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Artin glueings of frames as semidirect products

Artin glueings provide a way to reconstruct a frame from a closed sublocale and its open complement. We show that Artin glueings can be described as the weakly Schreier split extensions in the category of frames with finite-meet preserving maps. These extensions correspond to meet-semilattice homomorphisms between frames, yielding an extension bifunctor. Finally, we discuss Baer sums, the induced order structure on extensions and the failure of the split short five lemma.


Introduction
Let H = (|H|, OH) and N = (|N |, ON ) be topological spaces. We might ask which topological spaces G have H as an open subspace and N as its closed complement. This is solved by the so-called Artin glueing construction [1,13]. Let us briefly describe the intuition behind this construction.
In such a situation it is clear that |G| = |N | ⊔ |H|. Furthermore, each open U in G corresponds to a pair (U N , U H ) where U N = U ∩ N ∈ ON and U H = U ∩ H ∈ OH. This gives an alternative description of OG as the frame L G of such pairs with the meet and join operations corresponding to componentwise intersection and union respectively.
Given such a space G, the pairs occurring in L G are determined by a finite-meet preserving map α : OH → ON which sends each open U ∈ OH to the largest open V ∈ ON such that (V, U ) ∈ L G . A pair (V, U ) belongs to L G if and only if V ⊆ α(U ). In fact, any meet-preserving map α will determine such a space G in this way. We call the resulting space the Artin glueing of α.
The above discussion deals only with the lattices of open sets of the topological spaces and the construction works equally well for frames. The aim of this paper is to explore the commonalities between this construction and the semidirect product of groups.
Let us recall some basic properties of semidirect products. Given a group homomorphism α : H → Aut(N ), we can construct a group G satisfying: (1) H ≤ G and N ⊳ G, We see here a vague analogy between the semidirect products of groups and the Artin glueings of frames. In both cases we have objects H and N which we want to embed as complemented 'subobjects' (sublocales in the frame case) of some other object, with N normal in the group case and closed in the frame case. In both cases these constructions are entirely determined by certain structure-preserving maps involving N and H.
In order to make this analogy precise, we look at the characterisation of semidirect products of groups as the solutions to the split extension problem. A split extension of groups is a diagram of the form where k is the kernel of e, e is the cokernel of k, and s is a section of e. Here G will always be a semidirect product of H and N , and the maps k and s will be the appropriate inclusions into the semidirect product. Furthermore, for each group N there is a functor SplExt(−, N ) : Grp op → Set which sends a group H to the set of split extensions of H by N . This functor is naturally isomorphic to Hom(−, Aut(N )). For more details on this functor, see [3,4].
Split extensions of groups are very well behaved and the notion of pointed protomodular category (see [6,2]) provides a general setting in which they can be studied. Sometimes, however, only some of the split extensions in a category are well behaved and this motivates the more general idea of S-protomodularity [7], where S can be thought of as a collection of split extensions.
It is in this vein that we study Artin glueings. It is not possible to talk about extensions in the usual category of frames (or locales), as without zero morphisms it does not make sense to talk about kernels and cokernels. Instead we move to the category Frm ∧ which has frames as objects and finite-meet preserving maps as morphisms. Both Frm ∧ and Grp are full subcategories of the category of monoids. The split extensions we consider are related to the Schreier split extensions from the study of monoids (see [10,7]).
The relationship between Frm and Frm ∧ is similar to the relationship between the category Set of sets and functions and the category Rel of sets and relations. In particular, Frm ∧ is order-enriched and we can find the frame homomorphisms inside it as the left adjoints. In this way Frm ∧ provides a proarrow equipment for Frm. This category has been used alongside glueings in [11]. The category Frm ∧ can also be thought of as the category of injective meet-semilattices [8], though we are less sure of the implications of this.
