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Grothendieck quantaloids for allegories of enriched categories.

1 Introduction

A topos arising as the category of left adjoints in a locally ordered category, is the subject of P. Freyd and A. Scedov's [1990] study of allegories. More precisely, an allegory A is a modular locally ordered 2-category whose hom-posets have binary intersections; taking left adjoints (also known as "maps") in an allegory A thus produces a category Map(A); and the interesting case is where the latter category is in fact a topos. Thus, in Freyd and Scedrov's own words, allegories "are to binary relations between sets as categories are to functions between sets". In practice, those interesting allegories arise most often as universal constructions on much smaller sub-allegories which are easier to describe explicitly. Freyd and Scedrov [1990] (but see also [Johnstone, 2002, A3]) give several theorems to this effect.

In [1982], R. Walters proved that any small site (C, J) gives rise to a small quantaloid R(C, J) in such a way that the topos Sh(C, J) is equivalent to the category of Cauchy-complete symmetric R(C, J)-enriched categories and functors. But the latter category is further equivalent to the category of all symmetric R(C, J)-categories and left adjoint distributors, and the quantaloid SymDist(R(C, J)) of all symmetric R(C, J)-enriched categories and all distributors is modular. In other words, the topos Sh(C, J) is the category of maps in the allegory of symmetric R(C, J)-enriched categories and distributors-- which thus qualifies as an "interesting" allegory.

In this paper we shall explain more precisely how "sheaves via quantaloid-enrichment" fit with "toposes via allegories". To that end, we define in Section 2, for any involutive quantaloid Q, a new involutive quantaloid Rel(Q), to be thought of as the locally posetal 2-category of "Q-sheaves and relations", and a new category Sh(Q), to be thought of as the category of "Q-sheaves and functions". The objects of Rel(Q), resp. Sh (Q), are particular symmetric quantaloid-enriched categories, and the morphisms are distributors, resp. functors; the relation between the two is that Sh(Q) is the category of symmetric left adjoints in Rel(Q). For appropriate Q, these Q-sheaves are, among the Q-orders of [Stubbe, 2005b], precisely the symmetric ones.

We show in Section 3 that, if Q = R(C, J), then Sh (Q) is equivalent to Sh (C, J) and Rel(Q) is equivalent to Rel(Sh(C, J)); thus we recover and refine Walters' [1982] insight. More generally, we prove in Section 4 that Rel(Q) is equivalent to Rel(T) for some Grothendieck topos T (and thus Sh(Q) is equivalent to T) if and only if Q is a modular, locally localic and weakly semi-simple quantaloid; we call these Grothendieck quantaloids. In other words, these Grothendieck quantaloids are precisely those for which the Q-sheaves and relations form an "interesting allegory". Locales and inverse quantal frames [Resende, 2007, 2012] are examples of Grothendieck quantales. If L is a locale, then Sh(L) is in fact the topos of sheaves on L. And if O(G) is the inverse quantal frame associated to an etale groupoid G [Resende, 2007], then it follows from [Heymans and Stubbe, 2009b; Resende, 2012] that Sh(O(G)) is the classifying topos of that groupoid.

2 Sheaves on an involutive quantaloid

The new notions that we will present at the end of this section draw heavily on the theory of quantaloid-enriched categories. For self-containedness we present some preliminaries in the first couple of subsections. For more details and for the many appropriate historical references we refer to [Stubbe, 2005a; Heymans and Stubbe, 2011].

Enrichment, involution and symmetry

A quantaloid Q is, by definition, a category enriched in the symmetric monoidal closed category Sup of complete lattices and supremum-preserving functions; and a homomorphism F: Q [right arrow]71 of quantaloids is a Sup-enriched functor. An involution on a quantaloid Q is a homomorphism [(--).sup.[omicron]]: [Q.sup.[omicron]p] [right arrow] Q which is the identity on objects and satisfies [f.sup.[omicron][omicron]] = f for any morphism f in Q. The pair (Q, [(--).sup.[omicron]]) is then said to form an involutive quantaloid; we shall often simply speak of "an involutive quantaloid Q", leaving the notation for the involution understood. When both Q and 71 are involutive quantaloids, then we say that F: Q [right arrow] 71 is a homomorphism of involutive quantaloids when it is a homomorphism such that F([f.sup.[omicron]]) = [(Ff).sup.[omicron]].

Whenever a morphism /: A [right arrow] B in a quantaloid (or in a locally ordered category, for that matter) is supposed to be a left adjoint, we write [f.sup.*] for its right adjoint. A symmetric left adjoint in an involutive quantaloid Q is a left adjoint whose right adjoint is its involute: [f.sup.*] = [f.sup.[omicron]]. Precisely as we write Map(Q) for the category of left adjoints in Q, we write SymMap(Q) for the category of symmetric left adjoints.

A category A enriched in a quantaloid Q consists of a set [A.sub.0] of objects, each x [member of] [A.sub.0] having a type ta [member of] [Q.sub.0], and for any x, y [member of] [A.sub.0] there is a hom- arrow A(y,x): tx[right arrow]ty in Q, subject to associativity and unit requirements: A(z,y) [omicron] A(y,x) [less than or equal to] A(z,x) and [l.sub.tx] [less than or equal to] A(x,x) for all x,y,z [member of] [A.sub.0]. A functor F: A[right arrow]B between such Q-categories is an object-map x [??] Fx such that tx = t(Fx) and A(y, x) [less than or equal to] B(Fy, Fx) for all x, y [member of] A. Such a functor is smaller than a functor G: A[right arrow]B if [l.sub.tx] [less than or equal to] B(Fx, Gx) for every x [member of] A. With obvious composition one gets a locally ordered 2-category Cat(Q) of Q-categories and functors.

For two objects x,y [member of] A, the hom-arrows A(y, x) and A(x, y) go in opposite directions. Hence, to formulate a notion of "symmetry" for Q-categories, it is far too strong to require A(y,x) = A(x,y). Instead, at least for involutive quantaloids, a Q-category A is symmetric when A(x, y) = A[(y, x).sup.[omicron]] for every two objects x, y [member of] A [Betti and Walters, 1982]. We shall write SymCat(Q) for the full sub-2-category of Cat(Q) determined by the symmetric Q-categories (in which the local order is in fact symmetric, but not anti-symmetric).

A distributor [PHI]: A[??]B between Q-categories consists of arrows [PHI](y,x): tx[right arrow]ty in Q, one for each (x,y) [member of] [A.sub.0] x [B.sub.0], subject to two action requirements: B(y', y) [omicron] [PHI] (y, x) [less than or equal to] [PHI] (y', x) and [PHI] (y, x) [omicron] A(x, x') [less than or equal to] [PHI] (y, x') for all y, y' [member of] [B.sub.0] and x, x' [member of] [A.sub.0]. The composite of such a distributor with another Y: B[??]C is written as [PSI] [cross product] [PHI]: A[??]C, and its elements are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for x [member of] [A.sub.0] and z [member of] [C.sub.0]. Parallel distributors can be compared elementwise, and in fact one gets a (large) quantaloid Dist(Q) of Q-categories and distributors. Each functor F: A[right arrow]B determines an adjoint pair of distributors: B(--, F--): A[??]B, with elements B(y,Fx) for (x,y) [member of] [A.sub.0] x [B.sub.0], is left adjoint to B(F--, --): B[??]A in the quantaloid Dist(Q). These distributors are said to be represented by F. More generally, a (necessarily left adjoint) distributor [PHI}: A[??]B is representable if there exists a (necessarily essentially unique) functor F: A[right arrow]b such that [PHI] = B(--, F--). This amounts to a 2-functor

Cat(Q)[right arrow]Map(Dist(Q)): (f: A[right arrow]b) [??] (b(-,F-): A[right arrow]B). (1)

We write SymDist(Q) for the full subquantaloid of Dist(Q) determined by the symmetric Q-categories. It is easily verified that the involution f [??] f [omicron] on the base quantaloid Q extends to the quantaloid SymDist(Q): explicitly, if O: A[??]b is a distributor between symmetric Q-categories, then so is [PHI][omicron]: B[??]A, with elements [[PHI].sup.[omicron]](a,b) := [PHI][(b,a).sup.[omicron]]. And if F: A[right arrow]b is a functor between symmetric Q -categories, then the left adjoint distributor represented by F has the particular feature that it is a symmetric left adjoint in SymDist(Q). That is to say, the functor in (1) restricts to the symmetric situation as

SymCat(Q)[right arrow]SymMap(SymDist(Q)): (f: A[right arrow]b) [??] (b(-,F-): A[??]b), (2)

obviously giving a commutative diagram

[ILLUSTRATION OMITTED]

The full embedding SymCat(Q) [omicron][??] Cat(Q) has a right adjoint functor:

[ILLUSTRATION OMITTED]

This symmetrisation sends a Q-category A to the symmetric Q-category [A.sub.s] whose objects (and types) are those of A, but for any two objects x, y the hom-arrow is [A.sub.s] (y,x) := A(y,x) [conjunction] A[(x,y).sup.[omicron]]. A functor F: A[right arrow]B is sent to [F.sub.s]: [A.sub.s][right arrow][B.sub.s]: a [??] Fa. It is a result of [Heymans and Stubbe, 2011] that the inclusion SymMap(SymDist(Q)) [right arrow] Map(Dist(Q)) admits a right adjoint that makes the diagram

[ILLUSTRATION OMITTED]

commute if and only if, for each family [([f.sub.i]: X[right arrow][X.sub.i], [g.sub.i]: [X.sub.i][right arrow]X).sub.i[member of]I] of morphisms in Q,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Such an involutive quantaloid Q is said to be Cauchy-bilateral. We will encounter examples of Cauchy-bilateral quantaloids further on in this paper.

