Category Comp Research Articles

Localization, Isomorphisms and Adjoint Isomorphism in the Category Comp(A - Mod)

A and B are considered to be non necessarily commutative rings and X a complex of (A - B) bimodules. The aim of this
 paper is to show that:
 
 The functors \overline{EXT}^n_{Comp(A-Mod)}(X,-): Comp(A-Mod) \longrightarrow Comp(B-Mod) and Tor_n^{Comp(B-Mod)}(X,-): Comp(B-Mod) \longrightarrow Comp(A-Mod) are adjoint functors.
 The  functor S_C^{-1}() commute with  the functors X\bigotimes - , Hom^{\bullet}(X,-) and their corresponding derived functors  \overline{EXT}^n_{Comp(A-Mod)}(X,-) and  Tor_n^{Comp(B-Mod)}(X,-).

A generalization of Gelfand-Naimark-Stone duality to completely regular spaces

Gelfand-Naimark-Stone duality establishes a dual equivalence between the category KHaus of compact Hausdorff spaces and the category ubaℓ of uniformly complete bounded archimedean ℓ-algebras. We extend this duality to the category CReg of completely regular spaces. This we do by first introducing basic extensions of bounded archimedean ℓ-algebras and generalizing Gelfand-Naimark-Stone duality to a dual equivalence between the category ubasic of uniformly complete basic extensions and the category Comp of compactifications of completely regular spaces. We then introduce maximal basic extensions and prove that the subcategory mbasic of ubasic consisting of maximal basic extensions is dually equivalent to the subcategory SComp of Comp consisting of Stone-Čech compactifications. This yields the desired dual equivalence for completely regular spaces since CReg is equivalent to SComp.

On the Extension of Seminormal Functors to the Category of Tychonoff Spaces

Chigogidze proposed a construction of extending a normal functor from the category Comp to the category Tych. We can apply his scheme to seminormal functors and study the properties of the original functor which are preserved under extension. We introduce the concept of functor having an invariant extension from Comp to Tych because the very existence of this invariance plays a key role in the preservation of the properties of a seminormal functor in its extension. It is proved that the superextension functor λ has an invariant extension. We check that if a seminormal functor has an invariant extension then its extension preserves a point, the empty set, intersection and is a monomorphic functor. If this functor has finite degree then its extension is continuous and hence a seminormal functor in Tych. If the functor is of infinite degree then continuity may be lost. Namely, we show that the extension of λ for Tych is not continuous.

On the functor of semiadditive τ-smooth functionals

In the work we investigate categorical and topological properties of the functor OSτ of semiadditive τ-smooth functionals in the category Tych of Tychonoff spaces and their continuous mappings, which extends the functor OS of semiadditive functionals in the category Comp of compact spaces and their continuous mappings. It is proved that the functor OSτ is a normal functor in the category Tych of Tychonoff spaces.

Absolute Extensors and Binary Monads

Let \U0001d53d be a monad in the category Comp of compact Hausdorff spaces and continuous maps. An abstract convexity was constructed by Radul for each \U0001d53d-algebra of the monad \U0001d53d in the category Comp. It was proved that if the convexity of the monad \U0001d53d with some additional properties is binary then \U0001d53d has good topological properties, in particular, FX is an absolute extensor in the class of 0-dimensional spaces for each openly generated compactum X. We show in this paper that binarity is also a necessary condition.

F-Dugundji spaces, F-Milutin spaces and absolute F-valued retracts

For every functional functor F:Comp→Comp in the category Comp of compact Hausdorff spaces we define the notions of F-Dugundji and F-Milutin spaces, generalizing the classical notions of Dugundji and Milutin spaces. We prove that the class of F-Dugundji spaces coincides with the class of absolute F-valued retracts. Next, we show that for a monomorphic continuous functor F:Comp→Comp admitting tensor products each Dugundji compact space is an absolute F-valued retract if and only if the doubleton {0,1} is an absolute F-valued retract if and only if some points a∈F({0})⊂F({0,1}) and b∈F({1})⊂F({0,1}) can be linked by a continuous path in F({0,1}). We prove that for the functor Lipk of k-Lipschitz functionals with k<2, each absolute Lipk-valued retract is openly generated. On the other hand, the one-point compactification of any uncountable discrete space is not openly generated but is an absolute Lip3-valued retract. More generally, each hereditarily paracompact scattered compact space X of finite scattered height n=ht(X) is an absolute Lipk-valued retract for k=2n+2−1.

Two Cartesian closed categories of information algebras

The compact information algebra and the continuous information algebra are two special information algebras, which are algebraic structures modeling computation in many different contexts. We show that the set of all continuous mappings between two continuous information algebras also forms a continuous information algebra. Further, we obtain that the two categories COMP and CON, consisting of compact information algebras and continuous information algebras as objects respectively, continuous mappings as morphisms, are both Cartesian closed.

Convexities Generated by L-Monads

Let \(\mathbb{F}\) be a monad in the category Comp. We build for each \(\mathbb{F}\)-algebra a convexity in general sense (see van de Vel 1993). We investigate properties of such convexities and apply them to prove that the multiplication map of the order-preserving functional monad is soft.

