Updated 2020-07-18
M. Johnson and R. Rosebrugh
This paper develops a new composition of symmetric
lenses that preserves information which is important for implementing system
interoperation.
It includes a cut-down but realistic example of a multi-system business supply chain
and illustrates the new mathematical content with analysis of the systems, showing
how the new composition facilitates the engineering required to implement the
interoperations.
All of the concepts presented here are based on either pure category theory or
on experience in solving business problems using applied category theory.
Proceedings of Applied Category Theory 2020, July 2020.
Available here .
M. Johnson and R. Rosebrugh
Recently lens-like definitions of multidirectional transformations have
appeared, encompassing both a "propagation" form and a "wide span" form. These "multiary
lenses" raise a number of new challenges that are addressed in this paper. First, in common with classical symmetric lenses, they need to
be studied modulo equivalence relations that factor out differences in
hidden data and declare two lenses to be equivalent if their observable behaviours correspond. Then, modulo those equivalence relations
the precise relationship between the propagation style lenses and the
wide span lenses needs to be established. All of this is carried through
here. But more significantly, the compositions of multiary lenses pro-
posed to date have suffered various limitations: well-behaved amendment lenses do not necessarily compose to give well-behaved amendment
lenses; technical conditions on model spaces can interfere with composition;
junction conditions needed to be introduced because certain kinds
of compositions of multiary lenses might not always be defined; and in
any case, composition of multiary lenses comes with unusual challenges
and cannot be expected to form a category structure since when there
are more than two data sources there are not even single notions of
domain or codomain for multiary lenses. In this paper we introduce
a class of asymmetric amendment lenses called spg-lenses (stable put-
get lenses) which is more general than well-behaved amendment lenses,
but is closed under composition, and we show how to use spg-lenses
to capture, via wide spans, a wide class of mutidirectional transformations which compose well and form a well-known and long-standing
structure, a multicategory - a multicategory of multiary lenses.
Proceedings of the 8th International Workshop on Bidirectional Transformations, June 2019.
Available here .
M. Johnson and R. Rosebrugh
We characterize those symmetric d-lenses which are representable as cospans of d-lenses. Such a symmetric d-lens
must have unique corrs per pair of objects and satisfy two other technical conditions. When the d-lens is also "least
change" then the corresponding cospan consists of c-lenses.
Proceedings of the 7th International Workshop on Bidirectional Transformations, April 2018.
Available here .
M. Johnson and R. Rosebrugh
Asymmetric c-lenses are the special cases of asymmetric d-lenses (also
called delta lenses) whose updates satisfy a universal property which
in many applications ensures "least-change". There has therefore been
hope that symmetric c-lenses might characterize those symmetric d-
lenses which satisfy a similar universal property. This paper begins an
analysis of symmetric c-lenses and their relationship to symmetric d-
lenses and explains why the authors do not expect symmetric c-lenses,
that is, equivalence classes of spans of c-lenses, to be central to
developing universal properties for symmetric lenses. Instead, we consider
cospans of c-lenses and show that they generate symmetric c-lenses
with an appropriate universal property. That property is further
analysed and used to motivate proposed generalisations to obtain universal,
least-change, properties for symmetric d-lenses. In addition we explore
how to characterise the symmetric d-lenses that arise from cospans of
c-lenses among all symmetric d-lenses.
Proceedings of the 6th International Workshop on Bidirectional Transformations, April 2017, 39-53.
Available here .
M. Johnson and R. Rosebrugh
Bidirectional Transformations provide mechanisms for maintaining synchronization between updatable data sources.
Lenses are certain mathematically specified bidirectional transformations. As part of a
project to unify the treatment of symmetric lenses (of various kinds) as equivalence classes of spans of asymmetric lenses (of corresponding kinds),
we relate symmetric delta lenses with spans of asymmetric delta lenses. Because delta lenses are based on state spaces which are categories rather
than sets, there is further structure that needs to be accounted for. One of the main findings in this paper is that the required equivalence relation
among spans is compatible with, but coarser than, the one expected. The main result is an isomorphism of categories between a category whose
morphisms are equivalence classes of symmetric delta lenses (here called fb-lenses) and the category of spans of delta lenses modulo the new
equivalence.
The Journal of Object Technology
Volume 16, no. 1 (February 2017) , pp. 2:1-32.
Available here .
M. Johnson and R. Rosebrugh
There are many different types of lenses, but largely they fall into the
three classes of the title: set-based, delta-based and edit-based lenses.