Presheaves, Cauchy-completion and symmetric-completion

A (contravariant) presheaf on A is a distributor into A whose domain is a one-object category with an identity hom-arrow. Writing [*.sub.X] for the one-object Q-category whose single object * has type X [member of] [Q.sub.0] and whose single hom-arrow is the identity [l.sub.x], a presheaf is then typically written as [??]: [*.sub.X] [??]A. The set of presheaves on A is written P(A): it is a Q-category when we define that t[??]: [*.sub.X] [??]A) = X and P(A)([psi],[phi]) = [[psi],[phi]] (this being a lifting in the quantaloid Dist(Q), i.e. the value at p of the right adjoint to composition with [psi]). The Yoneda embedding of A into P(A) is the fully faithful functor of Q-enriched categories [Y.sub.A] : A[right arrow]V(A) that sends a [member of] A to the representable presheaf A(--, a): [*.sub.ta] [??]A. In fact, this procedure extends to a functor V : Cat(Q) [right arrow]Cat(Q), which is the free cocompletion KZ-doctrine on the category of Q-categories. (A covariant presheaf on A is a distributor <p: A [??] [*.sub.X]; they are not of much importance in this paper.)

A Q-category A is said to be Cauchy complete when each left adjoint distributor with codomain A is represented by a functor [Lawvere, 1973], that is, when for each Q-category B the functor in (1) determines an equivalence of ordered sets

Cat(Q)(B,A) [equivalent] Map(Dist(Q))(B,A).

This clearly implies that the functor in (1) restricts to a biequivalence of locally ordered 2-categories between [Cat.sub.cc] (Q), the full subcategory of Cat(Q) determined by the Cauchy complete Q-categories, and Map(Dist(Q)). Moreover, the full inclusion of [Cat.sub.cc] (Q) in Cat(Q) admits a left adjoint:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

That is to say, each Q-category A has a Cauchy completion [A.sub.cc]: it is the full subcategory of the presheaf category P(A) whose objects are the left adjoint presheaves on A. The Yoneda embedding [Y.sub.A]: A[right arrow]V(A) factors through [A.sub.cc], and the distributor induced by [Y.sub.A]: A[right arrow][A.sub.cc] turns out to be an isomorphism in Dist(Q). Therefore the quantaloid Dist(Q) is equivalent to its full subquantaloid [Dist.sub.cc] (Q) whose objects are the Cauchy complete Q-categories. As a result, there is an equivalence of locally ordered 2-categories

[Cat.sub.cc](Q) [equivalent] Map([Dist.sub.cc](Q)) [equivalent] Map(Dist(Q)). (4)

The Cauchy completion can of course be applied to a symmetric Q-category (assuming that Q is involutive), but the resulting Cauchy complete category need not be symmetric anymore: the functor [(-).sub.cc]: Cat(Q) [right arrow]Cat(Q) does not restrict to SymCat(Q) in general. However, its very definition suggests the following modification [Heymans and Stubbe, 2011]: a symmetric Q-category A is symmetrically complete if, for any symmetric Q-category B, the functor in (2) determines an equivalence of symmetrically ordered sets

SymCat(Q)(B, A) [equivalent] SymMap(SymDist(Q))(B, A).

This implies that the functor in (2) restricts to a biequivalence between [SymCat.sub.sc](Q) , the full subcategory of SymCat(Q) of its symmetrically complete objects, and SymMap(SymDist(Q)). Moreover, the full inclusion of [SymCat.sub.sc](Q) in SymCat(Q) admits a left adjoint:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Explicitly, for a symmetric Q-category A, its symmetric completion [A.sub.sc] is the full subcategory of the Cauch completion [A.sub.cc] (and thus also a full subcategory of the presheaf category P(A)) determined by the symmetric left adjoint presheaves. For similar reasons as above, there is an equivalence of involutive quantaloids between SymDist(Q) and its full subquantaloid [SymDist.sub.sc](Q) of symmetrically complete Q-categories, and therefore also an equivalence of categories

[SymCat.sub.sc](Q) [equivalent] SymMap([SymDist.sub.sc](Q)) [equivalent] SymMap(SymDist(Q)). (6)

Importantly, a result of [Heymans and Stubbe, 2011] says that, if Q is a Cauchy-bilateral quantaloid, then the symmetric-completion and the Cauchy-completion of any symmetric Q-category coincide, and the symmetrisation of a Cauchy complete Q-category is symmetrically complete. In fact, there is a distributive law of the monad [(-).sub.cc]: Cat(Q)[right arrow]Cat(Q) over the comonad [(-).sub.s]: Cat(Q)[right arrow]Cat(Q). This means in particular that there is a commutative diagram of adjunctions as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The equal signs in this diagram are the equivalences of (4) and (6); the bottom row is fully included in the top row, and can be obtained from it by 'symmetrisation'.

Universal constructions

An idempotent in a quantaloid Q is, of course, an endomorphism e: A[right arrow] A such that [e.sup.2] = e. Such an idempotent is said to split in Q when there exists a diagram

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

in Q. If [epsilon] is a class of idempotents in a quantaloid Q, then we write [Q.sub.[epsilon]] for the quantaloid obtained by splitting the idempotents in [epsilon]. An explicit description goes as follows: the objects of [Q.sub.[epsilon]] are the elements of [epsilon], and [Q.sub.[epsilon]] (e, f) = {x: A [right arrow]B \ fox = x = x [omicron] e} whenever e: A [right arrow] A and f: B [right arrow] B are in [epsilon]. Composition and local suprema in QE are as in Q, but the identity on an idempotent e is, obviously, e: e [right arrow] e itself. If all identities in Q are in [epsilon], then there is a fully faithful homomorphism of quantaloids

I: Q[right arrow][Q.sub.[epsilon]]: (x: A[right arrow]b) [??] (x: [1.sub.A][right arrow][1.sub.B])

which is the universal splitting in Q of idempotents in [epsilon]. Spelled out, this means that if F: Q [right arrow] 71 is a homomorphism of quantaloids, and the images of all idempotents in [epsilon] split in R, then there is an essentially unique homomorphism [bar.F]: [Q.sub.[epsilon]] [right arrow]R such that [bar.F] [omicron] I = F. Moreover, if F is fully faithful then so is [bar.F].

When Q is an involutive quantaloid, then we say that an idempotent e: A [right arrow] A in Q is symmetric when [e.sup.[omicron]] = e. It is straightforward that QE is then involutive too: the involute of x [member of] [Q.sub.[epsilon]] (e, f) is computed as in Q, for the symmetry of e and f make sure that x[omicron] [member of] [Q.sub.[epsilon]] (f, e). As before, it is surely the case that, whenever all identities in Q are in [epsilon], the involutive quantaloid QE has a universal property for the splitting of idempotents. Noting however that I: Q[right arrow] [Q.sub.[epsilon]] preserves the involution, we can point out a slightly more subtle universal property. Say that the splitting in the diagram in (7) is symmetric when q = [p.sup.[omicron]]. If F: Q[right arrow]R is a homomorphism of involutive quantaloids and the images of all idempotents in [epsilon] split symmetrically in R, then there is an essentially unique homomorphism [bar.F]: [Q.sub.[epsilon]] [right arrow] R of involutive quantaloids such that [bar.F] [omicron] I = F; in other words, if F preserves the involution then so does [bar.F]. And again, if F is fully faithful then so is [bar.F].

In any quantaloid Q, products and sums are the same thing, so they are usually referred to as direct sums. We write A = [[direct sum].sub.i[member of]I] [A.sub.i] for the direct sum of a family [([A.sub.i]).sub.i] of objects of Q, with injections [s.sub.i]: [A.sub.i] [right arrow]A and projections [p.sub.i]: A[right arrow][A.sub.i]; in fact, for A to be the direct sum of the [([A.sub.i]).sub.i], it is a necessary and sufficient condition that [p.sub.i] [omicron] [S.sub.j] = [[delta].sub.ij] and [V.sub.i][s.sub.i] [omicron] [p.sub.i] = [1.sub.A]. In these equations, [[delta].sub.ij]: [A.sub.j][right arrow][A.sub.i] is the "Kronecker delta": it is the identity morphism when i = j and the zero morphism otherwise. The universal direct sum completion of a small quantaloid Q exists, and can explicitly be described as the quantaloid Matr(Q) of matrices over Q. An object in Matr(Q) is a Q-typed set, i.e. a set A together with a type function t: A [right arrow] Qq, and a morphism between two such Q-typed sets is a matrix M: A[right arrow]B, i.e. a family M(b,a): ta[right arrow]tb of morphisms in Q, one for each (a,b) [member of] A x B. Of course, matrices can be composed: for M: A[right arrow]B and N: B [right arrow]C we have N [omicron] M: A [right arrow]C with elements

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The identity on a Q-typed set A is the matrix [[DELTA].sub.A]: A [right arrow] A all of whose elements are "Kronecker deltas". With elementwise supremum, this makes Matr(Q) a quantaloid; and whenever Q is involutive, so is Matr(Q) (for elementwise involution). There is a fully faithful homomorphism

J: Q[right arrow]Matr(Q): (f: X[right arrow]y) [right arrow] ((f) : {X}[right arrow]{y})

sending a morphism to the matrix between singletons in the obvious way (which preserves the involution on Q whenever there is one), which is the universal direct sum completion of Q.

Any Q-typed set A determines a Q-category A by putting [A.sub.0] = A and A(a',a) = [[DELTA].sub.A](a',a): this is precisely a discrete Q-category in the sense that the hom-arrow between two different objects is a zero morphism and every endohom-arrow is an identity morphism. A matrix between Q-typed sets is easily seen to be precisely a distributor between discrete Q-categories, so the quantaloid Matr(Q) is precisely the full subquantaloid of Dist(Q) of discrete Q-categories. A discrete Q-category is obviously symmetric, so whenever Q is an involutive quantaloid, Matr(Q) can also be considered as full involutive subcategory of SymDist(Q). Furthermore, a monad in Matr(Q) is exactly a Q-category, and (assuming that Q is involutive) a symmetric monad is a symmetric Q-category. In other words, both Dist(Q) and SymDist(Q) can be constructed from Matr(Q) by splitting a particular class of idempotents:

- Dist(Q) = Matr[(Q).sub.[epsilon]] for E the class of monads in Matr(Q),

- SymDist(Q) = Matr[(Q).sub.[epsilon]] for E the class of symmetric monads in Matr(Q).

Composing the various universal constructions we thus find how Dist(Q) and SymDist(Q) can be considered as completions of Q itself.

For any involutive quantaloid Q' it is a matter of fact that the process of splitting all monads in Q' can be broken down in two steps: first split all symmetric monads in Q', then split all anti-symmetric monads in the thusly obtained quantaloid. (A monad m: X[right arrow]X in an involutive quantaloid is said to be antisymmetric when m [conjunction] [m.sup.[omicron]] = [1.sub.X].) Applying this to Q' = Matr(Q) for a small involutive quantaloid Q, this exhibits how Dist(Q) is also a completion of SymDist(Q).