In this paper we concern ourselves with the collection of split extensions of the form described above, but where k and s are required to satisfy a Schreier-type condition (or equivalently, where s is required to be right adjoint to e). We find that G will always be an Artin glueing of H and N determined by the map k * s. As one might expect, there is a family of functors AdjExt(−, N ) here too, but now these extend to a bifunctor AdjExt, which is naturally isomorphic to Hom : Frm op ∧ × Frm ∧ → Set. The fact that hom-sets have a natural meet-semilattice structure gives a notion of Baer sum of the extensions. We also study the induced order structure on the extensions and the failure of the split short five lemma. These results will be extended to the topos case in later paper.
2. An extension problem in Frm ∧ 2.1. Adjoint extensions. The category Frm ∧ of frames and finite-meet preserving maps is enriched over meet-semilattices and so between any two frames L and M there is a largest meet-preserving map. This map is the constant 1 map, which sends each element of L to the top element of M . It is apparent that composing with this map on either side again yields a constant 1 map and so we see that these maps are the zero morphisms of our category. Due to this somewhat unfortunate conflict of terminology we use ⊤ L,M to refer to the zero morphism between L and M or just ⊤ when its meaning is unambiguous.
We can now define the kernel of a morphism f as the equaliser of f and ⊤ and the cokernel of f as the coequaliser of f and ⊤. Kernels do not always exist in this category; however, cokernels always do exist in the form described below.
Proposition 2.2. Let f : N → G be a morphism in Frm ∧ and let u = f (0). The cokernel of f is given by e : G ↓u, where e(x) = x ∧ u. Furthermore, e has a right adjoint section given by e * (y) = y u .
Proof. The right adjoint to e exists by well-known properties of frames. Since e is surjective, e * splits e by general properties of adjoints.
Clearly e composes with f to give ⊤ and so we need only check that it satisfies the universal property. Suppose g : G → X composes with f to give ⊤ N,X . In order to show that e is the cokernel of f we must show that g factors through e to give a unique map. Uniqueness is automatic, because e is epic.
To show that the meet-semilattice homomorphism g factors through the surjection e, it is enough to show that g(x) = g(y) whenever e(x) = e(y). But if e(x) = e(y), then x ∧ f (0) = y ∧ f (0) and so g( , which equals g(y) by running the same argument in reverse.
Definition 2.3. We say a morphism is a normal epimorphism if it occurs as the cokernel of some morphism. Dually, a monomorphism is a normal monomorphism if it occurs as the kernel of some morphism.
Proposition 2.2 shows that every normal epimorphism is of the form − ∧ u : G ↓u. Conversely, such a morphism is always a normal epimorphism as it is clearly seen to be the cokernel of the inclusion of ↑u ⊆ G. Note that normal epimorphisms in Frm ∧ are precisely the open frame quotients, which accords well with the idea that H should be an open sublocale of the glueing.
While Frm ∧ does not possess all kernels, this is not a problem for working with extensions, since kernels of normal epimorphisms always do exist.
↓u be a normal epimorphism in Frm ∧ . The kernel of e is given by k : ↑u G, where k(x) = x. The map k has a left adjoint k * (x) = x ∨ u which preserves finite meets.
Proof. Let f : X → G compose with e to give ⊤ X,H . The only elements sent by e to 1 lie in ↑u and so the image of f is contained in ↑u. The restriction of f to ↑u shows the existence condition for the universal property, while uniqueness follows since k is monic. It is then easy to see that x → x ∨ u provides a left adjoint to k.
It is well known that every normal monomorphism is the kernel of its cokernel. Thus the previous result implies that a map is a normal monomorphism if and only if it is of the form ↑u ֒→ G. In other words, the normal monomorphisms in Frm ∧ are precisely the right adjoints of closed frame quotients.
We now have a good understanding of both the kernel and cokernel maps in a split extension. It is only the splitting s that remains mysterious. We will need to impose some conditions on the splitting in order to obtain a well-behaved theory.
