Artin glueings of toposes as adjoint split extensions

Artin glueings of frames correspond to adjoint split extensions in the category of frames and finite-meet-preserving maps. We extend these ideas to the setting of toposes and show that Artin glueings of toposes correspond to a 2-categorical notion of adjoint split extensions in the 2-category of toposes, finite-limit-preserving functors and natural transformations. A notion of morphism between these split extensions is introduced, which allows the category Ext(H,N) to be constructed. We show that Ext(H,N) is contravariantly equivalent to Hom(H,N), and moreover, that this can be extended to a 2-natural contravariant equivalence between the Hom 2-functor and a naturally defined Ext 2-functor.


Introduction
Artin glueings of toposes were introduced in [1] and provide a way to view a topos G as a combination of an open subtopos G o(U ) and its closed complement G c(U ) . This situation may be described as the 'internal' view, but we might instead look at it externally. Here we have that Artin glueings of two toposes H and N correspond to solutions to the problem of which toposes G does H embed in as an open subtopos and N as its closed complement.
There is an analogy to be made with semidirect products of groups. We may either view a group as being generated in a natural way from two complemented subgroups (one of which is normal), or externally, view a semidirect product as a solution to the problem of how to embed groups H and N as complemented subobjects so that N is normal. Of particular importance to us is that semidirect products correspond to split extensions of groups (up to isomorphism).
Artin glueings of Grothendieck toposes decategorify to the setting of frames and in this algebraic setting the analogy to semidirect products has been made precise. In [11] it was shown that Artin glueings correspond to certain split extensions in the category of frames with finite-limitpreserving maps. While these results were proved in the setting of frames, it is not hard to see that the arguments carry over to Heyting algebras. It is this view that we now extend back to the elementary topos setting. We now recall the main results of [11].
In the category of frames with finite-meet-preserving maps, there exist zero morphisms given by the constant 'top' maps. This allows us to consider kernels and cokernels. Cokernels always exist and the cokernel of f : N → G is given by e : G → ↓f (0) where e(g) = f (0) ∧ g. This map has a right adjoint splitting e * sending h to h f (0) . Kernels do not always exist, but kernels of cokernels always do, and the kernel of e : G → ↓u is the inclusion of ↑u ⊆ G. The cokernel is readily seen to be the open sublocale corresponding to u and the kernel the corresponding closed sublocale. This immediately gives that the split extensions whose splittings are adjoint to the cokernel correspond to Artin glueings.
A notion of protomodularity for categories equipped with a distinguished class of split extensions was first introduced in [5]. In a similar way, in [11] (as well as the current paper) a distinguished role is played by those split extensions whose splittings are right adjoint to the cokernel.
With this correspondence established, the corresponding Ext functor was shown to be naturally isomorphic to the Hom functor. Each hom-set Hom(H, N ) has an order structure and this order structure was shown to correspond contravariantly in Ext(H, N ) to morphisms of split extensions. Finally, it was demonstrated how the meet operation in Hom(H, N ) naturally induces a kind of 'Baer sum' in Ext(H, N ).
In this paper all of the above results find natural generalisation to the topos setting after we provide definitions for the analogous 2-categorical concepts. (Also see the paper [16] by Niefield which appeared after we wrote this paper and discusses the relationship between (non-split) extensions and glueing in a very general context. The current paper uses split extensions and discusses the functorial nature of the construction for toposes in more detail.)

2-categorical preliminaries.
There are a number of 2-categories and 2-categorical constructions considered in this paper and so we provide a brief overview of these here. All the 2-categories we consider shall be strict.
Briefly, a (strict) 2-category consists of objects, 1-morphisms between objects and 2-morphisms between 1-morphisms. Phrased another way, instead of hom-sets between objects as is the case with 1-categories, for any two objects A and B we have an associated hom-category Hom(A, B).
Both 1-morphisms and 2-morphisms may be composed under the right conditions. If F : A → B and G : B → C are 1-morphisms, then we may compose them to yield GF : A → C. We usually represent this with juxtaposition, though if an expression is particularly complicated we may use G • F . As with natural transformations, there are two ways to compose 2-morphisms -vertically and horizontally. We may compose α : F → G and β : G → H vertically to give βα : F → H, sometimes written β • α. Orthogonally, if we have F 2 F 1 : A → C, G 2 G 1 : A → C, α : F 1 → G 1 and β : F 2 → G 2 , then we may compose α and β horizontally to form β * α : F 2 F 1 → G 2 G 1 . Vertical and horizontal composition are related by the so-called interchange law. For each object A there exists an identity 1-morphism id A and for each 1-morphism F there exists an identity 2-morphism id F . Just as one can reverse the arrows of a category B to give B op , one can reverse the 1-morphisms of a 2-category C to give C op . It is also possible to reverse the directions of the 2-morphisms yielding C co and when both the 1-morphisms and 2-morphisms are reversed we obtain C co op .
In this paper we will make extensive use of string diagrams. For an introduction to string diagrams for 2-categories see [14]. We will use the convention that vertical composition is read from bottom to top and horizontal composition runs diagrammatically from left to right.
We consider 2-functors between 2-categories defined as follows. (We follow the convention that 2-functors are not necessarily strict.) Definition 2.1. A 2-functor F between 2-categories C and D consists of a function F sending objects C in C to objects F(C) in D and for each pair of objects C 1 and C 2 in C a functor F C 1 ,C 2 : Hom(C 1 , C 2 ) → Hom(F(C 1 ), F(C 2 )), for which we use the same name. Additionally, for each pair of composable 1-morphisms (F, G) we have an invertible 2-morphism ω G,F : F(G) • F(F ) → F(G • F ) called the compositor, and for each object C in C we have an invertible 2-morphism κ A : Id F (A) → F(Id A ) called the unitor. This data satisfies the following constraints.