All this goes to show that both Dist(Q) and SymDist(Q) lead a "double life". On the one hand, they are concretely constructed quantaloids: their objects are (symmetric) Q-categories, and their morphisms are distributors. This makes it possible to compute with individual objects and morphisms of Dist(Q) (or SymDist(Q)). But on the other hand, Dist(Q) and SymDist(Q) are universal constructions on Q: first add all direct sums to Q, then split either all monads or only the symmetric ones. These universal properties thus say something about the collection of all objects and morphisms of Dist(Q) or SymDist(Q). The first approach is clearly rooted in the theory of quantaloid-enriched categories, whereas the second approach is close in spirit to allegory theory. Indeed, quoting P. Johnstone [2002, p. 138], "many allegories of interest may be generated by idempotent-splitting processes from quite small full sub-allegories". Of course, Dist(Q) or SymDist(Q) need not be allegories (neither of them is necessarily modular, see further), but they are both generated by universal processes from a quite small full sub-quantaloid, namely from Q itself.

Orders and sheaves over a base quantaloid

We now have everything ready to state the central definitions with which we shall work in this paper. First we recall a definition first given in [Stubbe, 2005b]:

Definition 2.1 Given a small quantaloid Q and a set E of idempotents in Q, we define

Ord(Q,[epsilon]) := [Cat.sub.cc]([Q.sub.[epsilon]]) and ldl(Q, [epsilon]) := [Dist.sub.cc]([Q.sub.[epsilon]])

for, respectively, the locally ordered 2-category of (Q, [epsilon])-orders and order functions, and the quantaloid of (Q, [epsilon])-orders and ideal relations. If E is taken to be the set of all idempotents in Q, then we write [Q.sub.si] instead of [Q.sub.[epsilon]], Ord(Q) instead of Ord(Q, [epsilon]), and Idl(Q) instead of Idl(Q, [epsilon]); we then simply speak of Q-orders (and order functions and ideal relations).

Next we present a new definition, intended as "symmetric" version of the previous definition. Because the term "symmetric Q-order" is technically inadequate (it suggests a Q-order with a symmetric hom, quod non), and the term "Q-set" already means something related-but-different in the literature (see e.g. [Higgs, 1973; Fourman and Scott, 1979; Borceux, 1994; Mulvey and Nawaz, 1995; Gylys, 2001; Johnstone, 2002; and others]), we opt to speak of "Q-sheaves":

Definition 2.2 Given a small involutive quantaloid Q and a set [epsilon] of symmetric idempotents in Q, we define

Sh(Q, [epsilon]) := [SymCat.sub.sc]([Q.sub.[epsilon]]) and Rel(Q, [epsilon]) := [SymDist.sub.sc] ([Q.sub.[epsilon]])

for, respectively, the category of (Q, [epsilon])-sheaves and functions, and the quantaloid of (Q, [epsilon])-sheaves and relations. If [epsilon] is taken to be the set of all symmetric idempotents in Q, then we write [Q.sub.ssi] instead of [Q.sub.[epsilon]], Sh(Q) instead of Sh(Q, [epsilon]), and Rel(Q) instead of Rel(Q, [epsilon]); we then simply speak of Q-sheaves (and functions and relations).

We shall explain at the end of Section 3 how, for so-called small quantaloids of closed cribles, the symmetry condition in the above definition is in fact equivalent to an appropriate discreteness condition.

From the general theory on (symmetric) Q-categories that we explained in the previous subsections, we can now conclude that:

Proposition 2.3 For any small quantaloid Q and any set [epsilon] of idempotents in Q, there is a biequivalence oflocally ordered 2-categories

Ord(Q, [epsilon]) [??] Map(ldl(q, [epsilon])): (F: A[right arrow]b) [right arrow] (b(-,F-): A[??]b).

For any small involutive quantaloid Q and any set [epsilon] of symmetric idempotents in Q, there is an equivalence of categories

Sh(Q, [epsilon]) [??] SymMap(Rel(G, [epsilon])): (F: A[right arrow]b) [right arrow] (b(-,F-): A[??]b).

If Q is an involutive quantaloid and [epsilon] a set of symmetric idempotents such that [Q.sub.[epsilon]] is Cauchy-bilateral, then both squares in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

commute, and the bottom row is obtained by "symmetrising" the top row.

Here is yet another result of the general theory of Q-categories:

Proposition 2.4 For any small (resp. involutive) quantaloid Q and any set [epsilon] of (resp. symmetric) idempotents in Q, there is an equivalence of (resp. involutive) quantaloids

Idl(Q, [epsilon]) [equivalent] Dist([Q.sub.[epsilon]]),resp. Rel(Q, [epsilon]) [equivalent] SymDist([Q.sub.[epsilon]]).

This proposition explains an important subtlety: each (Q, [epsilon])-order (or (Q, [epsilon])-sheaf) is Morita equivalent to a (symmetric) [Q.sub.[epsilon]]-category. This fact has often been used (implicitly) to forget about Cauchy completeness altogether: several definitions of "sheaf on an involutive quantaloid" that can be found in the literature, amount (in one form or another) to stating that a sheaf is a symmetric category, and a morphism of sheaves is a left adjoint distributor. (An example that springs to mind, is the formalism of projection matrices, on which we shall comment in more detail in Section 4.) However, we have deliberately opted to include the requirement of Cauchy (or symmetric) completeness in the definition of "sheaf" on a quantaloid Q, for it expresses precisely the "gluing condition" that one expects of such a notion (as well illustrated by [Walters, 1981]). But of course it comes in handy that, modulo Morita equivalence, this completeness can be swiped under the carpet.

The whole of Section 3 is devoted to showing that the topos of sheaves on a site (C, J) is equivalent to Sh(Q) when Q = R(C, J) is the small quantaloid of closed cribles in (C, J).

3 Sheaves on a site

For any small involutive quantaloid Q we stated in Definition 2.2 and Proposition 2.3 that

Sh(Q) := [SymCat.sub.sc]([Q.sub.ssi]) [equivalent] SymMap(SymDist([Q.sub.ssi])).

Walters [1982] showed that, for a small site (C, J),

Sh(C, J) [equivalent] [SymCat.sub.cc](R(C, J)) [equivalent] Map(SymDist(R(C, J))),

where R(C, J) is the so-called small quantaloid of closed cribles (which Walters originally called the bicategory of relations) constructed from (C, J). In this section we shall show that sheaves on a small site (C, J) (in the topos-theoretic sense) correspond with sheaves on the small involutive quantaloid R(C, J) (in the sense of our Definition 2.2).

More precisely, we shall prove that, if Q = R(C, J) is a small quantaloid of closed cribles, then for any set [epsilon] of symmetric idempotents in Q containing the identities, SymDist(Q) and SymDist([Q.sub.[epsilon]]) are equivalent modular quantaloids; and because each left adjoint in a modular quantaloid is necessarily a symmetric left adjoint, it follows that Sh(Q, [epsilon]) is equivalent to Map(SymDist(Q)), which in turn is equivalent to Sh(C, J) by Walters' [1982] result. To give our proof, we shall use the axiomatic description of R(C, J) due to [Heymans and Stubbe, 2012], for it allows us to prove our claim via elementary computations in involutive quantaloids, much in the line of Freyd and Scedrov's [1990] work on allegories (see also [Johnstone, 2002]). In the next subsection we recall the necessary results from our earlier work.

Axioms for a small quantaloid of closed cribles

First we recall some definitions:

Definition 3.1 A quantaloid Q is:

1. locally localic if, for all objects X and Y, Q(X, Y) is a locale,

2. map-discrete if, for any left adjoints f:X[right arrow]Y and g: X[right arrow]Y in Q, f < g implies f= g,

3. weakly tabular if for every q: X[right arrow]Y in Q,

q = V {[fg.sup.*] | (f,g) : X[??]Y is a span of left adjoints such that [fg.sup.*] [less than or equal to] q},

4. map-tabular if for every q: X [right arrow]Y in Q there is a span (f,g): X[??]Y of left adjoints in Q such that [fg.sup.*] = q and [f.sup.*]f [conjunction] [g.sup.*]g = [1.sub.dom(f)],

5. weakly modular if for every pair of spans of left adjoints in Q, say (f,g): X [??] Y and (m,n): X[??]Y, we have [fg.sup.*] [conjunction] [mn.sup.*] [less than or equal to] f([g.sup.*]n [conjunction] [f.sup.*]m)[n.sup.*],

6. tabular if it is involutive and if for every q: X[right arrow]Y in Q there exists a span (f,g): X[??]Y of left adjoints in Q such that [fg.sup.[omicron]] = q and [f.sup.[omicron]]f [conjunction] [g.sup.[omicron]]g = [l.sub.dom(f)],

7. modular if it is involutive and if for any f: X[right arrow]Y, g: Y[right arrow]Z andh: X[right arrow]Z in Q we have gf [conjunction] h [less than or equal to] g(f A [g.sup.[omicron]]h) (or equivalently, gf [conjunction] h [less than or equal to] (g A [hf.sup.[omicron]]) f).

The notions of modularity (1) and tabularity are cited from Freyd and Scedrov [1990] who give them in the context of allegories (2). Weak modularity, weak tabularity and map-tabularity were introduced in [Heymans and Stubbe, 2012] with the specific aim to axiomatise small quantaloids of closed cribles.

There are many useful relations between several of these notions; we recall some of these in the next lemma.

Lemma 3.2 1. In any modular quantaloid, all left adjoints are symmetric left adjoints.

2. Any modular quantaloid is map-discrete.

3. Any locally localic and modular quantaloid is Cauchy-bilateral.

4. A small quantaloid Q is weakly tabular if and only if Dist(Q) is map-tabular.

5. A small quantaloid Q is locally localic and modular if and only if Matr(Q) is modular.

The first two statements in the above lemma appear in [Freyd and Scedrov, 1990; Johnstone, 2002], the third is quoted from [Heymans and Stubbe, 2011], and the two other statements come from [Heymans and Stubbe, 2012]. Also the following result appears in the latter reference.

