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An introduction to quantaloid-enriched categories
No full textInternational audienceThis survey paper, specifically targeted at a readership of fuzzy logicians and fuzzy set theorists, aims to provide a gentle introduction to the basic notions of quantaloid-enriched category theory. We discuss at length the definitions of quantaloid, quantaloid-enriched category, distributor and functor, always giving several examples that – hopefully – appeal to the intended readership. To indicate the strength of this general theory, we explain in considerable detail how (co)limits are dealt with, and particularly how the Yoneda embedding of a quantaloid-enriched category in its free (co)completion comes to be. Our insistence on quantaloid-enrichment (rather than quantale-enrichment) is duly explained by examples requiring a notion of “partial elements” (sheaves, partial metric spaces). A final section glosses over some further topics, providing ample references for the interested reader
Categorical structures enriched in a quantaloid : categories and semicategories/
No full textThis thesis consists of two parts: a synthesis of the theory of categories enriched in a quantaloid; and a weakening of this theory for it to include semicategories describing ordered sheaves on a quantaloid.
A synthesis of, and supplements to, results in the literature concerning the theory of categories enriched in a quantaloid Q (as particular case of categories enriched in a bicategory) is contained in the first chapters. This theory is built with Q-categories, functors and distributors, and contains such notions as, for example, adjoint functors, weighted colimits, presheaves, Kan extensions, Cauchy completions and Morita equivalence, and so on. The literature does not provide an overview of these matters, so it was necessary to provide one here.
Then the necessary theory is developed to arrive at an elementary description of ``ordered sheaves on a quantaloid Q', henceforth referred to as Q-orders. As there is no ``topos of sheaves on a quantaloid', Q-orders cannot be defined as ordered objects in such a topos. Instead a description of Q-orders as categorical structures enriched in the quantaloid Q is proposed. The well-known ordered sheaves on a locale L (i.e.~ordered objects in the topos of sheaves on L) should of course be a particular example of the general theory, taking Q to be the (one-object suspension of) L. Then it turns out that the theory of Q-categories has to be weakened to include ``categories without units', i.e. Q-semicategories. But for Q-semicategories to admit a convenient distributor calculus, a ``regularity' condition has to be imposed. And for those regular Q-semicategories to admit a reasonable theory of Cauchy completions and Morita equivalence, the even stronger condition of ``total regularity' has to be imposed. The former notion has been studied before for semicategories enriched in a symmetric monoidal closed category; the latter notion is new, and is introduced via the intuitively clear idea of ``stability of objects'. The point is then that precisely the Cauchy complete totally regular Q-semicategories are the Q-orders; for a locale L they are indeed the ordered objects in the topos of sheaves on L. A (bi)equivalent description of those Q-orders can be given in terms of categories enriched in the split-idempotent completion of the quantaloid Q: a totally regular semicategory enriched in Q corresponds in a precise sense to a category enriched in the split-idempotent completion of Q. Applying this once more to a locale L instead of a quantaloid Q, these results thus deepen the work of the Louvain-la-Neuve school, and reconcile it with that of the Sydney school, on the description of (ordered) sheaves on a locale as enriched categorical structures.
The extended introduction gives a compact yet intuitive presentation of the developments contained in the thesis.(MATH 3)--UCL, 200
More on Q-modules
Get PDFA. Joyal and M. Tierney showed that the internal suplattices in the topos of
sheaves on a locale are precisely the modules on that locale. Using a totally different
technique, I shall show a generalization of this result to the case of (ordered) sheaves
on a (small) quantaloid. Then I make a comment on module-equivalence versus sheafequivalence,
using a recent observation of B. Mesablishvili and the notion of ‘centre’ of a
quantaloid
Towards “Dynamic Domains”: Totally Continuous Cocomplete Q-categories
No full textAbstractIt is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or more generally, a quantaloid. In fact, we are lead to consider cocomplete quantaloid-enriched categories as fundamental mathematical structure for a dynamic logic common to both computer science and physics. Here we explain the theory of totally continuous cocomplete categories as generalization of the well-known theory of totally continuous suplattices. That is to say, we undertake some first steps towards a theory of “dynamic domains”
The double power monad is the composite power monad
No full textInternational audienceIn the context of quantaloid-enriched categories, we rely essentially on the classifying property of presheaf categories to give a conceptual proof of a theorem due to Höhle: the double power monad and the composite power monad, on the category of quantaloid-enriched categories, are the same. Via the theory of distributive laws, we identify the algebras of this monad to be the completely codistributive complete categories, and the homomorphisms between such algebras are the bicontinuous functors. With these results we hope to contribute to the further development of a theory of Q-valued preorders (in the sense of Pu and Zhang)
Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories
No full text21 pagesInternational audienceWe study presheaves on semicategories enriched in a quantaloid: this gives rise to the notion of regular presheaf. A semicategory is regular when its representable presheaves are regular, and its regular presheaves then constitute an essential (co)localization of the category of all of its presheaves. The notion of regular semidistributor allows to establish the Morita equivalence of regular semicategories. Continuous orders and Omega-sets provide examples
Categorical structures enriched in a quantaloid: categories, distributors and functors
No full text46 pagesInternational audienceWe thoroughly treat several familiar and less familiar definitions and results concerning categories, functors and distributors enriched in a base quantaloid Q. In analogy with V-category theory we discuss such things as adjoint functors, (pointwise) left Kan extensions, weighted (co)limits, presheaves and free (co)completion, Cauchy completion and Morita equivalence. With an appendix on the universality of the quantaloid Dist(Q) of Q-enriched categories and distributors
'Hausdorff distance' via conical cocompletion
No full textMinor changesInternational audienceIn the context of quantaloid-enriched categories, we explain how each saturated class of weights defines, and is defined by, an essentially unique full sub-KZ-doctrine of the free cocompletion KZ-doctrine. The KZ-doctrines which arise as full sub-KZ-doctrines of the free cocompletion, are characterised by two simple "fully faithfulness" conditions. Conical weights form a saturated class, and the corresponding KZ-doctrine is precisely (the generalisation to quantaloid-enriched categories of) the Hausdorff doctrine of [Akhvlediani et al., 2009]
Q-modules are Q-suplattices
No full text12 pagesInternational audienceIt is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using enriched category theory and a lemma on KZ doctrines we prove (the generalization of) this fact in the case of ordered sheaves on a small quantaloid. Comparing module-equivalence with sheaf-equivalence for quantaloids and using the notion of centre of a quantaloid, we refine a result of F. Borceux and E. Vitale
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