Notice that for a split extension of groups, every element of G is of the form k(n)s(h) for some n ∈ N and h ∈ H. For split extensions of monoids this condition does not hold automatically, but when assumed explicitly gives rise to the class of weakly Schreier extensions [5,9]. This is the condition we will imposethat is, we assume that every element of the frame G is of the form k(n) ∧ s(h). In fact, as we will see below, there is a canonical choice of n and h, since h is uniquely determined and we may choose n to be as large as possible. The following result shows that under this condition, the splitting is uniquely determined by e. Proof. By propositions 2.2 and 2.4 we may take N to be of the form ↑u and H to be ↓u for some element u ∈ G. Then k(x) = x and e(x) = x ∧ u.
Suppose the extension is weakly Schreier and consider an element g ∈ G. By assumption, g = k(n) ∧ s(h) for some n ∈ N and some h ∈ H. Then e(g) = ek(n) ∧ es(h) = 1 ∧ h = h. Thus, g = k(n) ∧ se(g). In particular, g ≤ se(g) and so id G ≤ se. But es = id H ≤ id H and so s is right adjoint to e as required.
For the other direction suppose s is the right adjoint of e so that s(x) = e * (x) = x u . We must show that each element of g ∈ G can be expressed as k(n) ∧ e * (h) for some n ∈ N and h ∈ H. By the above we may take h = e(g), while k * (g) is the most natural candidate for n. Taking the meet yields But u ∧ g u ≤ g and thus kk * (g) ∧ e * e(g) = g as required.
This means that all the information of a weakly Schreier extension N G H k e s is contained in the normal epi e, since k can be recovered as its kernel and s as its right adjoint. Since every normal epi gives rise to a weakly Schreier extension, we have a complete classification of the weakly Schreier extensions in Frm ∧ . From now on, we will refer to this class of split extensions as the adjoint extensions.
Let S be the class of normal epimorphisms Frm ∧ equipped with their left adjoints. It is clear that all isomorphisms belong to S and we will show S is stable under pullback in proposition 3.1. Finally, for any such map, the adjoint and the kernel are jointly extremally epic by proposition 2.5. Thus, the pointed category Frm ∧ is S-protomodular in the sense of [5]. That is, it is almost an S-protomodular category in the sense of [7] except for the requirement of finite-completeness.
Notice also that any extension in Frm ∧ -that is, a diagram N G H k e in which e is the cokernel of k, and k is the kernel of e -gives rise to a unique adjoint extension and vice versa. Thus the adjoint extension problem coincides with the extension problem in Frm ∧ . However, as we will see later these concepts should not conflated, since the morphisms of extensions and (adjoint) split extensions are not the same.

Artin glueings.
In the introduction we gave an informal description of the Artin glueing construction. We now discuss them in more detail and explore the connection to adjoint extensions. The ideas behind the results of this section are well understood (for example, see [12]), but we prove them in our current context for completeness. Since meets in Gl(α) are computed componentwise, we see that π 1 and π 2 preserve finite meets and so are morphisms in Frm ∧ . In fact, they both have right adjoints in Frm ∧ . The right adjoint of π 1 is given by π 1 * (n) = (n, 1) and the right adjoint of π 2 is given by π 2 * (h) = (α(h), h).
With these right adjoints, Artin glueings give rise to adjoint extensions. is an adjoint extension.
Proof. It is enough to show that π 2 is a normal epi and that π 1 * is its kernel. Observe that H is isomorphic to ↓(0, 1) ⊆ Gl(α) via the map h → (0, h) and that this makes the following diagram commute.
Notice that α can be recovered from the glueing by considering the composite π 1 π 2 * = α. In fact, any gives rise a finite-meet preserving map k * e * and we may glue along this map to obtain another extension as above.
In order to compare the original adjoint extension to the one given by the glueing of k * e * we require a notion of morphism of adjoint extensions. It is apparent that isomorphisms of adjoint extensions are those for which the meet-preserving map is an isomorphism.
Proof. Without loss of generality we may assume N = ↑u, H = ↓u, k(x) = x, e(x) = x ∧ u and e * (x) = x u .