Theorem 3.3 For a small quantaloid Q, the following conditions are equivalent:

1. Q is a small quantaloid of closed cribles, i.e. there exists a small site such that Q[equivalent]R(C, J),

2. putting, for X [member of] Map(Q),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defines a Grothendieck topology J on Map(Q) for which Q [congruent to] R(Map(Q), J),

3. Q is locally localic, map-discrete, weakly tabular and weakly modular. In this case, Q carries an involution, sending q: Y[right arrow]X to

[q.sup.[omicron]] := V {[gf.sup.*] | (f,g)' Y[??]X is a span of left adjoints such that [fg.sup.*] [less than or equal to] q}, which makes Q also modular.

Splitting symmetric idempotents

In this subsection we study the properties of the involutive quantaloid SymDist(Q) when Q is a small quantaloid of closed cribles.

First we point out two useful conditions to determine whether a (small or large) involutive quantaloid Q' has symmetric splittings for its symmetric idempotents. The first lemma can be found in [Freyd and Scedrov, 1990, 2.162; Johnstone, 2002, Lemma A3.3.3] and the second is a corollary of [Freyd and Scedrov, 1990, 2.166 and 2.169; Johnstone, 2002, A3.3.6 and A3.3.12]. For completeness' sake we shall give proofs here too, specifically adapted to the situation at hand.

Lemma 3.4 If Q' is a modular quantaloid, then each splitting of a symmetric idempotent is necessarily a symmetric splitting.

Proof: First observe that for any f: X[right arrow]Y in a modular Q' we always have / [less than or equal to] [ff.sup.[omicron]]f: because f = [l.sub.Y]f [conjunction] f [less than or equal to] ([1.sub.Y] [conjunction] [ff.sup.[omicron]])f [less than or equal to] [ff.sup.[omicron]]f. Now suppose that e: A[right arrow]A, p: A[right arrow]B and q: B[right arrow]A satisfy e = [e.sup.2] = [e.sup.[omicron]] = qp and pq = [l.sub.B] in Q'. Then it follows that [q.sup.[omicron]] = [pqq.sup.[omicron]] [less than or equal to] [pp.sup.[omicron]][pqq.sup.[omicron]] = [pp.sup.[omicron]][q.sup.[omicron]] = p[(qp).sup.[omicron]] = [pe.sup.[omicron]] = pe = p, and similarly [p.sup.[omicron]] [less than or equal to] q.

Lemma 3.5 If Q' is a modular and tabular quantaloid in which all symmetric monads (3) split, then all symmetric idempotents in Q' have a (necessarily symmetric) splitting.

Proof: Let e: A [right arrow] A be a symmetric idempotent in Q': we shall exhibit a splitting. To that end, first consider a tabulation (f,g) of e [conjunction] [1.sub.A]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, f and g are left adjoints in Q' such that [fg.sup.[omicron]] = [1.sub.A] [conjunction] e and [g.sup.[omicron]]g [conjunction] [f.sup.[omicron]]f = [1.sub.B]. Because Q' is modular, we know moreover that f [??] [f.sup.[omicron]] and g [??] [g.sup.[omicron]]. It is useful to point out that g [less than or equal to] ef and f [less than or equal to] eg follow from these assumptions, and that, in turn, this implies that g [less than or equal to] eg and eg = ef.

Now define t := [(eg).sup.[omicron]](eg) = [g.sup.[omicron]]eg'- B[right arrow]B. Then clearly [t.sup.[omicron]] = t holds; it is furthermore easy to check that tt = [g.sup.[omicron]][egg.sup.[omicron]]eg < [g.sup.[omicron]]e[1.sub.A] eg = [g.sup.[omicron]]eg = t; and t = [(eg).sup.[omicron]]eg [greater than or equal to] g[omicron]g [greater than or equal to] [1.sub.B] follows from inequalities pointed out above. In sum, this says that t: B [right arrow] B is a symmetric monad. By assumption we can split t: there is a diagram

[ILLUSTRATION OMITTED]

such that t = [h.sup.[omicron]]h and [hh.sup.[omicron]] = [1.sub.[B.sup.t]] (where, again by modularity, h [??] [h.sup.[omicron]]).

Next, consider the diagram

[ILLUSTRATION OMITTED]

in which, by definition of t, we have t = [(eg).sup.[omicron]](eg). Using modularity of Q' and the tabulation (f, g) of [1.sub.A] [conjunction] e, we can compute that e = e[1.sub.A]e [conjunction] e [less than or equal to] e([1.sub.A] [conjunction] [e.sup.[omicron]][ee.sup.[omicron]])e = e([1.sub.A] [conjunction] e)e = e([fg.sup.[omicron]])e = (ef)[(eg).sup.[omicron]] = (eg)[(eg).sup.[omicron]]. But (eg)[(eg).sup.[omicron]] = [egg.sup.[omicron]]e [less than or equal to] ee = e follows immediately from g H g[omicron], hence we obtain e = (eg)[(eg).sup.[omicron]].

Composing these two diagrams produces a splitting in Q' for the symmetric idempotent e: A [right arrow] A, as required.

For any small involutive quantaloid Q, SymDist(Q) is a quantaloid in which all symmetric monads split: simply because it is the universal splitting of symmetric monads in Matr(Q). Below we shall furthermore prove that, whenever Q is a small quantaloid of closed cribles, SymDist(Q) is modular and tabular too.

Lemma 3.6 If Q is a locally localic quantaloid and E is a collection of idempotents in Q, then Qe is locally localic too.

Proof: If p: L[right arrow]L is an idempotent sup-morphism on a complete lattice, then p(L) [subset or equal to] L is a complete lattice too, with the same suprema as in L, but with p(x) [conjunction]' p(y) := p(p(x) A p(y)) as binary infimum and p(T) as emtpy infimum (i.e. top element). A simple computation shows that, if L is a locale, then so is p{L). This applies to [e.sub.2] [omicron] - [omicron] [e.sub.1]: Q(X, Y) [right arrow] Q(X, Y) for idempotents [e.sub.1]: X[right arrow]X and [e.sub.2]: Y[right arrow] Y in [epsilon], to show that [Q.sub.[epsilon]] {[e.sub.1], [e.sub.2]) is a locale whenever Q(X, Y) is; hence [Q.sub.[epsilon]] is locally localic whenever Q is.

Lemma 3.7 If Q isa modular quantaloid and E is a collection of symmetric idempotents in Q, then QE is modular too.

Proof: Local suprema, composition and involution in [Q.sub.[epsilon]] are the same as in Q. As pointed out in the above proof, the infimum of f,g: [e.sub.1] [right arrow][e.sub.2] in [Q.sub.[epsilon]] is f [conjunction]' g := [e.sub.2] (f [conjunction] g) [e.sub.1], but thanks to the modular law it is easily seen that

f [conjunction] g = [e.sub.2]f [conjunction] [ge.sub.1] [less than or equal to] e2([fe.sup.[omicron].sub.1] [conjunction] [e.sup.[omicron].sub.2]g)[e.sub.1] = [e.sub.2](f [conjunction] g)[e.sub.1],

whereas [e.sub.2](f [conjunction] g)[e.sub.1] [less than or equal to] f [conjunction] g is always valid, hence in this case the local binary infima in [Q.sub.[epsilon]] are the same as in Q. Thus it follows that [Q.sub.[epsilon]] is modular whenever Q is.

Proposition 3.8 If Q is a small, locally localic, modular quantaloid, then SymDist(Q) is modular.

Proof: Matr(Q) is modular by Lemma 3.2, so SymDist(Q) = [(Matr(Q)).sub.[epsilon]], with [epsilon] the collection of symmetric monads in Matr(Q), is modular too by Lemma 3.7.

Proposition 3.9 If Q is a small, weakly tabular, Cauchy-bilateral quantaloid, then SymDist(Q) is tabular.

Proof: From [Heymans and Stubbe, 2012, Proposition 3.5] we recall that a small quantaloid Q is weakly tabular if and only if Dist(Q) is map-tabular. The proof for the necessity goes as follows: Suppose that [PHI]: A[??]B is a distributor. We can assume without loss of generality that A and B are Cauchy complete, because every Q-category is isomorphic to its Cauchy completion in Dist(Q). Now define the Q-category R to be the full subcategory of A x B whose objects are those (a, b) [member of] A x B for which [1.sub.ta] [less than or equal to] [PHI] (a, b), and write T (resp. S) for the composition of the inclusion R [right arrow] A x B with the projection of A x B onto A (resp. onto B). By construction we then have B(S--, S--) A A(T--, T--) = R; and, relying on the weak tabularity of Q and the Cauchy completeness of A and B, a lenghty computation shows that [PHI] = A(--, T--) [cross product] B(S--, --). That is to say, the left adjoints A(-,T-): R[??]A and B(-,S-): R[??]B in Dist(Q) provide for a map-tabulation of [PHI]: A[??]B.

We now modify this proof to suit our needs. ForanyO: B[??]A in SymDist(Q) we must find [SIGMA]: R[??]A and [THETA]: R[??]B in SymDist(Q) such that [SIGMA] [cross product] [[THETA].sup.[omicron]] = [PHI] and [[SIGMA].sup.[omicron]] [cross product] [SIGMA] [conjunction] [[THETA].sup.[omicron]] [cross product] [THETA] = R. If Q is Cauchy- biateral then the Cauchy completion of a symmetric Q -category is again symmetric, hence any symmetric Q-category is isomorphic to its Cauchy completion in SymDist(Q) (and not merely in Dist(Q)). Therefore we may still suppose that A and B are Cauchy complete. Referring to the above, the category R is clearly symmetric whenever A and B are, and the left adjoint distributors represented by the functors S: R[right arrow]B and T: R[right arrow] A are evidently symmetric left adjoints. Thus the result follows.

In view of Theorem 3.3 we may now conclude from the above:

Theorem 3.10 If Q is a small quantaloid of closed cribles, then SymDist(Q) is a modular and tabular quantaloid in which all symmetric idempotents split symmetrically.

Change of base

This subsection is devoted to the proof of the fact that, when Q is a small quantaloid of closed cribles and [epsilon] is a class of symmetric idempotents in Q containing all identities, then also [Q.sub.[epsilon]] is a small quantaloid of closed cribles, and the involutive quantaloids SymDist(Q) and SymDist([Q.sub.[epsilon]]) are equivalent. To tackle this problem, we study the "change of base" homomorphism from SymDist(Q) to SymDist([Q.sub.[epsilon]]) which is determined by the universal property of splitting symmetric idempotents. Let us first recall the appropriate terminology.