This paper develops some of the general relationships between those
classes. The main results are that a category of set-based lenses is a
full subcategory of a category of delta-based lenses determined by send-
ing sets to codiscrete categories; that symmetric set-based lenses can
similarly be seen as symmetric delta-based lenses; that symmetric edit-
based lenses are able to be represented as symmetric delta-based lenses,
although not as a subcategory; and that symmetric edit-based lenses
can also be seen as spans of a new notion of asymmetric edit-based
lenses. The importance of the paper is that it provides a substantial
unification with concrete inter-conversions developed among the three
main approaches to lenses in both their symmetric and asymmetric
forms.
Proceedings of the 5th International Workshop on
Bidirectional Transformations, April, 2016, 1-13.
Available here .
M. Johnson and R. Rosebrugh
Spans are pairs of arrows with a common domain. Despite their symmetry, spans are frequently viewed as oriented transitions from one of
the codomains to the other codomain. The transition along an oriented span might be thought of as transitioning backwards along the
first arrow (sometimes called "leftwards") and then, having reached the
common domain, forwards along the second arrow (sometimes called
"rightwards"). Rightwards transitions and their compositions are well-understood. Similarly, leftwards transitions and their compositions can
be studied in detail. And then, with a little hand-waving, a span is "just"
the composite of two well-understood transitions - the first leftwards,
and the second rightwards.
In this paper we note that careful treatment of the sources, targets
and compositions of leftwards transitions can be usefully captured as
a monad L built using a comma category construction. Similarly
the sources, targets and compositions of rightwards transitions form
a monad R, also built using a comma category construction. Our main
result is the development of a distributive law, in the sense of Beck
but only up to isomorphism, distributing L over R. Such a distributive
law makes RL a monad, the monad of anchored spans, thus removing
the hand-waving referred to above, and establishing a precise calculus
for span-based transitions.
As an illustration of the applicability of this analysis we use the new
monad RL to recast and strengthen a result in the study of databases
and the use of lenses for view updates.
Proceedings of the
4th International Workshop on Bidirectional Transformations, July 2015, 31-42
Available here .
M. Johnson and R. Rosebrugh
As part of an ongoing project to unify the treatment of symmetric lenses
(of various kinds) as equivalence classes of spans of asymmetric lenses
(of corresponding kinds) we relate the symmetric delta lenses of Diskin
et al, with spans of asymmetric delta lenses. Because delta lenses are
based on state spaces which are categories rather than sets there is
further structure that needs to be accounted for and one of the main
findings in this paper is that the required equivalence relation among
spans is compatible with, but coarser than, the one expected. The
main result is an isomorphism of categories between a category whose
morphisms are equivalence classes of symmetric delta lenses (here called
fb-lenses) and the category of spans of delta lenses modulo the new
equivalence.
Proceedings of the 4th International Workshop on Bidirectional Transformations,
July 2015, 1-15
Available here .
M. Johnson and R. Rosebrugh
Corresponding to the variety of notions of asymmetric lens,
various notions of symmetric lens have been proposed. A common theory of the various
asymmetric and symmetric lenses should result from a study of spans of
asymmetric lenses. In order to define a category whose arrows are
spans of asymmetric lenses, the fact that a cospan of asymmetric lenses
may not have a pullback must be dealt with. In this article, after resolving that
problem we develop the functors which exhibit a category whose arrows are spans
of well-behaved lenses as a retract of a category whose arrows are the corresponding symmetric
lenses. We relate them to the symmetric lenses of Hofmann, Pierce and Wagner.
Proceedings EDBT/ICDT Workshops 2014,
112-118
Available here .
M. Johnson and R. Rosebrugh
We compare the delta lenses of Diskin et al. with
with the c-lenses, known to be equivalent to opfibrations, already studied by the authors.
Contrary to expectation a c-lens is a d-lens but not conversely. This result is surprising
because delta-lenses appear to provide the same information as c-lenses, and some more besides,
suggesting that the implication would be the reverse - a d-lens would appear to be a
special kind of c-lens. The source of the surprise can be traced to the way the two concepts
deal differently with morphisms in a certain base comma category.
Both c-lenses and delta-lenses are important because they extend the notion of lens to take
account of the information available in known transitions between states and this has
important implications in practice.
Electronic Communications of the EASST,
57(2013), 1-18
Available here .
M. Johnson and R. Rosebrugh
Many authors have argued, for good reasons, that
in a range of applications the lens put-put law is too strong.