Categorical Models for Fairness: Completion vs Delay

We study two different approaches to semantics for fairness within thecategorical framework of presheaf models for concurrency [1, 2]. The first approach, used in e.g. dataflow models, is based on representing finite or infinite completed observations. A completed observation is the result of an infinite computation. A finite completed observation is then the result of an infinite computation that only produces finitely many actions, i.e. at some point it stops producing output but stays active (as opposed to being deadlocked). The second approach is that of e.g. Milner's SCCS with finite delay [6], having an observable delay action and representing also finite incomplete observations, but only infinite completed observations.More precisely, we consider (separated) presheaves over respectively the category Comp of incomplete finite, and completed infinite and finite observations, and the category Inf of incompleted finite and completed infinite observations, possibly containing delays. The latter model was applied previously by the author [5, 4] to give a denotational semantics of SCCS with finite delay shown to be fully abstract with respect to Hennessy and Stirlings extended bisimulation [3]. One of the motivations for studying finite delay is the ability to encode a fair asynchronous parallel operator. However, the model contains a lot more processes than just those obtained from the encoding. Moreover, the introduction of delay actions makes the modelling of synchronous communication less obvious. This raises the questions of identifying the behaviours of fair processes and giving a “delay free” representation of the processes, both answered by the present paper. We apply the first model to give a denotational semantics for a CSP-like process language, FCSP, with a fair asynchronous parallel operator with synchronous communication, an idle process (the identity for the fair parallel operator) and a deadlocked process (represented by the incomplete observation with no actions). The canonical notion of bisimulation obtained from span of open maps is sensitive to both fairness and deadlock.We give a syntactic encoding of FCSP into SCCS with finite delay. Benefiting from the categorical presentation we show that, semantically, the encoding corresponds to the inverse image of the geometric morphism induced by the functor gamma from Inf to Comp which removes delay actions and adds the idle completion, if the sequence contained infinitely many delay actions but only finitely many non-delay actions.

Characterizations of algebraic and varietal categories of topological spaces

In this paper we continue our considerations of algebraic categories of spaces [8,9]. Especially, various separation and compactness properties are investigated. We disprove the wide-spread feeling, that there are only trivial examples besides categories of compact Hausdorff spaces. In fact, every functor F : Set→ Top, which is left adjoint to the underlying set functor F ( Set)→ Set, serves as left adjoint for the underlying set functor of some algebraic subcategory of Top. Hence free spaces are free algebras! However, the Sierpinski space as well as the reals are never contained in any algebraic category of spaces. But there is an algebraic category of Hausdorff spaces cogenerated by the first uncountable ordinal space. This one admits no varietal extension in Top. The trivial and the compact cases are characterized by mild additional assumptions, generalizing results in [5,8,9]. For instance, the only closed hereditary, productive, and varietal subcategories of Top are the categories Comp 2 of compact Hausdorff, Ind of indiscrete, and Triv of at most one point spaces.

Category COMP and complementizerukubain Xhosa

The contents and the status of the category COMP in syntax have been investigated, tracing its history from the transformational hypothesis to its current status where it shares in the [Xbar]-theory as C″. The results of this investigation have been applied to Xhosa.

Functor exp n c , absolute retracts and Hilbert space

Recently, along with the problem of studying the properties of spaces of the form F(X), where X is compact, and F is a covariant functor from the category Comp of compacta and continuous maps to itself, more and more attention has been given to the problem of preservation by functors of certain properties of the fibers of maps. We note that the second problem is more difficult than the first as a rule. This is clear, if only from the fact that the first problem, in the case of preservation of points by the functors, can be treated as the problem of investigatin the properties of the constant map. Closely related to this them~ are questions concerning thepreservation (or change) of properties of a map after applying some covariant functor to it. At the Prague topology symposium in 1981 V. V. Fedorchuk, starting one of the first investigations in this domain, formulated some results, found in recent years, and also indicated directions of possible investigations [I]. In particular, fiberwise versions of the assertions the classical theorems of Wojdyslawski [2] and Curtis--Schori [3, 4] were cited. In [5] the author investigated properties of the functor exp~ : Comp -+ Comp. In this note we investigate conditions under which, depending on the properties of the space X and the map /: X-+ Y, the space (exp$ /)-i {y} is an absolute neighborhood retract of the class ~ of all metrizable spaces and an 12-manifold. We also study questions of the preservation by the functor exp~ of certain properties of the fibers of maps. Investigating the noncompact case we first prove theorems on spaces of the form (exp$ f)-1 {y} and afterwards, using the fiberwise versions found as a preliminary, we formulate results relating to spaces of the form exp$ X. Theorems 3 and 6 give conditions on X, which are necessary and sufficient for the space exp~ X to be an A (N) R (~) and an 12-manifold (based on Hilbert space), respectively. We note that in investigating the noncompact case, in Theorems 4 and 5 we are able to get rid of the condition of existence for the map f of so-called triviality regions. In the analogous theorems relating to the compact case it still does not seem possible to get rid of the indicated condition.