Let Q and Q' be small involutive quantaloids and F: Q [right arrow] Q! be a homomorphism that preserves the involution. It is easily seen that a symmetric Q-category A determines a symmetric Q'-category FA by putting:

--objects: [(FA).sub.0] = [A.sub.0] with types [t.sub.FA]a = F(ta) in [Q'.sub.0],

--hom-arrows: (FA)(a',a) = F(A(a',a)) for all objects a,a'.

Similarly for distributors, and F so determines a homomorphism [bar.F]: SymDist(Q) [right arrow]SymDist(Q') of involutive quantaloids that makes the diagram

[ILLUSTRATION OMITTED] (8)

commute: [bar.F] is the change of base homomorphism induced by F. (Of course, I denotes the canonical inclusion of Q in SymDist(Q), and similarly for I'.)

Now we recall a necessary and sufficient condition for the "change of base" induced by some F: Q [right arrow] Q! to be an equivalence. As it is straightforward to verify that F: Q [right arrow] Q' is fully faithful if and only if the change of base [bar.F] is fully faithful, we need to take a closer look at the essential surjectivity on objects of [bar.F].

Lemma 3.11 Let F: Q[right arrow]Q' be a homomorphism of small involutive quantaloids. The change of base F: SymDist(Q)[right arrow]SymDist(Q/) is an equivalence of involutive quantaloids ifand only ifthere exists a fully faithful homomorphism of involutive quantaloids G: Q! [right arrow] SymDist(Q) making the diagram below essentially commutative:

[ILLUSTRATION OMITTED]

Proof: First suppose that [bar.F] is an equivalence. Considering the commutative square in (8), the required fully faithful G is obtained by composing I' with the pseudo-inverse of [bar.F].

Conversely, suppose that a fully faithful G exists such that G [omicron] F [congruent to] I. Because G and I are fully faithful, so is F, and thus also [bar.F]. Size issues apart, also I and G induce a change of base, and we end up with an essentially commutative diagram

[ILLUSTRATION OMITTED]

The homomorphisms [bar.I], [bar.F] and [bar.G] are fully faithful, because I, F and G are. If we show that [bar.I] is essentially surjective on objects, then it is an equivalence, and hence so is [bar.F].

To see that [bar.I] is indeed essentially surjective on objects, one can do as follows. Given C in SymDist(SymDist(Q)), let us explicitly write the hom-arrow from an object x [member of] C to an object y [member of] C as [[GAMMA].sub.y,x]: [A.sub.x] [??] [A.sub.y]; these morphisms in SymDist(Q) satisfy the conditions that make C a symmetric category: [A.sub.x] [less than or equal to] [[GAMMA].sub.x,x], [V.sub.y[member of]C] [[GAMMA].sub.z,y] [cross product]] [[GAMMA].sub.y,x] [less than or equal to] [[GAMMA].sub.z,x] and [[GAMMA].sub.x,y] = [[GAMMA].sub.y,x][omicron] x (for all x, y [member of] C). With these data, we define a symmetric Q-category A as follows:

--objects: [A.sub.0] := [[??].sub.x[member of]C] [A.sub.x], with inherited types,

--hom-arrows: for u [member of] [A.sub.x] and v [member of] [A.sub.y], A(v, u) := [[GAMMA].sub.y,x] (v, u).

Regarding A now as an object in SymDist(SymDist(Q)), via the change of base 7, we further define a distributor [GAMMA]: [bar.I](A) [??]C by:

--distributor-elements: for u [member of] [A.sub.x] and y [member of] C, [GAMMA](y, u) := [[GAMMA].sub.y,x](--, u).

It is then a fact that [GAMMA] [cross product] [[GAMMA].sup.[omicron]] = C and [[GAMMA].sup.[omicron]] [cross product] [GAMMA] = [bar.I](A). All verifications are long but straightforward computations.

In the exact same situation as in the above lemma, we can sometimes say more:

Lemma 3.12 In the situation of Lemma 3.11, if G is fully faithful and SymDist(Q) is modular and tabular then Q' is modular and weakly tabular.

Proof : Modularity of Q' follows straightforwardly from the modularity of SymDist(Q) and the fully faithful homomorphism Q'[right arrow]SymDist(Q) of involutive quantaloids.

To deduce the weak tabularity of Q' from the tabularity and modularity of SymDist(Q), we first make a helpful observation. Given any O: A[??]B in SymDist(Q),let

[ILLUSTRATION OMITTED]

be a tabulation; then, in particular, [PHI] = [THETA] [cross product] [[SIGMA].sup.[omicron]] and [SIGMA] [??][[SIGMA].sup.[omicron]]. Now consider the family

[(C(-,c): [*.sub.tc][??]c).sub.c[member of]C]

of all representable presheaves on C, each of which is a left adjoint in SymDist(Q). Precomposing both [SIGMA]: C[??]A and [THETA]: C[??]B with these thus gives a family, indexed by the c [member of] C,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of spans of left adjoints in SymDist(Q), whose domains are in the image of the canonical embedding Q [right arrow] SymDist(Q), such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, if [PHI]: A[??]B is in the image of G: Q'[right arrow]SymDist(Q), then-because the image of Q [right arrow] SymDist(Q) is contained in the image of G - it admits a weak tabulation by spans of left adjoints in the image of G. By fully faithfulness of G, Q' is weakly tabular.

The above results apply in particular when Q is a small quantaloid of closed cribles and when we put Q' = [Q.sub.ssi]: they show that splitting the symmetric idempotents in a small quantaloid of closed cribles is "harmless" for the theory of sheaves. In fact, instead of splitting all symmetric idempotents, we can choose to split only those in a class [epsilon] of symmetric idempotents containing all identities.

Theorem 3.13 If Q is a small quantaloid of closed cribles and [epsilon] is a class of symmetric idempotents in Q containing all identities, then also [Q.sub.[epsilon]] is a small quantaloid of closed cribles and the inclusion Q [right arrow] [Q.sub.[epsilon]] induces an equivalence SymDist(Q) [equivalent] SymDist([Q.sub.[epsilon]]) of involutive quantaloids.

Proof: If Q is a small quantaloid of closed cribles, then it is locally localic, hence so is [Q.sub.[epsilon]], by Lemma 3.6. The other results follow from the commutative diagram

[ILLUSTRATION OMITTED]

of fully faithful functors, and the fact that SymDist(Q) is modular and tabular.

Of course, taking [epsilon] to be the class of all symmetric idempotents in Q, we find that [Q.sub.ssi] is a small quantaloid of closed cribles such that SymDist(Q) [equivalent] SymDist([Q.sub.ssi]). But taking [epsilon] to be the class of all symmetric monads in Q, or the class of all symmetric comonads (4), produces other important examples.

Walters' theorem revisited

We now have everything ready to make the following extension to the result of [Walters, 1982]. As is customary, we write Rel(T) for the quantaloid of internal relations in a topos T. The next theorem excludes all confusion with our earlier notation Rel(Q).

Theorem 3.14 For any small site (C, J), any small quantaloid Q [equivalent] R(C, J) and any set [epsilon] of symmetric idempotents in Q containing all identities, we have the following equivalences:

1. Sh(Q,[epsilon]) [equivalent] [SymCat.sub.cc](Q) [equivalent] Sh(C, J),

2. Rel(Q,[epsilon]) [equivalent] SymDist(Q) [equivalent] Rel(Sh(C, J)).

Proof: This proof relies on Walters' [1982, p. 101] theorem that the topos Sh (C, J) is biequivalent to the bicategory [SymCat.sub.cc] (R(C, J)) (Walters' insistence on the term biequivalence stresses the fact that a single morphism in the category Sh(C, J) gets identified with an equivalence class of morphisms in the bicategory [SymCat.sub.cc](R(C, J)) whose homs are symmetric preorders), on Freyd and Scecrov's [1990,2.148] theorem that any tabular allegory A is equivalent (as allegory) to the allegory Rel(Map(A)) of internal relations in the regular category Map (A), and on the particular properties of SymDist(Q), for Q a small quantaloid of closed cribles, that we summarised in Theorems 3.10 and 3.13.

Because Q is a small quantaloid of closed relations, so is [Q.sub.[epsilon]] (see Theorem 3.13); one particular consequence is that, for symmetric categories enriched in either quantaloid, the symmetric completion coincides with the Cauchy completion (cf. Lemma 3.2 and Proposition 2.4). Furthermore, again by Theorem 3.13, SymDist(Q) is equivalent to SymDist([Q.sub.[epsilon]]). All this justifies the following equivalences of involutive quantaloids:

Rel(Q,[epsilon]) := [SymDist.sub.sc]([Q.sub.[epsilon]]) = [SymDist.sub.cc]([Q.sub.[epsilon]]) [equivalent] SymDist([Q.sub.[epsilon]]) [equivalent] SymDist(Q).

By Theorem 3.10 we know that SymDist(Q) is a modular quantaloid, hence so are its equivalents; all left adjoints in the above involutive quantaloids are therefore symmetric left adjoints, by Lemma 3.2. Taking (symmetric) left adjoints therefore produces the following equivalences of categories (or rather, biequivalences of 2-categories which are locally symmetrically ordered):

Sh(Q, [epsilon]) := [Cat.sub.sc]([Q.sub.[epsilon]]) [equivalent] SymMap([SymDist.sub.sc]([Q.sub.[epsilon]])) [equivalent] Map(SymDist(Q)) [equivalent] [SymCat.sub.cc](Q).

Invoking at this point Walters' theorem, this proves (1). But because the involutive quantaloid SymDist(Q) is not only modular but also tabular (see again Theorem 3.10), Freyd and Scedrov's theorem proves it to be equivalent to the involutive quantaloid of internal relations in Map(SymDist(Q)), which in turn proves (2).

The theorem above thus says two things about a small quantaloid of closed cribles Q and a set [epsilon] of symmetric idempotents in Q containing all identities: firstly, that the category Sh(Q, [epsilon]) := [SymCat.sub.sc]([Q.sub.[epsilon]]) is the category of sheaves on a site; secondly, that this category Sh (Q, [epsilon]) admits, up to equivalence, the simpler description [SymCat.sub.cc](Q). (And similar for Rel(Q, [epsilon]).) Choosing [epsilon] to be the set of all symmetric idempotents in Q, we find:

Corollary 3.15 If Q isa small quantaloid of closed cribles, then Sh (Q) is a Grothendieck topos and Rel(Q) is its category of relations.