On the other hand, the present authors have shown
that very well behaved lenses, which do
satisfy the put-put law by definition, are
algebras for a certain monad, and that this viewpoint
admits fruitful generalisations of the lens concept to
a variety of base categories. In the algebra approach
to lenses, the put-put law corresponds to the associativity
axiom, and so is fundamentally important.
Thus we have a dilemma. The put-put law seems inappropriate
for many applications, but is fundamental to the
mathematical development that can support an extended
range of applications.
In this paper we resolve this dilemma. We outline
monotonic put-put laws and introduce a new mixed put-put
which appears to be immune to many of the objections to
the classical put-put law, but which supports a very
satisfactory mathematical foundation.
Electronic Communications of the EASST,
49(2013), 1-13
Available here .
R. Rosebrugh, N. Sabadini and R. F. C. Walters
We consider commutative Frobenius algebras in braided strict monoidal categories in the study of the circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras. For example, we show how Armstrong's description of knot colourings and knot groups fit into this context.
Theory and Applications of Categories, Vol. 26, 2012, 743-767.
Available in pdf .
Michael Johnson, Robert Rosebrugh and R.J. Wood
This article is extends the "lens'' concept for view updating in Computer
Science beyond the categories of sets and ordered sets. It is first shown that a
constant complement view updating strategy also corresponds to a lens for a
categorical database model. A variation on the lens concept
called c-lens is introduced, and shown to correspond to the
categorical notion of Grothendieck opfibration. This variant guarantees a universal solution to
the view update problem for functorial update processes.
Mathematical Structures in Computer Science, Vol. 22, 2012, 25-42.
Available in
pdf.
F. Marmolejo, R. Rosebrugh and R. J. Wood
In 1978, Street and Walters defined a locally small category K to be
totally cocomplete if its
Yoneda functor Y has a left adjoint X. Such a K is totally
distributive if X has a left adjoint W. Small powers of the category
of small sets are totally distributive, as are certain sheaf categories.
A locally small category K is
small cocomplete if it is a P-algebra, where P is the small-colimit
completion monad on Cat. In 2007, Day and Lack showed
that P lifts to R-algebras, where R is the small-limit
completion monad on Cat. It follows that there is a distributive
law RP to PR and we say that K is completely distributive
if K is a PR-algebra, meaning that K is small cocomplete, small
complete, and the P structure preserves small limits. Totally distributive
implies completely distributive. We show that there is a further supply of
totally distributive categories provided by categories of interpolative
bimodules between small taxons as introduced by Koslowski in 1997.
Available in
pdf.
Michael Johnson, Robert Rosebrugh and R.J. Wood
The correspondence between view update
translations and views with a constant complement reappears more generally as the
correspondence between update strategies and meet complements in the order based
setting of S. Hegner. We show that these two theories of database view updatability
are linked by the notion of lens which is an algebra for a monad. We generalize
lenses from the category of sets to consider them in categories with finite products, in
particular the category of ordered sets.
Journal of Universal Computer Science, Vol. 16, 2010, 729 - 748.
Available in
pdf
.
F. Marmolejo, R. Rosebrugh and R. J. Wood
The 2-category of constructively completely distributive lattices
is shown to be bidual to a 2-category of generalized orders
that admits a monadic schizophrenic object biadjunction over the
2-category of ordered sets.
Theory and Applications of Categories, Vol. 22, 2009, pp
1-23
Available in
dvi
and
ps
and
pdf
.
M. Johnson and R. Rosebrugh
LNCS, 5140, 232-237, 2008.
Available in
pdf
.
M. Johnson and R. Rosebrugh
LNCS, 5140, 238-252, 2008.
Available in
pdf
.
Michael Johnson and Robert Rosebrugh.
Invited book chapter.
Available in
pdf
.
M. Johnson and R. Rosebrugh
Maintainability and modifiability of information system software
can be enhanced by the
provision of comprehensive support for views, since view
support allows application programs to continue to operate
unchanged when the underlying information system is modified.
Supporting views depends upon a solution to the view
update problem.
This paper presents a new treatment of view updates for
formally specified semantic data models
based on the category
theoretic sketch data model. The sketch data model has been the basis of a number of
successful major information system consultancies.
We define view updates by a universal property in
models of the formal specification, and explain why this
indeed gives a complete and correct treatment of view updatability,
including a solution to the view update problem.
However, a definition of updatability which is based on models
causes some inconvenience in applications, so we prove that in
a variety of circumstances updatability is guaranteed independently
of the current model. This is done first with a very general criterion, and then
for some specific cases relevant to applications.
