We characterize injective objects, injective hulls and essential embeddings in the category of symmetric categories enriched in a small, integral and involutive quantaloid. In particular, injective partial metric spaces are precisely formulated.
We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Q-enriched categories. For such quantaloids, which we call Cauchy-bilateral quantaloids, it follows that the Cauchy completion of any symmetric Q-enriched category is again symmetric. Examples include Lawvere's quantale of non-negative real numbers and Walters' small quantaloids of closed cribles.
Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid Q, which we call `Q-order'. This requires a theory of semicategories enriched in the quantaloid Q, that admit a suitable Cauchy completion. There is a quantaloid Idl(Q) of Q-orders and ideal relations, and a locally ordered category Ord(Q) of Q-orders and monotone maps; actually, Ord(Q)=Map(Idl(Q)). In particular is Ord(Omega), with Omega a locale, the category of ordered objects in the topos of sheaves on Omega. In general Q-orders can equivalently be described as Cauchy complete categories enrich…
We 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.
It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is total and cototal, and so is the category of Q-distributors and Q-Chu transforms.
Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and order-cocomplete. Bearing this in mind, we analyze how Sup-valued homomorphisms on Q are related to Q-categories. With an appendix on action, representation and variation.
We 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.
For any small involutive quantaloid Q we define, in terms of symmetric quantaloid-enriched categories, an involutive quantaloid Rel(Q) of Q-sheaves and relations, and a category Sh(Q) of Q-sheaves and functions; the latter is equivalent to the category of symmetric maps in the former. We prove that Rel(Q) is the category of relations in a topos if and only if Q is a modular, locally localic and weakly semi-simple quantaloid; in this case we call Q a Grothendieck quantaloid. It follows that Sh(Q) is a Grothendieck topos whenever Q is a Grothendieck quantaloid. Any locale L is a Grothendieck qua…
Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor's exponentiability follows from the works of Niefield [1980] and Dyckhoff and Tholen [1987].
Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of Q-categories factors through both of these functors.
Based on Garner's discovery that topological categories are total categories enriched in a quantaloid, this paper presents a series of results related to initial and final density in categorical topology via (co)density in quantaloid-enriched categories, focusing on (co-)Sierpiński objects, Galois correspondences and their fixed points.
With quantaloids carefully constructed from multi-adjoint frames, it is shown that multi-adjoint concept lattices, multi-adjoint property-oriented concept lattices and multi-adjoint object-oriented concept lattices are derivable from Isbell adjunctions, Kan adjunctions and dual Kan adjunctions between quantaloid-enriched categories, respectively.
Representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid. In a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis (FCA) and rough set theory (RST), which not only extend the realm of formal contexts to multi-typed and multi-valued ones, but also provide a general approach to construct various kinds of representation theorems. Besides incorporating several well-known representation theorems in FCA and RST as well as formulating new ones, it i…
This dissertation is devoted to a study of adjunctions concerning categories enriched over a quantaloid Q (or Q-categories for short), with the following types of adjunctions involved: (1) adjoint functors between Q-categories; (2) adjoint distributors between Q-categories; (3) adjoint functors between categories consisting of Q-categories. For a small quantaloid Q and a distributor between Q-categories, two adjunctions between the Q-categories of contravariant and covariant presheaves are presented. These adjunctions respectively extend the fundamental construction of Isbell adjunctions and K…
In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categorical topology into the context of quantaloid-enriched category theory, a field that previously drew its motivation and applications from other domains, such as quantum logic and sheaf theory.
For any small quantaloid $\Q$, there is a new quantaloid $\D(\Q)$ of diagonals in $\Q$. If $\Q$ is divisible then so is $\D(\Q)$ (and vice versa), and then it is particularly interesting to compare categories enriched in $\Q$ with categories enriched in $\D(\Q)$. Taking Lawvere's quantale of extended positive real numbers as base quantale, $\Q$-categories are generalised metric spaces, and $\D(\Q)$-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-e…
In 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].