Symmetric vs. discrete

In this subsection we wish to make a remark on the symmetry axiom that we used in Definition 2.2 of Q-sheaves. In any locally ordered category K, an object D is said to be discrete when, for any other object X [member of] K, the order K(X, D) is symmetric. In [Heymans and Stubbe, 2012] we showed that, for a Cauchy-bilateral quantaloid Q, every symmetric and Cauchy complete Q-category is a discrete object of [Cat.sub.cc](Q). In general the converse need not hold, but:

Proposition 3.16 If Q is a small quantaloid of closed cribles, then a Cauchy complete Q-category is discrete in [Cat.sub.cc](Q) if and only if it is symmetric.

Proof: Suppose that A is a discrete object in [Cat.sub.cc](Q); we seek to prove that A(y, x) = A[(x, y).sup.[omicron]] for any x, y [member of] A. Relying in particular on the weak tabularity of Q, it is sufficient to show that, for any span (f,g): ty[??]tx of left adjoints in Q,

[fg.sup.*] [less than or equal to] A(x,y) [??] [fg.sup.*] [less than or equal to] A[(y, x).sup.[omicron]].

But, because A is Cauchy complete, for any such span (f, g) we can consider the tensors x [cross product] f and y [cross product] g in A, and writing U = dom(f) = dom(g) we indeed have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the second equivalence is due to the discreteness of A and the last equivalence holds because [f.sup.*] = [f.sup.[omicron]] due to the modularity of Q.

If Q is a small quantaloid of closed cribles, then so is [Q.sub.ssi], and the Cauchy competion and symmetric completion of a symmetric [Q.sub.ssi]-enriched category coincide. Thus we find:

Corollary 3.17 If Q is a small quantaloid of closed cribles, then Sh(Q) is the full subcategory of discrete objects of Ord(Q) and Rel(Q) is the full subquantaloid of discrete objects of Idl(Q).

That is to say, whereas we defined the objects of Sh(Q) as the symmetric objects in Ord(Q), we now find that they are exactly the discrete objects.

4 Grothendieck quantaloids and quantales

In the previous section we showed that, for Q a small quantaloid of closed cribles, Sh(Q) is a Grothendieck topos and Rel(Q) is its category of relations. Given this result, it is a natural to ask whether this is the case for other involutive quantaloids too; and if so, for which ones. Precisely, we wish to find necessary and sufficient conditions on Q for Rel(Q) to be the category of relations in a topos.

Definition 4.1 A small involutive quantaloid Q is called a Grothendieck quantaloid (if Q has only one object we speak of a Grothendieck quantale) if there exists a topos T such that there is an equivalence Rel(T) [equivalent] Rel(Q) of involutive quantaloids.

A sufficient condition on Q is being a small quantaloid of closed cribles. On the other hand, the internal relations in a topos form a modular quantaloid, and Q is a full subquantaloid of Rel(Q), so a necessary condition will be the modularity of Q. To establish a precise necessary-and-sufficient condition, we first point out a connection with projection matrices.

Definition 4.2 If Q is a small involutive quantaloid, then ProjMatr(Q) := Matr[(Q).sub.ssi] is the involutive quantaloid of projection matrices (5).

Straightforwardly extending the terminology for quantales [Resende, 2012], we shall say that a quantaloid Q is stably Gelfand if it is an involutive quantaloid in which [ff.sup.[omicron]]f [less than or equal to] f implies f [less than or equal to] [ff.sup.[omicron]]f for any morphism f: X[right arrow]Y. Any modular quantaloid is trivially stably Gelfand, as seen in the proof of Lemma 3.4.

Lemma 4.3 1. For a stably Gelfand quantaloid Q there is an equivalence ProjMatr(Q) [equivalent] SymDist([Q.sub.ssi]) [equivalent] Rel(Q) of involutive quantaloids.

2. If Q is a small quantaloid of closed cribles, then ProjMatr(Q) and SymDist(Q) are equivalent involutive quantaloids.

3. A small involutive quantaloid Q is a Grothendieck quantaloid ifand only ifthere exists a topos T such that there is an equivalence Rel(T) [equivalent] ProjMatr(Q) of involutive quantaloids.

Proof: (1) Let P: X[right arrow]X be a symmetric idempotent in Matr(Q); that is to say, X is a Q-typed set, and P is a collection of Q-morphisms P(x', x) : tx[right arrow]tx', one for each (x, x') [member of] X x X, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From this it is clear that P(x,x') [omicron] P(x,x')[omicron] [omicron] P(x, x') [less than or equal to] P(x, x'), so that by hypothesis the converse inequality holds too. The computation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then shows that P(x, x) = P(x, x) [omicron] P(x, x) ; and similarly for P(x, x) = P(x, x) [omicron] P(x, x). In other words, each P(x, x) is an object of [Q.sub.ssi], and each P(x, x) is a morphism in [Q.sub.ssi] from P(x', x') to P(x,x). As a consequence, we can define a symmetric [Q.sub.ssi] -category P whose [Q.sub.ssi] -typed object set is X with types tx := P(x, x), and whose hom-arrows are P(x, x') := P(x, x'). Note that the [Q.sub.ssi]-category P is normal in the sense of [Stubbe, 2005b]: all of its endo-hom-arrows are identities. Furthermore, if both P: X[right arrow]X and Q: Y[right arrow]Y are projection matrices, and M: P[right arrow]Q is a morphism in ProjMatr(Q), i.e. a matrix M: X[right arrow] Y such that Q [omicron] M = M = M [omicron] P, then we can define a distributor P[??]Q with elements [PHI](y, x) = M(y, x). In fact, each distributor between P and Q arises in this way. In short, the correspondence P[right arrow]P extends to an equivalence of involutive quantaloids between ProjMatr(Q) and the full involutive subquantaloid of SymDist([Q.sub.ssi]) of the normal symmetric [Q.sub.ssi]-categories (compare with [Stubbe, 2005b, Lemma 6.1]). But furthermore, a long but straightforward computation shows that each symmetric [Q.sub.ssi] -category is Morita equivalent with a normal symmetric [Q.sub.ssi] -category: so SymDist([Q.sub.ssi]) is equivalent to its full involutive subquantaloid of normal objects (compare with [Stubbe, 2005b, Lemma 6.2]). Taken together, all this proves that the correspondence P [right arrow] P extends to an equivalence of involutive quantaloids between ProjMatr(Q) and SymDist([Q.sub.ssi]). Finally, by Proposition 2.4 the latter is furthermore equivalent to Rel(Q) := [SymDist.sub.sc]([Q.sub.ssi]) (as involutive quantaloid).

(2) Holds by Theorem 3.14, taking [epsilon] to be the set of all symmetric idempotents in Q.

(3) If Q is a Grothendieck quantaloid, then Rel(Q) [equivalent] Rel(T) for some topos T, so Q is modular because it is a full involutive subquantaloid of Rel(Q). If, on the other hand, we assume that ProjMatr(Q) [equivalent] Rel(T) for some topos T, then again Q is modular, now because it is a full involutive subquantaloid of ProjMatr(Q). In either case, Q is certainly stably Gelfand, so ProjMatr(Q) [equivalent] Rel(Q) by the first statement in this Lemma, which proves ProjMatr(Q) [equivalent] Rel(T) [equivalent] Rel(Q) in either case.

We shall now recall some notions that [Freyd and Scedrov, 1990, 2.216(1), 2.225] introduced in the context of allegories, but that we adopt here for quantaloids. (In fact, the property that we call 'weak semi-simplicity' was not given a name by Freyd and Scedrov [1990].)

Definition 4.4 A morphism q: X[right arrow]Y in an involutive quantaloid Q is:

1. simple if [qq.sup.[omicron]] [less than or equal to] 1Y,

2. semi-simple if there are simple morphisms fand g such that q = [fg.sup.[omicron]],

3. weakly semi-simple if q = V{[fg.sup.[omicron]] | [fg.sup.[omicron]] [less than or equal to] q with f and g simple}.

And an involutive quantaloid Q is (weakly) (semi-)simple ifeach ofits morphisms is.

The next lemma can be found in [Freyd and Scedrov, 1990, 2.16(10)], but we spell out its proof for later reference.

Lemma 4.5 A modular quantaloid Q is semi-simple if and only if [Q.sub.ssi] is tabular.

Proof: Suppose that [Q.sub.ssi] is tabular. Given a morphism q: X[right arrow]Y in Q, it can be included in [Q.sub.ssi] as q: lx[right arrow]ly, so let q = [fg.sup.[omicron]] be a tabulation in [Q.sub.ssi]. If Q is modular then so is [Q.sub.ssi] (by Lemma 3.7) so every left adjoint is a symmetric left adjoint. From this it is straightforward that both fand g are simple morphisms in Q such that q = [fg.sup.[omicron]].

Conversely, suppose first that Q is only semi-simple, and let q [member of] [Q.sub.ssi] (r, p); that is to say, r: X[right arrow]X and p: Y[right arrow]Y are symmetric idempotents in Q, and q: X[right arrow]Y is a morphism in Q satisfying pq = q = qr. Now let b: Z[right arrow]X and a: Z [right arrow] Y be simple morphisms in Q such that [ab.sup.[omicron]] = q. It is then straightforward to check that pa : [l.sub.Z][right arrow]p and rb: [l.sub.Z][right arrow]r are simple morphisms in [Q.sub.ssi] such that (pa)[(rb).sup.[omicron]] = q. In other words, [Q.sub.ssi] is semi-simple whenever Q is. Now the other hypothesis says that Q is also modular; by Lemma 3.7 we know that [Q.sub.ssi] is modular too. It thus remains to prove that [Q.sub.ssi] is tabular when it is semi-simple and modular. So again, let q [member of] [Q.sub.ssi] (r,p) and suppose now that x: e[right arrow]p and y: e[right arrow]r are simple morphisms in [Q.sub.ssi] such that q = [xy.sup.[omicron]]. Simplicity of x and y makes z := [x.sup.[omicron]]x A [y.sup.[omicron]]y [member of] [Q.sub.ssi] (e, e) a symmetric morphism satisfying zz [less than or equal to] z; it follows from the modular law that it is therefore a symmetric idempotent, and furthermore that [xzy.sup.[omicron]] = [xy.sup.[omicron]]. Choosing a (necessarily symmetric) splitting of z in [Q.sub.ssi] , say a w [member of] [Q.sub.ssi] (f, e) such that z = [ww.sup.[omicron]] and w[omicron]w = f, it is then tedious but routine to check that x' := xw and y' := yw are left adjoints in [Q.sub.ssi] that tabulate q.