We include some detail about the sketch data model,
noting that it involves extensions of
algebraic data specification techniques.
This has appeared as:
Theoretical Computer Science 388 (2007), 109-129.
We show that the generic symmetric monoidal category with a commutative
separable algebra which has a $\Sigma$-family of actions is the category of
cospans of finite $\Sigma$-labelled graphs restricted to finite sets as
objects, thus providing a syntax for automata on the alphabet $\Sigma$. We use
this result to produce semantic functors for $\Sigma$-automata.
This has appeared as:
Theory and Applications of Categories, Vol. 15, 2005, pp
164-177
Available in dvi and ps and pdf .
In the early 1990's the authors proved that the full subcategory of `sup-lattices' determined by the constructively completely distributive (CCD) lattices is equivalent to the idempotent splitting completion of the bicategory of sets and relations. Having many corollaries, this was an extremely useful result. Moreover, as the authors soon suspected, it specializes a much more general result.
Let D be a monad on a category C in which idempotents split. Write kar(C_D) for the idempotent splitting completion of the Kleisli category. Write spl(C^D) for the category whose objects are pairs ((L,s),t), where (L,s) is an object of the Eilenberg-Moore category for D, and t is a homomorphism that splits s, with spl(C^D)(((L,s),t),((L',s'),t'))=C^D((L,s)(L',s')).
The main result is that kar(C_D) is isomorphic to spl(C^D). We also show how this implies the CCD
lattice characterization theorem and consider a more general context.
This has appeared as:
Theory and Applications of Categories, Vol. 13, 2004, pp
172-183
Available in dvi and ps and pdf .
R. Rosebrugh, N. Sabadini and R.F.C. Walters
The context of this article is the program to study the bicategory of spans of graphs as an algebra of processes, with applications to concurrency theory. The objective here is to study functorial aspects of reachability, minimization and minimal realization. The compositionality of minimization has application to model-checking. Mathematical Structures in Computer Science, 14(2004), 685-714.
Available in dvi , ps and pdf .
M. Johnson and R. Rosebrugh
Partial information is common in real-world databases. Yet the theoretical foundations of data models are not designed to support references to missing data (often termed nulls). Instead, we usually analyse a clean data model based on assumptions about complete information, and later retrofit support for nulls. The sketch data model is a recently developed approach to database specification based on category theory. The sketch data model is general enough to support references to missing information within itself (rather than by retrofitting). In this paper we explore three approaches to incorporating partial information in the sketch data model. The approaches, while fundamentally different, are closely related, and we show that under certain fairly strong hypotheses they are Morita equivalent (that is they have the same categories of models, up to equivalence). Despite this equivalence, the query languages arising from the three approaches are subtly different, and we explore some of these differences. Has appeared as ENTCS, Volume 78, 1-18, 2003.
Available in pdf .
R. Rosebrugh and R. J. Wood
For the 2-monad $((-)^2,I,C)$ on CAT, with unit $I$ described by
identities and multiplication $C$ described by composition, we show that a
functor $F : {\cal K}^2 \rightarrow \cal K$ satisfying $FI_{\cal K} =
1_{\cal K}$ admits a unique, normal, pseudo-algebra structure for $(-)^2$
if and only if there is a mere natural isomorphism $F F^2 \rightarrow F
C_{\cal K}$. We show that when this is the case the set of all natural
transformations $F F^2 \rightarrow F C_{\cal K}$ forms a commutative
monoid isomorphic to the centre of $\cal K$.
This has appeared as:
Theory and Applications of Categories, Vol. 10, 2002, pp
134-147
Available in dvi and ps and pdf .
R. Rosebrugh and R. J. Wood
This article shows that the distributive laws of Beck in the bicategory of
sets and matrices, wherein monads are categories, determine
M. Johnson and R. Rosebrugh.
ENTCS,
Volume 61, 6, 1-13, 2002.
[ ps.gz (~50k) | ps (~130k) ]
© Elsevier Science
M. Johnson and R. Rosebrugh.
Proceedings of the Tenth OOPSLA Workshop on Behavioral Semantics, Tampa, Florida, October 2001, 121-132.
M. Johnson and R. Rosebrugh.
Has appeared as Proceedings of the IEEE Conference on Software Maintenance, 32--39, 2001.