(3) If F : X → Y is a 1-morphism, then we have the unit axiom ω F, There is a notion of 2-natural transformation between 2-functors defined as follows.
Next for each object X ∈ X , ρ must respect the identity Id X . For this we need the following diagram to commute.
Finally, they must satisfy the following 'naturality' condition for F, F : X → Y and α : F → F .
When each component ρ X is an equivalence, we call ρ a 2-natural equivalence.
One 2-functor of note is the 2-functor Op : Cat co → Cat which sends a category C to its opposite category C op . We will use this 2-functor in Section 5 to help compare two 2-functors of different variances.
Limits and colimits have 2-categorical analogues, which will be used extensively throughout this paper. A more complete introduction to these concepts can be found in [13]. In particular, we will make use of 2-pullbacks and 2-pushouts, as well as comma and cocomma objects, which we describe concretely below. (1) Let T : X → B and S : X → C be 1-morphisms and let ψ : F T → GS be a 2-morphism. Then there exists a 1-morphism H : X → P and invertible 2-morphisms ν : P G H → T and µ : (2) If H, K : X → P are 1-morphisms and α : P G H → P G K and β : P F H → P F K are 2-morphisms satisfying that ϕK • F α = Gβ • ϕH, then there exists a unique 2-morphism γ : H → K such that P G γ = α and P F γ = β.
A 2-pullback is defined similarly, except both ϕ and ψ are required to be invertible, and is represented as follows.
We now recall the definition of a fibration of categories.
In fact, we will also need the notion of a fibration in other 2-categories, such as the 2-category Cat lex of finitely-complete categories and finite-limit-preserving functors. The general definitions of fibrations, morphisms of fibrations and 2-morphisms of fibrations can be found, for example, in [7,. However, it is not hard to see that the fibrations in Cat lex are simply the finite-limit-preserving functors which are fibrations in Cat.

2.2.
Elementary toposes and Artin glueings. By a topos we mean an elementary toposthat is, a cartesian-closed category admitting finite limits and containing a subobject classifier. The usual 2-category of toposes has 1-morphisms given by geometric morphisms and 2-morphisms given by natural transformations.
For Grothendieck toposes, the subobjects of the terminal object may be imbued with the structure of a frame. Moreover, a geometric morphism between two Grothendieck toposes induces a locale homomorphism between their locales of subterminals. This induces a functor from the category of Grothendieck toposes into the category of locales, which is in fact a reflector. The open subtopos E o(U ) corresponding to a subterminal U has a reflector given by the exponential functor (−) U . It is not hard to see that this topos is equivalent to the slice topos E/U , which in turn can be thought as the full subcategory of the objects in E admitting a map into U . From this point of view, the reflector E : E → E/U maps an object X to the product X × U . We denote its right adjoint by E * = (−) U and write θ and ε for the unit and counit respectively. Note that E * E(G) = (G × U ) U ∼ = G U . In addition to a right adjoint, E also has a left adjoint E ! , which is simply the inclusion of E/U into E.
The closed subtopos E c(U ) has reflector K * : E → E c(U ) given on objects by the following pushout.
is given by the universal property of the pushout in the following diagram. Here the left, front and top faces commute and so a diagram chase determines that p G 1 f and p G 2 id U indeed form a cocone.
We denote the right adjoint of K * by K and write ζ and δ for the unit and counit respectively.
As expected, E o(U ) and E c(U ) are complemented subobjects. Given toposes H and N we can ask for which toposes G may H be embedded as an open subtopos and N its closed complement. This is solved completely by the Artin glueing construction. For any finite-limit-preserving functor F : H → N we may construct the category Gl(F ) whose objects are triples (N, H, ) in which N ∈ N , H ∈ H and : N → F (H) and whose morphisms are pairs (f, g) making the following diagram commute.
The category Gl(F ) is a topos, and moreover, the obvious projections π 1 : Gl(F ) → N and π 2 : Gl(F ) → H are finite-limit preserving. The projection π 2 has a right adjoint π 2 * sending objects H to (F (H), H, id F (H) ) and morphisms f to (F (f ), f ). This map π 2 * is a geometric morphism and, in particular, an open subtopos inclusion. Similarly, π 1 has a right adjoint sending objects N to (N, 1, !) and morphisms f to (f, !) where ! : N → 1 is the unique map to the terminal object. This is itself a geometric morphism, and indeed, a closed subtopos inclusion.
Remarkably, Artin glueings may be viewed as both comma and cocomma objects in the category of toposes with finite-limit-preserving functors. We provide a proof of the latter in Section 4.
One sees immediately that π 1 π 2 * = F . This suggests a way to view any open or closed subtopos as corresponding to one in glueing form. If K : G c(U ) → G and E * : G/U → G are respectively the inclusions of open and closed subtoposes, then there is a natural sense in which these maps correspond to π 1 * : G c(U ) → Gl(K * E * ) and π 2 * : G/U → Gl(K * E * ) respectively. This fact is well known, though a new proof will be provided in Section 3.2.
We now note that the maps π 1 : Gl(F ) → N and π 2 : Gl(F ) → H are fibrations. By the above argument, these results apply equally to the inverse image maps of open and closed subtoposes. This is likely well known, though we were unable to find explicit mention of this in the literature. where N , and P f are defined by the following pullback.