The proof of the previous lemma can be tweaked to obtain the following 'weak' variant:

Lemma 4.6 A modular quantaloid Q is weakly semi-simple if and only if [Q.sub.ssi] is weakly tabular.

Proof: This is a straightforward adaptation of the previous proof: instead of working with single pairs of simple morphisms we must work with families of pairs of simple morphisms.

Suppose that [Q.sub.ssi] is weakly tabular. A morphism q: X[right arrow]Y in Q can be viewed as a morphism q: [1.sub.x] [right arrow] ly in [Q.sub.ssi], so consider its weak tabulation in [Q.sub.ssi]:

q = V {[fg.sup.*] | f and g are left adjoints in [Q.sub.ssi] such that [fg.sup.*] [less than or equal to] q}.

If Q is modular then it follows (as in the previous proof) that all the f's and g's in the above expression are simple in Q and exhibit q's weak semi-simplicity.

Conversely, suppose first that Q is weakly semi-simple. If q: r[right arrow]p is a morphism in [Q.sub.ssi] (between symmetric idempotents r: X[right arrow]X and p: Y[right arrow]Y, say) then at least we know that q: X[right arrow] Y is weakly semi-simple in Q:

q = {[ab.sup.[omicron]] | a and b are simple morphisms in Q such that [ab.sup.[omicron]] [less than or equal to] q}.

As in the previous proof, each such pair (a, b) of simple morphisms in Q determines a pair (pa, rb) of simple morphisms in [Q.sub.ssi] , and the lot of them exhibit q's weak semi-simplicity in [Q.sub.ssi]. Thus [Q.sub.ssi] is weakly semi-simple whenever Q is. Adding the hypothesis that Q is modular, we must prove that [Q.sub.ssi] is in fact weakly tabular. So again, let q: r[right arrow]p be a morphism in [Q.sub.ssi], and suppose now that

q = V{[xy.sup.[omicron]] | x and y are simple morphisms in [Q.sub.ssi] such that [xy.sup.[omicron]] [less than or equal to] q}.

Each of the pairs (x, y) of simple morphisms in [Q.sub.ssi] can be transformed, as in the previous proof, into a pair (x', y') of left adjoint morphisms in [Q.sub.ssi], and the lot of them provide for a weak tabulation of q.

Much like Theorem 3.3 contains an axiomatic description of small quantaloids of closed cribles, we can now give an axiomatisation of Grothendieck quantaloids. In a sense, this is a refined analysis of the notion of 'weak semi-simplicity'.

Theorem 4.7 For a small involutive quantaloid Q, the following are equivalent:

1. Q is weakly semi-simple,

2. Matr(Q) is semi-simple,

If Q is modular then this is also equivalent to:

3. [Q.sub.ssi] is weakly tabular.

If Q is modular and locally localic then this is also equivalent to:

4. [Q.sub.ssi] is a small quantaloid of closed cribles,

5. ProjMatr(Q) is tabular,

6. there exists a small site (C, J) such that Rel(Q) [equivalent] Rel(Sh(C, J)),

7. Q is a Grothendieck quantaloid.

In fact, the small site (C, J) of which statement (6) speaks, is the site associated (as in Theorem 3.3) with the small quantaloid of closed cribles [Q.sub.ssi] of which statement (4) speaks.

Proof: (1 [??] 2) Let M: A[right arrow]B be a morphism in Matr(Q): we must find semisimple matrices F: C[right arrow]B and G: C[right arrow]A such that M = [FG.sub.[omicron]]. Each element of M, that is, each Q-arrow M(b, a) : ta [right arrow] tb, is weakly semi-simple by assumption; thus

M(b,a) = V{[fg.sup.[omicron]] | [fg.sup.[omicron]] [less than or equal to] M(b,a) with f and g simple}. (9)

For each (a, b) [member of] A x B we define the set

[C.sub.(a,b)] = {(f, g) | [fg.sub.[omicron]] [less than or equal to] M(b, a) with f and g simple morphisms in Q}

and furthermore we define C to be the coproduct of the [C.sub.(a,b)'s.] The constant functions [C.sub.(a,b)] [right arrow]A: (f,g) [right arrow] a and [C.sub.(a,b)][right arrow]B: (f,g) [right arrow] b therefore uniquely define functions [alpha]: C[right arrow]A and [beta]: C[right arrow]A; and putting the type of (f,g) [member of] C to be the domain of f (= the domain of g) makes C an object of Matr(Q).

With the aid of the identity matrices [[DELTA].sub.A]: A [right arrow] A and [[DELTA].sub.B]: B [right arrow] B we now define two Q-matrices, F: C [right arrow] B and G: C [right arrow]A, to have as elements

F(b, (f,g)) = [[DELTA].sub.B](b, [beta](f,g)) [omicron] f and G(a, (f,g)) = [[DELTA].sub.A](a,[alpha](f,g)) [omicron] g.

Simplicity of all f's and g's makes sure that F and G are simple matrices, and the formula M = FG[omicron] precisely coincides with (9).

(2 [??] 1) Any q: X[right arrow]Y in Q may be viewed as a one-element matrix (q): {X}[right arrow]{Y} between singletons (with obvious types). By hypothesis there are simple matrices F: C[right arrow]{Y} and G: C[right arrow]{X} such that (q) = [FG.sup.[omicron]]. The simplicity of F and G implies that, for each c [member of] C, the morphisms [f.sub.c] := F(Y, c): tc[right arrow]Y and [g.sub.c] := G(c,X): X[right arrow]tc are simple morphisms in Q; and (q) = [FG.sup.[omicron]] expresses precisely that q = [V.sub.c[member of]C] [f.sub.c][g.sup.[omicron].sub.c], showing q to be weakly semi-simple in Q.

(1 [??] 3) This is the contents of Lemma 4.6

(3 [??] 4) If Q is a locally localic and modular quantaloid, then so is [Q.sub.ssi] (by Lemmas 3.6 and 3.7); in particular, [Q.sub.ssi] is map-discrete and weakly modular too (see Lemma 3.2). Thus [Q.sub.ssi] is weakly tabular if and only if it is a small quantaloid of closed cribles (cf. Theorem 3.3).

(2 [??] 5) From Lemmas 3.2 and 3.7 we know that Matr(Q) is modular. Lemma 4.5 does the rest, since ProjMatr(Q) = [(Matr(Q)).sub.ssi].

(4 [??] 6) If [Q.sub.ssi] [equivalent] R(C, J) for some small site (C, J), then Rel(Q) [equivalent] Rel([Q.sub.ssi]) is equivalent to Rel(Sh(C, J)) by Theorem 3.14.

(6 [??] 7) Is evident.

(7 [??] 5) If Q is a Grothendieck quantaloid, then ProjMatr(Q) - which by Lemma 4.3 is equivalent to Rel(Q) - is equivalent to the allegory of internal relations in a topos, so it is most certainly tabular (see e.g. [Freyd and Scedrov, 1990, 2.142]).

Quantaloids vs. quantales

The theorem above thus says that a Grothendieck quantaloid is precisely a modular, locally localic and weakly semi-simple quantaloid. There is an easier criterion than weak semi-simplicity when dealing with Grothendieck quantales rather than quantaloids. Using the term quantal frame to mean a quantale whose underlying sup-lattice is a locale [Resende, 2007], we can state it as:

Theorem 4.8 A Grothendieck quantale is a modular quantal frame with a weakly semisimple top (i.e. T = V{[fg.sup.[omicron]] | f, g simple}).

Proof: One implication is trivial. For the other, let q [member of] Q; we must show that it is weakly semi-simple. For any two simple elements of Q, f and g say, the modular law and the simplicity of f and g allow us to compute that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The element h := f ([f.sup.[omicron]] qg [conjunction] 1) is simple, because it is smaller than the simple element f. In other words, this shows that, for any pair (f, g) of simple elements, there exists a simple element h such that q [conjunction] [fg.sup.[omicron]] = [hg.sup.[omicron]]. Using the remaining hypotheses, we can thus compute that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so Q is indeed weakly semi-simple.

As an application of the "change of base" principles that we developed in Section 3, we shall now show how every Grothendieck topos is equivalent to a category of Q-sheaves, with Q a Grothendieck quantale.

First recall that two small quantaloids Q and R are said to be Morita-equivalent when the (large) quantaloids of modules [[Q.sup.op],Sup] and [[R.sup.op],Sup] are equivalent. B. Mesablishvili [2004] proved that for any small quantaloid Q there is a Morita-equivalent quantale [Q.sup.m]; he uses abstract V-category theoretic arguments to prove his claim. Unraveling his arguments, we can give an explicit construction of [Q.sup.m]: it is Matr(Q)([Q.sub.0], [Q.sub.0]), the quantale of endo-matrices with elements in Q on the Q-typed set of objects of Q (where, of course, the type of an object X [member of] Q is X).

Given a morphism f: A [right arrow] B in a small quantaloid Q, we shall write [M.sub.f] [member of] [Q.sup.m] for the matrix all of whose elements are zero, except for the element indexed by (A, B) [member of] [Q.sub.0] x [Q.sub.0], which is equal to f. The function f[right arrow] [M.sub.f] is easily seen to preserve composition and suprema (but evidently not the identities, so it is not a quantaloid homomorphism). However, if E is a class of idempotents in [Q.sup.m] containing all of {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}, and we split these idempotents in [Q.sup.m], then we obtain a homomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is easily seen to be fully faithful and injective on objects. If Q is a small involutive quantaloid, then it is straightforward to define an involution on the quantale [Q.sup.m] as well, which the function f [right arrow] [M.sub.f] preserves. If all elements of E are symmetric (which is automatic for the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), then the above homomorphism is not only fully faithful and injective on objects, but also preserves the involution.