[ ps.gz (~73k) | ps (~165k) ]
© IEEE
M. Johnson and R. Rosebrugh.
The authors have developed a new approach to database interoperability using the sketch data model. An important question is: What are the algorithms that support updates in the sketch data model? The question arises since the sketch data model uses EA-sketches to specify data structures, and these include constraint and other information not normally supported by relational database systems. In this paper we answer the question by providing a formal definition of insert update together with an algorithm which provably achieves updates. The algorithm treats data and constraints on an equal categorical footing. Further exactness properties (limits and colimits) can aid specification, and we provide algorithms for updates of EA sketched databases with finite limits. The sketch data model is being used in industry for designing interoperations for computer supported cooperative work and CASE tools are under development. In press for CSCWD2001, the Fifth International Conference on Computer Supported Cooperative Work in Design, London, Ontario, 2001.
Available in pdf .
M. Johnson and R. Rosebrugh.
Information system software productivity can be increased by improving the maintainability and modifiability of the software produced. This can be achieved by the provision of support for views. Supporting views depends on a solution to the view update problem. This paper presents a new treatment of view updates. The formal specification technique we use is based on category theory and has been the basis of a number of successful major information system consultancies. We define view updates by a universal property in a subcategory of models of the formal specification, and explain why this indeed gives a comprehensive treatment of view updatability, including a solution to the view update problem. A definition of updatability which is based on models causes inconvenience in applications, so we prove that in a variety of circumstances updatability is guaranteed independently of the current model. The solution to the view update problem can be seen as requiring the existence of an initial model for a specification. LNCS, 2021, 534-549, 2001.
Available in pdf .
C.N.G. Dampney, M. Johnson and R. Rosebrugh.
This paper describes the sketch data model and investigates the view update problem (VUP) in the sketch data model paradigm. It proposes an approach to the VUP, and presents a range of examples to illustrate the scope of the proposed technique. We define under what circumstances a view update can be propagated to the underlying database. Unlike many previously proposed approaches the definition is succinct and consistent, without ad hoc exceptions, and the propagatable updates form a broad class. The examples demonstrate that under a range of circumstances a view schema can be shown to have propagatable views in all states, and thus state-independence can frequently be recovered. Proceedings of the twelfth Australasian Database Conference ADC2001, 29--36, IEEE Press, 2001.
Available in pdf .
M. Johnson and R. Rosebrugh.
Computer supported cooperative work (CSCW) depends more and more upon database interoperability. The design of interbusiness CSCW when the businesses are already operating independent systems depends either upon effective reverse engineering, or upon sufficiently rich semantic models and good database management system support for logical data independence. This paper takes the second approach presenting a rich semantic data model that the authors have been developing and have used successfully in a number of major consultancies, and a new approach to logical data independence and view updatability based on that model. We show how these approaches support database interoperability for business-to-business transactions, and, for CSCW within an organisation, how they support federated databases. Proceedings of CSCW2000, the Fourth International Conference on Computer Supported Collaborative Work, IEEE Hong Kong, 161-166, 2000.
F. Marmolejo, R. Rosebrugh and R. J. Wood
We pursue distributive laws between monads, particularly in the context of KZ-doctrines, and show that a very basic distributive law has (constructively) completely distributive lattices for its algebras. Moreover, the resulting monad is shown to be also the double dualization monad (with respect to the subobject classifier) on ordered sets. Journal of Pure and Applied Algebra 168(2002), 209-226 (Coimbra Volume).
R. Rosebrugh and R. J. Wood
We extend the concept of constructive complete distributivity so as to make it applicable to ordered sets admitting merely {\em bounded} suprema. The KZ-doctrine for bounded suprema is of some independent interest and a few results about it are given. The 2-category of ordered sets admitting bounded suprema over which non-empty infima distribute is shown to be bi-equivalent to a 2-category defined in terms of idempotent relations. As a corollary we obtain a simple construction of the non-negative reals. Appear in Applied Categorical Structures, 9(2001), pp 437-456.
Available in dvi and ps and pdf .
M. Johnson, R. Rosebrugh and R. J. Wood
Entity-Relationship-Attribute ideas are commonly used to specify and design information systems. They use a graphical technique for displaying the objects of the system and relationships among them. The design process can be enhanced by specifying constraints of the system and the natural environment for these is the categorical notion of sketch. Here we argue that the finite-limit, finite-sum sketches with a terminal node are the appropriate class and call them EA sketches. A model for an EA sketch in a lextensive category is a `snapshot' of a database with values in that category. The category of models of an EA sketch is an object of models of the sketch in a 2-category of lextensive categories. Moreover, modelling the {\em same} sketch in certain objects in other 2-categories defines both the query language for the database and the updates (the dynamics) for the database. This article has appeared in Theory and Applications of Categories 10(2002), 94-112.