The cartesian property of (P f , f ) follows from the universal property of the pullback. To see that this map satisfies the universal property, suppose that we have a morphism (g 1 , g 2 ) : (A, B, k) → (N, H, ) which is mapped by π 1 to f h. We must show there is a unique map h such that (g 1 , g 2 ) = (f, id H )h and π 1 (h) = h. These constraints imply that h = (h, g 2 ).
To see that (h, g 2 ) is a morphism in Gl(F ), we consider the following diagram. The left-hand square commutes as π 1 (g 1 , g 2 ) = f h and the right-hand square commutes since (g 1 , g 2 ) is a morphism.

Adjoint extensions
In generalising the frame results to the topos setting, it is clear that the appropriate 2-category to consider is Top lex , the 2-category of toposes, finite-limit-preserving functors and natural transformations. (This is the horizontal bicategory of the double category of toposes considered in [15].) For convenience, we will assume that 1 always refers to a distinguished terminal object in a topos, and 0 a distinguished initial object.
We will now introduce the necessary concepts in order to discuss extensions of toposes and show how Artin glueings can be viewed as adjoint extensions. In particular, the definition of extension will require notions of kernel and cokernel.
3.1. Zero morphisms, kernels and cokernels. The definition of extensions requires a notion of zero morphisms. Let us now define these in the 2-categorical context. • Z contains an object of each hom-category, • Z is an ideal with respect to composition (as in [9]) -that is, g ∈ Z =⇒ f gh ∈ Z, • Z is closed under 2-isomorphism in the sense that if f ∈ Z and f ∼ = f then f ∈ Z, • for any parallel pair f 1 , f 2 of morphisms in Z, there is a unique 2-morphism ξ : Definition 3.2. A zero object in a 2-category is an object which is both 2-initial and 2-terminal.
Lemma 3.3. A 2-terminal or 2-initial object in a pointed 2-category is always a zero object. Furthermore, any 2-category with a zero object has a unique pointed 2-category structure where the zero morphisms are those 1-morphisms which factor through the zero object up to 2-isomorphism.
Remark 3.4. Definition 3.1 is a categorification of pointed-set-enriched categories. Pointed categories are often instead defined as those having a zero object. The previous lemma shows that our definition agrees with a definition in terms of zero objects when the category has a terminal or initial object.
Definition 3.5. The 2-cokernel of a morphism f : A → B in a pointed 2-category is an object C equipped with a morphism c : B → C such that cf is a zero morphism and which is the universal such in the following sense.
(1) If t : B → X is such that tf is a zero morphism, then there exists a morphism h : C → X such that hc is isomorphic to t.
The 2-kernel of a morphism in a pointed 2-category C is simply the 2-cokernel in C op .
A similar notion was defined for groupoid-enriched categories in [8].
Note that these may also be defined in terms of 2-pushouts or 2-coequalisers involving the zero morphism (in the same way kernels and cokernels can be defined in the 1-categorical setting).
Although there would usually be coherence conditions that need to be satisfied, the uniqueness of natural isomorphisms between zero morphisms eliminates all of them in the case of 2-cokernels.
Remark 3.6. Note that the condition (2) for 2-kernels is simply the statement that the 2-kernel map is a fully faithful 1-morphism. Moreover, since an adjoint of a fully faithful morphism is fully faithful in the opposite 2-category, we have that when a putative 2-cokernel c : B → C has a (left or right) adjoint d, then condition (2) for 2-cokernels is equivalent to d being fully faithful.
We can now consider how these concepts behave in our case of interest. Note that Top lex has a zero object, the trivial topos. Then zero morphisms in Top lex are precisely those functors which send every object to a terminal object.
In Top lex , 2-cokernels of morphisms F : N → G always exist and are given by the open subtopos corresponding to F (0).
Proof. We know that E lies in Top lex and so we begin by showing that that EF is a zero morphism. The terminal object in G/F (0) is F (0) and so consider the following calculation.
Next suppose that T : G → X is such that T F is a zero morphism. We claim that T E * : H → X when composed with E is naturally isomorphic to T .
Observe that The final condition for the 2-cokernel holds immediately because E has a full and faithful adjoint.
Unfortunately, 2-kernels do not always exist in Top lex . However, they do exist in the larger 2-category Cat lex of finitely-complete categories and finite-limit-preserving functors.
Proposition 3.8. Let F : G → H be a morphism in Cat lex . The 2-kernel of F , which we write as Ker(F ), is given by the inclusion into G of the full subcategory of objects sent by F to a terminal object.
Proof. Since F preserves finite limits and sends each object in Ker(F ) to a terminal object, it is clear that Ker(F ) is closed under finite limits. Naturally, the inclusion is a finite-limit-preserving functor.
It is clear that F K is a zero morphism. We must check that if T : X → G is such that F T is a zero morphism, then it factors through Ker(F ). Note that since F T is a zero morphism, all objects (and morphisms) in its image lie in Ker(F ). Thus, it is easy to see that T factors through Ker(F ). The uniqueness condition of the universal property is immediate, as the inclusion of Ker(F ) is full and faithful.
We will only be concerned with 2-kernels of 2-cokernels. The following proposition shows that these do always exist in Top lex .
Proposition 3.9. Let U be a subterminal of a topos G and consider E : G → G/U defined as in Proposition 3.7. Then the kernel of E is given by K : G c(U ) → G, the inclusion of the closed subtopos corresponding to U .
Proof. Since Top lex is a full sub-2-category of Cat lex , it suffices to show that the closed subtopos G c(U ) is equivalent to Ker(E), the full subcategory of objects sent by E to a terminal object.
The reflector K * : G → G c(U ) sends an object G to the following pushout.