Furthermore, [Q.sup.m] is, by definition, a full subquantaloid of Matr(Q), which in turn is a full subquantaloid of SymDist(Q); let us write the full inclusion as J: [Q.sup.m][right arrow]SymDist(Q). In case symmetric idempotents split symmetrically in SymDist(Q), there is a fully faithful homomorphism J': [([Q.sup.m]).sub.[epsilon]][right arrow]SymDist(Q) of involutive quantaloids. This is in particular the case when Q is a small quantaloid of closed cribles, which leads us to:

Proposition 4.9 If Q is a small quantaloid of closed cribles, [Q.sup.m] is its Morita-equivalent quantale and [epsilon] is a class of symmetric idempotents in [Q.sup.m] containing all of {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}, then also [([Q.sup.m]).sub.[epsilon]] is a small quantaloid of closed cribles and the inclusion Q [right arrow] [([Q.sup.m]).sub.[epsilon]] induces an equivalence SymDist(Q)[right arrow]SymDist([([Q.sup.m]).sub.[epsilon]]) of involutive quantaloids.

Proof: If Q is a small quantaloid of closed cribles, then it is in particular locally localic and modular. Hence Matr(Q) is locally localic, implying that [Q.sup.m] is locally localic (as a one-object quantaloid), and therefore also [([Q.sup.m]).sub.[epsilon]] is locally localic. Moreover, it is straightforward to compute that the diagram

[ILLUSTRATION OMITTED]

of involutive quantaloids and homomorphisms that preserve the involution commutes up to natural isomorphism. Because J' is fully faithful, the results in Lemmas 3.11 and 3.12 apply, and prove the proposition. ?

If Q is a small quantaloid of closed cribles, then SymDist([Q.sub.ssi]) [equivalent] SymDist(Q) by Proposition 3.13, which is further equivalent to SymDist([([Q.sup.m]).sub.ssi]) by Proposition 4.9. This produces the following:

Corollary 4.10 If Q is a small quantaloid of closed cribles, then

1. Sh(Q) [equivalent] Sh ([Q.sup.m]),

2. Rel(Q) [equivalent] Rel([Q.sup.m]).

This implies that [Q.sup.m] is a Grothendieck quantale.

This result says in particular that any Grothendieck topos can equivalently be described as a category of sets with an equality relation taking truth-values in a Grothendieck quantale.

Examples

We end this paper with some examples, the first two of which clearly illustrate the difference between 'small quantaloids of closed cribles' and 'Grothendieck quantaloids'.

Example 4.11 (Closed cribles) As remarked before, each small quantaloid Q of closed cribles is a Grothendieck quantaloid, and Sh(Q) is equivalent to the topos of sheaves on the site canonically associated with Q.

Example 4.12 (Locales) A locale (L, V, [conjunction], T) with its trivial involution is a Grothendieck quantale, but it is not a small quantaloid of closed cribles (because it is not weakly tabular). Upon splitting the (symmetric) idempotents in L one obtains a small quantaloid of closed cribles; the site associated with the latter (as in Theorem 3.3) is exactly the canonical site (L, J) (for which [([x.sub.i]).sub.i] [member of] J(x) if and only if [V.sub.i] [x.sub.i] = x). Thus Sh(L) - in the sense of Definition 2.2 - is equivalent to the "usual" topos of sheaves on L.

Our next example is somewhat more involved. First we must recall from [Stubbe, 2005b] that the 2-category Ord(Q) of Definition 2.1 is equivalent to the 2-category [TRSCat.sub.cc](Q) of "Cauchy complete totally regular Q-semicategories and totally regular semifunctors"; and in [Heymans and Stubbe, 2009a] it is shown to be further equivalent to the 2-category Map([Mod.sub.lpg](Q)) of "locally principally generated Q-modules" and left adjoint module morphisms. It is not difficult to deduce, from the symmetrisation of Q-orders qua [Q.sub.ssi] -enriched categories that we proposed in Definition 2.2, the appropriate symmetrisations of Q-semicategories and of Q-modules, thus producing as many different but equivalent descriptions of Q-sheaves. In fact, in [Heymans and Stubbe, 2009b] we already studied the symmetric variant of locally principally generated Q-modules, albeit only for involutive quantales (and not quantaloids): the so-called "locally principally symmetric" objects in [Mod.sub.lpg](Q) form the subcategory [Mod.sub.lpg,lps](Q). In [Heymans and Stubbe, 2009b, Example 3.7(4)] we showed that, for any involutive quantale Q, the involutive quantaloid ProjMatr(Q) is equivalent to the involutive quantaloid Hilb(Q) of so-called "Q-modules with Hilbert structure" (and module morphisms between them). (The proof also appears in [Resende, 2012, Lemma 4.26, Theorem 4.29].) And we furthermore proved that, when Q is a modular quantal frame, then Hilb(Q) is further equivalent to the involutive quantaloid [Mod.sub.lpg,lps](Q) [Heymans and Stubbe, 2009b, Theorems 3.6 and 4.1]. Theorem 4.7 says in particular that a Grothendieck quantale is necessarily a modular quantal frame, so together with Lemma 4.3 this shows that in this case all of the involutive quantaloids Rel(Q), ProjMatr(Q), Hilb(Q) and [Mod.sub.lpg,lps](Q) are equivalent. Taking left adjoints in either of these therefore produces equivalent Grothendieck toposes

Sh(Q) [equivalent] Map(Rel(Q)) [equivalent] Map(ProjMatr(Q)) [equivalent] Map(Hilb(Q)) [equivalent] Map([Mod.sub.lpg,lps](Q)).

Example 4.13 (Inverse quantal frames) An inverse quantal frame Q is a modular quantal frame such that T = V{p [member of] Q | [p.sup.[omicron]]p V [pp.sup.[omicron]] [less than or equal to] 1}. It follows trivially from Theorem 4.8 that inverse quantal frames are Grothendieck quantales. There is a correspondence up to isomorphism between inverse quantal frames and etale groupoids [Resende, 2007]: for every etale groupoid G there is an inverse quantal frame Q = O(G) (as locale it is the object of morphisms of G, and its quantale multiplication stems from the composition law in G); and for every inverse quantal frame Q there is an etale groupoid G such that Q [congruent to] O(G). Moreover, [Resende, 2012, p. 62-65] proves that Map(Hilb(O(G))) is equivalent to the classifying topos BG of the etale groupoid G. Consequently,

Sh(O (G)) [equivalent] Map(Rel(O(G))) [equivalent] Map (ProjMatr(O (G))) [equivalent] Map(Hilb(O(G))) [equivalent] Map ([Mod.sub.lpg,lps](O(G)))

are all equivalent descriptions of the topos BG in terms of "sheaves on an involutive quantale".

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(1) In fact, J. Riguet [1948, p. 120] discovered much earlier what he called the Dedekind formula (in French, la relation de Dedekind) for relations between sets: if R [subset or equal to] E x F, S [subset or equal to] F x G and T [subset or equal to] E x G, then SR [intersection] T [subset or equal to] (S [intersection] TR[omicron])(R n S[omicron]T) (where R[omicron] is the opposite relation of R, etc.). Whereas it is obvious that the Dedekind formula implies the modular law, it is not difficult to see that the converse holds too: SR [intersection] T = (SR [intersection] T) [intersection] (SR [intersection] T) [subset or equal to] (SR [intersection] T) [intersection] (S [intersection] [TR.sup.[omicron]])R [subset or equal to] (S [intersection] [TR.sup.[omicron]])[((S [intersection] [TR.sup.[omicron]]).sup.[omicron]](SR [intersection] T) [intersection] R) [subset or equal to] (S [intersection] [TR.sup.[omicron]])(S[omicron]T [intersection] R). All this can, of course, be done in any involutive locally ordered 2-category, and indeed Riguet certainly understood that the importance of the Dedekind formula went beyond the calculus of relations: he explains that the term relation de Dedekind was deliberately so chosen because "elle contient comme cas particulier la relation entre ideaux dans un anneau decouverte par Dedekind".

(2) Freyd and Scedrov [1990] define an allegory A to be a locally posetal 2-category, equipped with an involution A [right arrow] [A.sup.op]: f [right arrow] [f.sup.[omicron]] (which fixes the objects, reverses the arrows, and preserves the local order), in which the modular law holds. Johnstone [2002] calls an allegory geometric when its hom-posets are complete lattices and composition distributes over arbitrary suprema. Thus, a geometric allegory is exactly the same thing as a modular quantaloid.

(3) In the context of allegories, Freyd and Scedrov [1990] use the term equivalence relation (and Johnstone [2002] speaks simply of an equivalence) for what we call a symmetric monad; when it splits, then it does so symmetrically (because an allegory is modular), and they say that the equivalence relation is effective. If all equivalence relations in an allegory split, they say that the allegory is effective.

(4) In a modular quantaloid Q, an arrow c: A[right arrow]A is a symmetric comonad if and only if it satisfies c [less than or equal to] 1A. Indeed, if c [less than or equal to] 1A then [c.sup.[omicron]] [less than or equal to] [1.sub.A] too, hence c [less than or equal to] [cc.sup.[omicron]]c (cf. the proof of Lemma 3.4) implies that c [less than or equal to] cc but also that c [less than or equal to] [c.sup.[omicron]], and by involution also [c.sup.[omicron]] [less than or equal to] c. Therefore, particularly in the context of allegories, the term coreflexive is often used.

(5) In [Freyd and Scedrov, 1990, 2.226], in the context of allegories rather than quantaloids, this construction is referred to as the systemic completion.

Hans Heymans, Isar Stubbe

Department of Mathematics and Computer Science, University of Antwerp,Middelheimlaan 1, 2020 Antwerpen, Belgium, email: hans.heymans@ua.ac.be

Laboratoire de Mathematiques Pures et Appliquees, Universite du Littoral-Cote d'Opale, 50 rue F. Buisson, 62228 Calais, France, email: isar.stubbe@lmpa.univ-littoral.fr

Received by the editors December 2011 - In revised form in June 2012.

Communicated by W. Tholen.

2010 Mathematics Subject Classification : 06F07,18B10,18B25,18D05,18D20.
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Author:Heymans, Hans; Stubbe, Isar
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Date:Dec 1, 2012
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