Consider the maps f : G → Gl(k * e * ) and f ′ : Gl(k * e * ) → G given by f (g) = (k * (g), e(g)) and f ′ (n, h) = k(n) ∧ e * (h). Notice that the map f is well defined because g ≤ e * e(g) and so k * (g) ≤ k * e * e(g). Clearly both f and f ′ preserve finite meets. π 2 * f f ′ We claim these maps are inverses. First note f ′ f (g) = kk * (g) ∧ e * e(g) = g, where the final equality follows as in the proof of proposition 2.5. Next we have f f ′ (n, h) = (k * (k(n) ∧ e * (h)), e(k(n) ∧ e * (h))) = (k * k(n) ∧ k * e * (h), ek(n) ∧ ee * (h)). Notice that ek(n) = 1 and ee * (h) = h and so the second component is h as required. Next observe that k * k(n) = n and n ≤ k * e * (h), since (n, h) ∈ Gl(k * e * ). This gives that the meet in the first component is n as required. Thus G and Gl(k * e * ) are isomorphic in Frm ∧ .
So, similarly to the case of groups where the split extensions could be identified with maps α : H → Aut(N ), we see that the adjoint extensions between H and N correspond to finite-meet preserving maps β : H → N .
Using proposition 2.5, we can also state theorem 2.9 in terms of weakly Schreier extensions. This description of weakly Schreier extensions is generalised to the case of monoids in [9].

Extension functors
3.1. Functoriality of adjoint extensions. In the category of groups (or any protomodular category with semidirect products -see [6] and [3]) there is a functor SplExt(−, N ) for each object N which sends an object H to the set of isomorphism classes of split extensions of H by N . This functor acts on morphisms by pullback as described below.  Proof. It is sufficient to compute the pullback in the category ∧-SLat of meet-semilattices and to observe that it is a frame. By theorem 2.9 we can take e to be π 2 : Gl(k * e * ) → H. Since ∧-SLat is algebraic, the pullback of π 2 and f is given by   Proof. By theorem 2.9 we may take k to be the map π 1 * : N → Gl(k * e * ). We claim that the pushout of π 1 * and f is given by Gl(f k * e * ) with the injections π ′ 1 * : N ′ → Gl(f k * e * ) and f ′ : Gl(k * e * ) → Gl(f k * e * ), where f ′ (n, h) = (f (n), h), as in the diagram below.
Note that the map ℓ as defined above does indeed preserve finite meets. So we need only show that p and q factor through ℓ as required. For q we have the simple equality ℓπ ′ 1 * (n) = ℓ(n, 1) = p(1, 1) ∧ q(n) = q(n). For p we must show that ℓ(f (n), h) = p(n, h). We know that ℓ(f (n), h) = p(f k * e * (h), h) ∧ q(f (n)). Because p and q form a cocone, we have that q(f (n)) = pπ 1 * (n) = p(n, 1). Substituting this in we get ℓ(f (n), h) = p(f k * e * (h), h) ∧ p(n, 1) = p(f k * e * (h), h) ∧ (n, 1)) = p(n, h) as required. . We then have the following corollary as above. To show the families (AdjExt(−, N )) N ∈Frm∧ and (AdjExt(H, −)) H∈Frm∧ yield a bifunctor, we only need that AdjExt(H ′ , g)AdjExt(f, N ) = AdjExt(f, N ′ )AdjExt(H, g) and to set AdjExt(f, g) to be their common value. By corollaries 3.2 and 3.4 we know that each family is isomorphic to hom functors, which clearly satisfy the condition and hence AdjExt is a bifunctor naturally isomorphic to Hom. We record this in the following theorem.  AdjExt(f, N ) is given by pullback along f and AdjExt(H, g) is given by pushout along g as above. This is naturally isomorphic to Hom : Frm op ∧ × Frm ∧ → Set.

3.2.