First we show that K * (G) lies in Ker(E). We know that E preserves colimits and so we obtain the following pushout in G/U .
But note that p is an isomorphism, since it is the pushout of an identity morphism and thus EK * (G) ∼ = U and K * (G) lies in Ker(E).
Finally, we must show that K * fixes the objects of Ker(E). First observe that U is the initial object in Ker(E), since if X is an object in Ker(E) then we have There is precisely one morphism in Hom G/U (U, U ), since U is the terminal object in G/U . Now consider the following candidate pushout diagram where G lies in Ker(E).
To see that the square commutes, note that by assumption G lies in Ker(E) and so G × U ∼ = U . Therefore, G × U is initial in Ker(E) and there is a unique map into G.
We now have This gives that G is the pushout and hence fixed by K * .

Adjoint extensions and Artin glueings.
We are now in a position to define our main object of study: adjoint split extensions.
equipped with a natural isomorphism ε : EE * → Id H is called an adjoint split extension if K is the 2-kernel of E, E is the 2-cokernel of K, E * is the right adjoint of E and ε is the counit of the adjunction.
Remark 3.11. Propositions 3.7 and 3.9 suggest that every adjoint split extension is equivalent to an extension arising from a closed subtopos and its open complement (in a sense that will be made precise in Definition 4.1).
The above situation is precisely the setting in which Artin glueings are studied and it is well known that in this case G is equivalent to an Artin glueing Gl(K * E * ). We will present an alternative proof of this result from the perspective of extensions.
We begin by showing that Artin glueings can be viewed as adjoint split extensions in a natural way.
Proposition 3.12. Let F : H → N be a finite-limit-preserving functor. Then the diagram Proof. We first note that Gl(F ) is a topos. This is a fundamental result in the theory of Artin glueings of toposes and a proof can be found in [18].
By Proposition 3.8, it is immediate that π 1 * is the 2-kernel of π 2 . To see that π 2 is the 2-cokernel of π 1 * , we first observe that the slice category of Gl(F ) by the subterminal (0, 1, !) = π 1 * (0) is equivalent to H. The objects of Gl(F )/(0, 1, !) are isomorphic to those the form (0, H, !) (since every morphism into an initial object in N is an isomorphism) and its morphisms of the form (!, f ). The following proposition is shown for Grothendieck toposes in [1], but deserves to be more well known. Here we prove it for general elementary toposes. (It also follows easily from the theory of Artin glueings, but here we will use it to develop that theory.) Recall that we use θ for the unit of the open subtopos adjunction E E * and ζ for the unit of the closed subtopos adjunction K * K.
Proof. First note that the diagram commutes by the naturality of ζ. Recall that, setting is the pushout of G and U along the projections π 1 : G × U → G and π 2 : G × U → U . Now we can rewrite the relevant pullback diagram as follows.
Here the ι maps are injections into the pushout and c is the unit of the exponential adjunction, which intuitively maps elements of G to their associated constant functions.
Let us express P in the internal logic. We have where ∼ denotes the equivalence relation generated by f ∼ * for * ∈ U . Explicitly, we find that Finally, commuting the subobject and the quotient we arrive at where the equivalence relation is generated by (f, g) ∼ (f, * ) for * ∈ U . Note that the union is now disjoint.
The map r : We can define a candidate inverse by s : P → G by (f, g) → g and (f, * ) → f ( * ), which can seen to be well-defined, since if (f, g) and (f , * ) are elements of the disjoint union with f = f and * ∈ U , then f ( * ) = c(g)( * ) = g.
. Thus, r and s are inverses as required.
Remark 3.14. The non-classical logic in the above proof can be hard to make sense of. It can help to consider the cases where U = 0 and U = 1. In the former case, G U contains no information and G + G×U U ∼ = G, while in the latter case the opposite is true. The general case 'interpolates' between these.
The next result will play a central role in this paper.
An enjoyable diagram chase around the cube shows that p is the unique morphism making the cube commute by the pullback property of the front face. Since pullbacks in the functor category are computed pointwise, this yields the desired result.
Remark 3.16. It is remarked in [11] that adjoint extensions of frames can be viewed as weakly Schreier split extensions of monoids as defined in [4]. Propositions 3.13 and 3.15 can be viewed as a categorified version of the weakly Schreier condition for the topos setting, though it is as yet unclear how the weakly Schreier condition might be categorified more generally. It would also be interesting to see how the general theory of (S-)protomodular categories [2,6] might be categorified. Another potential example of 2-dimensional protomodularity can be found in [12].
We can now prove the main result of this section. For now we will treat equivalences of adjoint extensions without worrying too much about coherence, which we will discuss in more detail when we define morphisms of extensions in Section 4.
be an adjoint split extension. Then it is equivalent to Proof. We denote the unit of π 2 π 2 * by θ : Id Gl(K * E * ) → π 2 * π 2 and observe that its component at (N, H, ) can be given explicitly by We denote the unit of π 1 π 1 * by ζ : Id Gl(K * E * ) → π 1 * π 1 and its components are given by Consider the functor Φ : . This is a morphism of extensions in the sense that we have natural isomorphisms α : π 1 * → ΦK, β : π 2 Φ → E and γ : π 2 * → ΦE * given by We must show that Φ is an equivalence. We claim that the following pullback in Hom( We shall make extensive use of Proposition 3.15 in order to prove this. To see that Φ Φ ∼ = id G note that composition with Φ on the right preserves limits and thus can be represented as the following pullback in Hom(G, G).