For a quantaloid $\mathcal{Q}$, considered as a bicategory, Walters introduced categories enriched in $\mathcal{Q}$. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad $\mathbb{T}$ over the discrete presheaf monad of the small quantaloid $\mathcal{Q}$ the primary data of the theory, rather than the lax monad extensions of $\mathbb{T}$ to the category of $\mathcal{Q}$-relations that they equivalently describe. The central piece of the paper establishes a…
In this communication we generalize some recent results of Rump to categories enriched in a commutative quantale V. Using these results, we show that every quantale-enriched multicategory admits an injective hull. Finally, we expose a connection between the Isbell adjunction and the construction of injective hulls for topological spaces made by Banaschewski in 1973.
For a small quantaloid $\mathcal{Q}$, a $\mathcal{Q}$-closure space is a small category enriched in $\mathcal{Q}$ equipped with a closure operator on its presheaf category. We investigate $\mathcal{Q}$-closure spaces systematically with specific attention paid to their morphisms and, as preordered fuzzy sets are a special kind of quantaloid-enriched categories, in particular fuzzy closure spaces on fuzzy sets are introduced as an example. By constructing continuous relations that naturally generalize continuous maps, it is shown (in the generality of the $\mathcal{Q}$-version) that the categor…
It 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-know…
In the enriched setting, the notions of injective and projective model structures on a category of enriched diagrams also make sense. In this paper, we prove the existence of these model structures on enriched diagram categories under local presentability, accessibility, and "acyclicity" conditions, using the methods of lifting model structures from an adjunction introduced by Garner, Hess, Kedziorek, Riehl, and Shipley.
Chu connections and back diagonals are introduced as morphisms for distributors between categories enriched in a small quantaloid $\mathcal{Q}$. These notions, meaningful for closed bicategories, dualize the constructions of arrow categories and the Freyd completion of categories. It is shown that, for a small quantaloid $\mathcal{Q}$, the category of complete $\mathcal{Q}$-categories and left adjoints is a retract of the dual of the category of $\mathcal{Q}$-distributors and Chu connections, and it is dually equivalent to the category of $\mathcal{Q}$-distributors and back diagonals. As an ap…
Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relation $\mathfrak{x}\to x$ between ultrafilters and points of a topological space $X$ as arrows in $X$. Naturally, this point of view opens the door to the use of concepts and ideas from (enriched) Category Theory for the investigation of (for instance) topological spaces. In this paper we study cocompleteness, adjoint functors and Kan extensions in the context …
Continuous lattices were characterised by Martin Escardo as precisely the objects that are Kan-injective w.r.t. a certain class of morphisms. We study Kan-injectivity in general categories enriched in posets. For every class H of morphisms we study the subcategory of all objects Kan-injective w.r.t. H and all morphisms preserving Kan-extensions. For categories such as Top_0 and Pos we prove that whenever H is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock-Zoeberlein monad. However, this does not generalise to proper classes: we present a class of conti…
Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:0804.0326 [math.CV]] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the category of injective spaces and left adjoints over $\mathsf{Set}$. In this paper we generalise these results, studying cocompleteness with respect to a given class of distributors. We show …
The derived category $D[C,V]$ of the Grothendieck category of enriched functors $[C,V]$, where $V$ is a closed symmetric monoidal Grothendieck category and $C$ is a small $V$-category, is studied. We prove that if the derived category $D(V)$ of $V$ is a compactly generated triangulated category with certain reasonable assumptions on compact generators or $K$-injective resolutions, then the derived category $D[C,V]$ is also compactly generated triangulated. Moreover, an explicit description of these generators is given.
Using generalized enriched categories, in this paper we show that Rosický's proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over $\mathsf{Set}$, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiability of $(\mathbb{T},V)$-categories and show that, under suitable conditions, every $(\mathbb{T},V)$-injective category is exponentiable in $(\mathbb{T},V)\text{-}\mathsf{Cat}…
