for every with $f\in S$ . We produce and for every isomorphisms $\psi ^f{\colon }f^{\ast } M\cong M_f$ such that

for every with $f\in S$ . We construct the presheaf

where $Z(\underline {h})$ is the composite

Z is indeed a functor, by the cocycle condition. Moreover, it is straightforward to show that $f^{\ast } Z\cong M_f$ for every . However, Z is not a sheaf. So we define , where $Z^+$ is the plus construction of Z and hence $Z^{++}$ is the sheafification of Z. It is straightforward to check that $(f^{\ast } Z)^+\cong f^{\ast } (Z^+)$ , by the explicit plus construction. Thus, using that $f^{\ast } Z\cong M_f$ , we define $\psi ^f$ to be the composite

$$ \begin{align*}f^{\ast} (Z^{++})\cong (f^{\ast} Z^+)^+\cong (f^{\ast} Z)^{++}\cong (M_f)^{++}\cong M_f,\end{align*} $$

where the last isomorphism is given by the fact that $M_f$ is a sheaf. It is then straightforward to show that the isomorphisms $\psi ^f$ satisfy the required condition.

Proof It suffices to prove that the characteristic morphism z for $i(\varphi )$ with respect to $\widetilde {\omega }$ produced in Remark 4.16 (and Theorem 4.14) factors through $\ell $ . Indeed such factorization only depends on the isomorphism class of z, by the Yoneda lemma, as any presheaf isomorphic to a sheaf is a sheaf. By Remark 4.16, z is the 2-natural transformation that corresponds with the functor

So it suffices to prove that such functor is a sheaf. Let $f{\colon }D\to C$ in and R a covering sieve on $f{\colon }D\to C$ , i.e., $R\in J(D)$ . Consider then a matching family

on R for $z_C(\operatorname {id}_{C})$ . We need to show that there is a unique $X\in {(\varphi _D)}_f$ such that $H(g)(X)=X_g$ for every . Notice that m is also a matching family on $R\in J(D)$ for H, as ${(\varphi _{D'})}_{f\circ g}\subseteq H(D')$ and the action of $z_C(\operatorname {id}_{C})$ on morphisms is given by the action of H. Since , also . As H is a stack, it then needs to be a sheaf. So there is a unique $X\in H(D)$ such that $H(g)(X)=X_g$ for every . It remains to prove that $\varphi _D(X)=f$ . Since is separated, it suffices to prove that $\varphi _D(X)\circ g=f\circ g$ for every . But by naturality of $\varphi $

$$ \begin{align*}\varphi_D(X)\circ g=\varphi_{D'}(H(g)(X))=\varphi_{D'}(X_g)=f\circ g.\end{align*} $$

We thus conclude that $z_C(\operatorname {id}_{C})$ is a sheaf.

Proof By Theorem 3.24, the restriction $\omega _J$ of the good 2-classifier $\widetilde {\omega }$ in along is a good 2-classifier in with respect to the property of having small fibres. We can apply Theorem 3.24 thanks to Remarks 5.3 and 5.9, Propositions 5.8 and 5.10, and Theorem 4.14.