Note that π 1 Φ = K * , π 2 Φ = E and that θ Φ(G) = (K * θ G , id E(G) ) which of course gives that Kπ 1 θ Φ = KK * θ. After making these substitutions into the diagram above, we have the pullback square occurring in Proposition 3.15, which by the universal property gives that ΦΦ is naturally isomorphic to Id G .
Together with Proposition 3.12 this shows that adjoint split extensions and Artin glueings are essentially the same. The equivalence Φ so defined is natural in a sense that will become clear later in Section 4.2.

The category of extensions
It follows from Theorem 3.17 that equivalence classes of adjoint extensions between H and N are in bijection with those of Hom(H, N ). However, the extensions have a natural 2-categorical structure and so we would like to know how this relates to the categorical structure of Hom(H, N ).

Morphisms of extensions.
Definition 4.1. Suppose we have two adjoint extensions, with the same kernel and cokernel objects and associated isomorphisms ε 1 and ε 2 respectively. Consider the following diagram where α, β and γ are natural isomorphisms and Ψ is a finitelimit-preserving functor.
The morphisms compose in the obvious way: by composing the functors and pasting the natural transformations together by juxtaposing the squares from the diagram above. Horizontal and vertical composition of 2-morphisms is given by the corresponding operations of the natural transformations τ . It is not hard to see that this gives a strict 2-category Ext (H, N ).  Proof. Any such β must satisfy ε 2 = ε 1 (βE 1 * )(E 2 γ). But this can be rewritten as ε 2 (E 2 γ −1 ) = ε 1 (βE 1 * ), which shows that β and γ −1 are mates with respect to the adjunctions E 1 E 1 * and E 2 E 2 * and hence determine each other.
We now show that β so defined is an isomorphism. As the mate of γ −1 , we can express β as (ε 2 E 1 )(E 2 γ −1 E 1 )(E 2 Ψθ 1 ). Now since ε 2 and γ −1 are isomorphisms, we need only show E 2 Ψθ 1 is an isomorphism. This map occurs in the pullback obtained by applying E 2 Ψ to the pullback square of Proposition 3.15.
Now observe that E 2 ΨK 1 ∼ = E 2 K 2 ∼ = 1 is a zero morphism and hence so are E 2 ΨK 1 K * 1 and E 2 ΨK 1 K * 1 E 1 * E 1 . Therefore, the bottom arrow of the above diagram is an isomorphism, and as the pullback of an isomorphism, E 2 Ψθ 1 is an isomorphism too.
Finally, we show that the condition on β for 2-morphisms of extensions is automatic. Simply observe the following string diagrams.
Here the first diagram represents β (E 2 τ ) and the last diagram represents β. In moving from the first diagram to the second we shift τ above θ 1 and to move from the second diagram to the third we use γ = (τ E 1 * )γ.
Proof. Suppose τ : Ψ → Ψ is a 2-morphism of extensions. Then we have τ K 1 = α α −1 and τ E 1 * = γ γ −1 . Now by composing the pullback square of Proposition 3.15 with Ψ and Ψ and using the naturality of τ we have the following commutative cube in Hom Top lex (G 1 , G 2 ).
The universal property of the pullback on the front face then gives that τ is uniquely determined by τ K 1 K * 1 and τ E 1 * E 1 , and hence by τ K 1 = α α −1 and τ E 1 * = γ γ −1 . Thus, the morphism τ is unique if it exists.
We can also attempt to use a similar cube to construct τ without assuming it exists a priori by replacing τ K 1 with α α −1 and τ E 1 * = γ γ −1 in the above diagram. However, in order to obtain a map τ from the universal property of the pullback, we require that the right-hand face commutes.
Note that this square commutes if and only if the similar square obtained by inverting the horizontal morphisms commutes. But this latter square is precisely the square we need to commute to obtain a 2-morphism in the opposite direction. Uniqueness then shows that these two 2-morphisms compose to give identities.
Finally, observe that commutativity of this square is the required equality stated above whiskered with E 1 . This is equivalent to the desired condition, since E 1 is essentially surjective. This will justify working with Ext(H, N ) (viewed as a 1-category) going forward.
Lemma 4.6. From a morphism of extensions (Ψ, α, β, γ) we can form an associated natural trans- where δ 2 is the counit of the K * 2 K 2 adjunction and which is depicted below. Two parallel morphisms of extensions are isomorphic if and only if their corresponding natural transformations are equal.
We can now move all the unprimed variables to the left and primed variables to the right by multiplying both sides of this equation on the left by α −1 K * 1 E 1 * and on the right by γ to obtain (α −1 K * 1 E 1 * )(Ψζ 1 E 1 * )γ = (α −1 K * 1 E 1 * )(Ψ ζ 1 E 1 * )γ . These are the mates of the desired natural transformations with respect to the adjunction K * 2 K 2 .

4.2.
The equivalence of categories. In this section we show that the categories Ext(H, N ) and Hom(H, N ) op are equivalent. This requires showing that isomorphism classes of morphisms of extensions correspond to natural transformations. We have already seen that each isomorphism class has an associated natural transformation. We will now further explore this relationship, making use of the following folklore result. Proof. We first check the 2-categorical condition. Consider two finite-limit-preserving functors U, V : G → X and natural transformations µ : U E * → V E * and ν : U K → V K such that (V ζE * )µ = (νK * E * )(U ζE * ). We must find a unique ω : U → V such that ωE * = µ and ωK = ν.
We use Proposition 3.15 to express U and V as pullbacks and then as in Lemma 4.4 we find that there is a unique map ω : U → V with ωE * = µ and ωK = ν as long as the following diagram commutes.