Available in dvi and ps and pdf .
R. Rosebrugh, N. Sabadini and R.F.C. Walters
The context of this article is the program to develop monoidal bicategories with a feedback operation as an algebra of processes, with applications to concurrency theory. The objective here is to study reachability, minimization and minimal realization in these bicategories. In this setting the automata are 1-cells in contrast with previous studies where they appeared as objects. As a consequence we are able to study the relation of minimization and minimal realization to serial composition of automata using (co)lax (co)monads. We are led to define suitable behaviour categories and prove minimal realization theorems which extend classical results. Mathematical Structures in Computer Science, 8(1998), 93-116.
Available in dvi and ps and pdf .
M. Fleming, R. Gunther and R. Rosebrugh
We describe a program which facilitates storage and manipulation of finitely-presented categories and finite-set valued functors. It allows storage, editing and recall of finitely-presented categories and functors. Several tools for testing properties of objects and arrows, and the computation of right and left kan extensions are included. The program is written in ANSI C and is menu-based. Use of the program requires a basic knowledge of category theory. Journal of Symbolic Computation 35(2003), 127-135.
M. Fleming, R. Gunther and R. Rosebrugh
A guide to use of a C program which allows storage and manipulation of finitely presented categories and functors, including Kan extensions of finite-set valued functors.
Available in dvi and pdf files, and here is an abridged version
M. Fleming, R. Gunther and R. Rosebrugh
A technical manual for users of the program described in the preceding abstract.
R. Rosebrugh and R. J. Wood
For an adjoint string V -| W -| X -| Y : B --> C, with Y fully faithful, it is frequently, but not always, the case that the composite VY underlies an idempotent monad. When it does, we call the string distributive. We also study shorter and longer `distributive' adjoint strings and how to generate them. These provide a new construction of the simplicial 2-category, Delta. This article has appeared in Theory and Applications of Categories 1(1995), 119-145.
R. Rosebrugh and R. J. Wood
If a category B with Yoneda embedding Y : B ---> CAT(B^op,set) has an adjoint string, U -| V -| W -| X -| Y, then B is equivalent to set. This paper has appeared in Proceedings of the American Mathematical Society 122(1994), 409-413.
R. Rosebrugh and R.J. Wood
A description of relational databases in categorical terminology given here has as intended application the study of database dynamics, in particular we view (i) updates as database objects in a suitable category indexed by a topos; (ii) L-fuzzy databases as database objects in sheaves. Indexed categories are constructed to model the databases on a fixed family of domains and also all databases for a varying family of domains. Further, we show that the process of constructing the relational completion of a relational database is a monad in a 2-category of functors. This paper has appeared in the Proceedings of the 1991 Category Theory Meeting, published by the AMS.
Here are the dvi and pdf files.
R. Rosebrugh and R. J. Wood
A complete lattice L is constructively completely distributive, ccd, when the sup arrow from down-closed subobjects of L to down-closed subobjects of L has a left adjoint. The Karoubian envelope of the (bi-)category of relations is equivalent to the (bi-)category of ccd lattices and sup-preserving arrows. The equivalence restricts to an equivalence between ideals and ``totally algebraic'' lattices. Both equivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to an equivalence with ccd lattices in a topos. This paper has appeared in Applied Categorical Structures 2(1994), 119-144.
Here are the dvi and pdf files.
R. Rosebrugh and R.J. Wood
A complete lattice L is constructively completely distributive, (CCD)(L), if the sup map defined on down closed subobjects has a left adjoint. We characterize preservation of this property by left exact functors between toposes using a ``logical comparison transformation''. The characterization is applied to (direct images of) geometric morphisms to show that local homeomorphisms (in particular, product functors) preserve (CCD) objects, while preserving (CCD) objects implies openness. This paper has appeared in the Canadian Mathematical Bulletin 35(1992), 537-547.
Here are the dvi and pdf files.
R. Rosebrugh and R.J. Wood
A complete lattice L is constructively completely distributive, (CCD)(L), if the sup map defined on down-closed subobjects has a left adjoint. It was known that in boolean toposes (CCD)(L) is equivalent to (CCD)(L\op). We show here that the latter property for all L (sufficiently, for Omega) characterizes boolean toposes. This paper has appeared in the Mathematical Proceedings of the Cambridge Philosophical Society, 110(1991) pp. 245-249.