Additional structure on the set of adjoint extensions. Since Frm ∧ is enriched over meet-semilattices, the natural isomorphism of theorem 3.5 induces a meet-semilattice structure on AdjExt (H, N ). On the other hand, the extensions of H by N have a natural category structure using the morphisms defined in definition 2.8. It is natural to ask how these two structures might be related. Let us start by considering the first notion in more detail.
Recall that an extension is a diagram N G H k e in which k is the kernel of e and e is the cokernel of k. As discussed at the end of section 2.1, every extension in Frm ∧ admits a unique splitting that turns it into an adjoint extension. Thus far we have mainly focused on describing the elements of AdjExt(H, N ) as isomorphism classes of adjoint extensions in order to make an analogy with split extensions of groups. But thinking of them in terms of extensions instead allows us to make a different analogy -this time to extensions of abelian groups. This shift of perspective makes sense because isomorphisms of extensions and adjoint extensions agree.
In an abelian category, the Ext functor admits a natural abelian group structure. Here the binary operation is called the Baer sum of extensions. More generally, in any category with biproducts every object has a unique commutative monoid structure and so any product-preserving Set-valued functor on such a category factors through the category of commutative monoids. When applied to the Ext functor of an abelian category, this yields the Baer sum operation. In our situation, AdjExt preserves finite products (as it is isomorphic to Hom) and this construction endows AdjExt(H, N ) with the structure of a meet-semilattice.
Of course, as mentioned above we can obtain the same result more directly by applying the isomorphism AdjExt ∼ = Hom and using the natural meet-semilattice structure on the hom-sets. Explicitly, this gives that N N × H H π 1 * π 2 is the top element and the meet of extensions N G 1 H k 1 e 1 and We can then apply the following proposition to give a particularly concrete description of the meet. Proposition 3.6. Let α, β : H → N be meet-semilattice homomorphisms. Then Gl(α ∧ β) is given by the intersection of Gl(α) and Gl(β) as sub-meet-semilattices of N × H in the obvious way.
Proof. Simply observe that where the final intersection is taken in N × H.
It is also interesting to consider the ordering of extensions induced by this meet-semilattice structure.
Corollary 3.7. If α ≤ β, then there is an obvious inclusion Gl(α) ֒→ Gl(β). Ext(N, H), we can view G 1 as a subset of G 2 . In fact, we obtain a morphism of extensions in the following sense. The meet-semilattice structure on Hom(G, G ′ ) induces a meet-semilattice structure on the hom-sets of the category Ext(H, N ) and so, in particular, the category is order-enriched.

So whenever an extension
Compare this to the category of adjoint extensions alluded to at the start of this section.
Definition 3.9. The category AdjExt(H, N ) has adjoint extensions as objects and morphisms given by diagrams of the form below as in definition 2.8.
Note that, unlike morphisms of extensions, morphisms of adjoint extensions are required to preserve the splittings. So while there is a correspondence between the objects, they give different categories. In particular, the map given in corollary 3.7 need not be a morphism of adjoint extensions, but in the following theorem we will see that it has a right adjoint which is. Proof. Let α, β : H → N be meet-semilattice homomorphisms such that α ≤ β. Then there is a corresponding inclusion i : Gl(α) ֒→ Gl(β) as in corollary 3.7. This inclusion is a frame homomorphism and thus has a right adjoint i * : Gl(β) → Gl(α). This adjoint sends (n, h) to (n ∧ α(h), h) and can be seen to be a morphism of adjoint extensions.
This procedure defines the action on morphisms of a contravariant functor from the underlying poset of AdjExt(H, N ) to AdjExt(H, N ). This functor is essentially surjective by theorem 2.9 and it is automatically faithful, since AdjExt(H, N ) is a poset. We now show it is full.
Since f is in the form described above, we may conclude that it is in the image of the functor once we show that α ≤ β. To see this note that (α(h), h) = f (β(h), h) = (β(h) ∧ α(h), h).
In the above proof we used the characterisation of equivalences of categories in terms of essentially surjective fully faithful functors for simplicity. However, it also possible to construct the 'inverse' equivalence explicitly without appealing to the axiom of choice.