But commutativity of this diagram is simply the assumed condition whiskered with E on the right. Now we show the 1-categorical condition. Suppose we have finite-limit-preserving functors T 1 : H → X and T 2 : N → X and a natural transformation ϕ : T 1 → T 2 K * E * . We must construct a finite-limit-preserving functor L : G → X and natural isomorphisms τ 1 : LE * → T 1 and τ 2 : Suppose we are given such a functor L and natural isomorphisms τ 1 , τ 2 and consider the following pullback diagram.
Here the bottom trapezium commutes by the naturality of τ 2 K * and the right trapezium commutes since (ϕτ 1 )E = (τ 2 K * E * • LζE * )E by assumption. Note that the left edge of the large square is the mate of τ 2 with respect to K * K and the top edge is the mate of τ 1 with respect to E E * . Now without assuming L exists to start with, we can use the outer pullback diagram to define it and we may recover τ 1 and τ 2 as the mates of the resulting pullback projections.
Observe that precomposing the pullback with K turns the right-hand edge into an isomorphism between zero morphisms. Hence the left-hand morphism (τ 2 K * )(Lζ)K is an isomorphism as well. Since τ 2 is given by composing this with the isomorphism T 2 δ, we find that τ 2 is an isomorphism. On the other hand, precomposing the pullback with E * turns the bottom edge into an isomorphism (as N E * − → G is a reflective subcategory). It follows that τ 1 is also an isomorphism.
Finally, we show that ϕ be can recovered in the appropriate way. The commutativity of the pullback square gives ϕE • τ 1 E • Lθ = T 2 K * θ • τ 2 K * • Lζ. The result of whiskering this with E * on the right and composing with T 2 K * E * ε is depicted in the string diagram below.
The desired equality follows after using the triangle identities to 'pull the wires straight'.
Finally, Γ H,N and Γ −1 H,N form an equivalence since the unit and counit are isomorphisms.
With this in mind we may now consider the full subcategory of Ext(H, N ) whose objects are only those extensions of the form N Gl(F ) H π 1 * π 2 π 2 * for some F : H → N . It is evident that this full subcategory is equivalent to Ext(H, N ) and for the remainder of the paper we will choose to perform calculations in this subcategory for simplicity. We will discuss how this can be done coherently when we investigate the Ext 2-functor in Section 5.
We can now give a concrete description of the behaviour of morphisms of extensions. Suppose that (Ψ, α, β, γ) : Γ H,N (F 1 ) → Γ H,N (F 2 ) is a morphism of adjoint extensions as in the following diagram. Since Ψ preserves finite limits, we can use Proposition 3.13 to completely determine its behaviour. Every object in Gl(F 1 ) can be written as the following pullback diagram, where objects in the category are represented by the green arrows pointing out of the page and the pullback symbol has elongated into a wedge.
Note that the front and back faces are pullback squares in N and the other faces correspond to morphisms in Gl(F 1 ).
We may now study how Ψ acts on this pullback diagram. Observe that the bottom face corresponds to the morphism π F 1 1 * ( ) where : N → F 1 (H). It is then sent by Ψ to the morphism represented in the diagram below.
The pullback of these two faces will then give the image of (N, H, ) under Ψ. The pullback diagram is given by the large cuboid in the diagram below. Here we have factored this pullback as in the similar pullback diagram in the proof of Proposition 4.7.
The bottom face of the bottom left cube and the right face of the top right cube are the commutative squares considered above. These have also been extended by the identity maps in the bottom right cube, so that the bottom and right-hand faces of the full cuboid are as required for the pullback in question.
The bottom left cube commutes by the naturality of α, while the top right cube commutes by the definition of ψ = Γ −1 H,N (Ψ) as in Proposition 4.8. Since α and γ are isomorphisms, the top left cube is also a pullback. Recall that the front and back faces are then also pullbacks. Since the top face of the top left cube must commute, we find that the green arrow we seek is given by p ψ H ( ) . Hence, Ψ(N, H, ) is isomorphic to (N × F 1 (H) F 2 (H), H, p ψ H ( )).
Of course, every natural transformation ψ : F 2 → F 1 yields a morphism of extensions defined by such a pullback. For the associated natural isomorphisms we may take β to be the identity and α to be ( α, id) where α is defined by the diagram below.
Finally, we take γ = ( γ, id) where γ is specified by the diagram below.
We now end this section with what is perhaps a surprising result about morphisms of extensions. Proof. Let ψ : F 2 → F 1 be the natural transformation associated to Ψ. We can construct a functor Ψ * : Gl(F 2 ) → Gl(F 1 ) which sends (N, H, ) to (N, H, ψ H ) and leaves morphisms 'fixed' in the sense that (f, g) : , which may be seen to be a morphism in Gl(F 1 ) using the naturality of ψ.
We claim that Ψ * is left adjoint to Ψ. To see this we consider the candidate counit ε N,H, = (ε , id H ), where ε is defined as in the following pullback diagram.
We must show that given a morphism (f, g) : there exists a unique morphism ( f , g) : g). We will construct this map using the following diagram.
Here the maps out of N 1 form a cone as we have ψ H 2 F 2 (g) 1 = F 1 (g)ψ H 1 1 = 2 f , where the first equality follows from naturality of ψ and the second from the fact that (f, g) is a morphism in Gl(F 1 ).
By the universal property we have that l 2 f = F 2 (g) 1 , which means that ( f , g) is a morphism from (N 1 , H 1 , 1 ) to (N 2 , H 2 , 2 ) in Gl(F 2 ). It is immediate from the diagram that (ε 2 , id H 2 ) • ( f , g) = (f, g) and it is also not hard to see that this is the unique such morphism. Thus, Ψ * is indeed left adjoint to Ψ.
Finally, we must show that Ψ * preserves finite limits. This follows immediately from the fact that finite limits in the glueing may be computed componentwise.
Remark 4.14. Notice that Ψ * is in fact a morphism of non-split extensions in the sense that it commutes with the kernel and cokernel maps up to isomorphism. However, it does not commute with the splittings unless Ψ is the identity. Ext(H, N ). In [11] it was shown for frames H and N that there was something akin to a Baer sum of extensions in Ext(H, N ). It is natural to ask if something analogous occurs in the category Ext (H, N ). Indeed, it is not hard to see via the equivalence with Hom(H, N ) op that Ext (H, N ) has all finite colimits. The following functor will help us compute these colimits. For the action on morphisms, let (Ψ, α, β, γ) be a morphism of extensions and let ψ : K * 2 E 2 * → K * 1 E 1 * be the corresponding natural transformation. H, ψ H ) and (f, g) to (f, g). It is immediate that the necessary diagram for this to be a morphism in the slice category commutes. We must demonstrate that this cone satisfies the universal property. Suppose we have some other cone (Ξ i : C → Gl(K * i E i * )) i∈J and consider the following diagram in Cat where Ψ = D(f ) for some morphism f : i → j in J .

Colimits in
Since each Ξ i is a morphism in Cat/(N ×H) we have that it commutes with the ! maps. This means that the Ξ maps all agree on the first two components. If we assume that Ξ k (C) = (N C , H C , k C ), then Ξ i = Ψ * Ξ j gives i C = ψ H C j C . Now consider the following diagram in N where we make use of the aforementioned limiting cone in Hom(H, N ).
Here we use the universal property of R componentwise at H C to produce the map C . This allows us to construct a map S : C → Gl(R) with S(C) = (N C , H C , C ). As for morphisms, now note that each Ξ k sends f : C → C to the 'same' pair (f 1 , f 2 ) and we define S to act on morphisms in the same way. The pair (f 1 , f 2 ) can be seen to be a morphism in Gl(R) from S(C) to S(C ) by considering the above diagram in the functor category and then using the naturality of : π 1 Ξ k → Rπ 2 Ξ k . This morphism S is the desired map and is easily seen to be unique.
From the above it is clear that M preserves limits and that every limiting cone of M D is isomorphic to one of the form Hom(H, N ). For M to create limits, it remains to show that every cone of D which maps to a limiting cone of M D is isomorphic to one of the above form. This follows since M is conservative.
Notice that the limit diagram was embedded into the slice category so that each Ξ in the proof would agree on the first two components. If the limit diagram is connected, this will happen automatically and so we obtain the following corollary. A disconnected (co)limit is the subject of the following example.
. Since products in a slice category correspond to pullbacks, we may construct this coproduct using the following pullback in Cat.
If ! P is the composite morphism from P to N × H, then the coproduct extension may be recovered as N P H (π 1 ! P ) * π 2 ! P (π 2 ! P ) * .

The Ext functor
Given that we have established that Ext(H, N ) is equivalent to Hom(H, N )  In other contexts (for instance, see [3]) the extension functor can be obtained from a fibration. In the protomodular setting, we start from the 'fibration of points' sending split epimorphisms to their codomain. In the more general setting of S-protomodularity (see [6]) we consider only a certain subclass of split epimorphisms. This suggests we consider a 2-fibration sending open subtopos adjunctions to the codomain of their inverse image functors.
A categorification of the Grothendieck construction (see [7]) gives that 2-fibrations correspond to 3-functors into 2Cat. Fortunately, aside from motivation, we will largely be able to avoid 3-functors for the same reasons that Ext(H, N ) is essentially a 1-category (Corollary 4.5).
While the paradigmatic example of a fibration is the codomain fibration, which maps from the whole arrow category to the base category, the domain of the analogous 2-fibration is restricted to the category of fibrations. See [7] for more details on 2-fibrations.
The fibre 3-functor Top co op lex → 2Cat corresponding to the 2-fibration Cod : Fib Top lex → Top lex can be described as follows (omitting the description of the coherence data for simplicity): • On objects it sends a topos E to the slice 2-category of finite-limit-preserving fibrations from toposes to E.
• On 1-morphisms it sends a finite-limit-preserving functor T : E → E to the 2-functor T corresponding to pulling back along T . -Then we may use the fibration property of E to lift the natural transformation τ S (E) to a natural transformation into P S . Explicitly, for an object X ∈ D S , the morphism τ S (E)(X) : T (S (E)(X)) → S(S (E)(X)) can be lifted to a morphism in D with codomain P S (X). These lifted morphisms assemble into a natural transformation τ from a new functor L : D S → D to P S . This functor sends a morphism f : X → Y to the morphism obtained by factoring P S (f )τ X through τ Y as shown in the diagram below. L(X) E -Finally, consider the 2-pullback of E and T and note that the maps L : D S → D and S (E) : D S → E form a cone as shown below.
Thus, we may factor these through P T and T (E) respectively to obtain a functor from D S to D T . This is the desired functor τ E : S (E) → T (E).
The coherent set of 2-isomorphisms for the 2-natural transformation can also be obtained by the cartesian property of the lifted maps and universal property of the 2-pullback.
We can then easily modify this to describe the fibre 3-functor for the 2-fibration of (open) points. Moreover, to obtain Ext(−, N ) : Top co op lex → 2Cat we restrict to inverse image functors of open subtoposes (equipped with right adjoint splittings) with fixed kernel object N . The above discussion restricts easily to this case, since these functors are stable under pullback along finite-limit-preserving functors and the relevant morphisms can be shown to be morphisms of extensions. To see this we will use the following folklore result, which we prove here for completeness.
Proof. We must first describe Q. Consider the following comma object diagrams.
Now we paste our candidate 2-pullback square to our other comma object diagrams as follows.
Since π F 1 Q = π F T 1 , we see that the pasting diagram above is just the comma object diagram corresponding to Gl(F T ). It follows that the upper square is a 2-pullback in a similar manner to (the converse direction of) the pullback lemma.
We will now describe the Ext 'functor' explicitly and at the same time demonstrate its relationship to Hom.
as in the following diagram.
Gl(F T ) The functor Ext(T, N ) acts on morphisms via the universal property of the 2-pullback. Let Ψ be a morphism of extensions corresponding to the natural transformation ψ : F 2 → F 1 and consider the following diagram, noting that T π F 1 T N N F 2 T (H) This is readily seen to be the same pullback which determines Γ H ,N (ψT ) (N, H, ).
Finally, we discuss how Ext(−, N ) acts on 2-morphisms. We follow the construction outlined above for the codomain 2-fibration. Let τ : T → T be a natural transformation. Then we describe the natural transformation Ext(τ, N ) : Ext(T , N ) → Ext(T, N ) componentwise.
Without loss of generality we may describe each component at extensions of the form Γ H,N (F ). Consider the following diagram.
As discussed in the case of the codomain fibration, we may define a functor L τ as follows. We   N , H, ). It remains to show that this gives a morphism of split extensions, but it is clear from the above pullback diagram that this functor is the morphism of split extensions corresponding to τ F (which itself is equal to Hom(τ, N ) F ).
The morphisms Ext(τ, N ) Γ H,N (F ) define the desired natural transformation Ext(τ, N ). (Naturality follows from the interchange law or from the general theory of the codomain 2-fibration.) The 2-functor Ext(−, N ) composes strictly, and for the unitors observe that Ext(Id H , N ) is equal to Γ H,N Γ −1 H,N : Ext(H, N ) → Ext(H, N ) so that we make take as our unitors the unit of the adjunction Γ H,N . The necessary 2-functor axioms can then easily be seen to hold.
Remark 5.2. The above argument also proves that the 2-functor sending adjoint extensions with fixed kernel N to their cokernels is a 2-fibration. Proof. The equality in point (2) is clear by inspection of the definition of Ext(T, N ). The proof of the coherence conditions is easy. In particular, the first coherence condition is satisfied because each morphism of the diagram is the identity. Similarly, for the second condition again all morphisms are the identity (though marginally more work is required to show that the unitor at H whiskered with Γ H,N is in fact the identity).
We turn our attention to the functor Ext(H, −) : Top co lex → Cat. We could not find an elegant description of this in terms of a 2-fibration. However, we believe a reasonable definition can be given by dualising our arguments for Ext(−, N ).
Naturally, for an object N we have that Ext(H, N ) is just the category of adjoint split extensions. Thus, fixing particular pushouts we can describe Ext(H, S) concretely as sending an extension As mentioned above, Ext(H, S) should act on morphisms by the universal property of the pushout. Let Ψ = (Ψ, α 1 , β 1 , γ 1 ) be a morphism of extensions and consider the 2-cocone given by P 2 Ψπ F 1 1 * 1 * Id N S as in the following pasting diagram (where we omit the 2-morphisms do avoid clutter).
Here the front, back and left faces commute strictly and the top face has associated invertible 2-morphism α −1 1 .
The left-hand square commutes by naturality of σ and the right-hand square commutes since (f, g) is morphism in Gl(F ). Now the σ N arrange into a natural transformation σ : L σ → P S .
If things are to behave dually, we should have L σ factor through P S . By the naturality of σ we have that the following diagram commutes. We might now hope to take Ext(H, σ) F (H) to be this resulting factor Γ H,N (σF ) * . However, this map goes in the 'wrong' direction. We can remedy this by taking the right adjoint and setting Ext(H, σ) F (H) = Γ H,N (σF ).
In order to specify Ext(H, −) completely, it only remains to discuss the compositors and unitors. As before we have that Ext(H, −) composes strictly and we take the unitors to be the unit of the adjunction Γ * H,N Γ H,N in Theorem 4.12.
We can express the relationship between Ext(H, −) and Hom(H, −) as follows. Proof. Just as before, the equality in point (2) is clear by inspection of the definition of Ext(H, S) and the necessary coherence conditions hold, because each involved morphism is an identity.
A bifunctor theorem for 2-functors was discussed in [10] and gives the precise conditions that allow two families of 2-functors M B : C → D and L C : B → D to be collated into a bifunctor P : B × C → D for which P (B, −) is isomorphic to M B and P (−, C) is isomorphic to L C . These conditions are that L C (B) = M B (C) and that for each f : B 1 → B 2 in B and g : C 1 → C 2 in C there exists an invertible 2-morphism χ f,g : L C 2 (g)M B 1 (f ) → M B 2 (f )L C 1 (g) satisfying certain coherence conditions reminiscent of those for distributive laws for monads. Such families together with χ are called a distributive law of 2-functors. Thus, we may apply the results of [10] to arrive at the 2-functor (Ext, ω, κ) : Top op lex ×Top lex → Cat defined below.  N ). It is shown in [10] that 'morphisms between distributive laws' can also be collated to give 2-natural transformations between the corresponding bifunctors. The 2-natural equivalences in Theorems 5.3 and 5.5 can be collected into a 2-natural equivalence Γ : Ext → Hom op provided that Γ N H = Γ H N and the Yang-Baxter equation holds. These conditions are immediate in our setting and so we obtain the following theorem.