Equilogical Spaces Research Articles

On generalized equilogical spaces

In this paper we carry the construction of equilogical spaces into an arbitrary category X topological over Set, introducing the category X-Equ of equilogical objects. Similar to what is done for the category Top of topological spaces and continuous functions, we study some features of X-Equ as (co)completeness and regular (co-)well-poweredness, as well as the fact that, under some conditions, it is a quasitopos. We achieve these various properties of the category X-Equ by representing it as a category of partial equilogical objects, as a reflective subcategory of the exact completion Xex, and as the regular completion Xreg. We finish with examples in the particular cases, amongst others, of ordered, metric, and approach spaces, which can all be described using the (T,V)-Cat setting.

Frames and topological algebras for a double-power monad

We study the algebras for the double power monad on the Sierpinski space in the cartesian closed category of equilogical spaces and produce a connection of the algebras with frames. The results hint at a possible synthetic, constructive approach to frames via algebras, in line with that considered in Abstract Stone Duality by Paul Taylor and others.

On a Generalization of Equilogical Spaces

We use the theory of triposes to prove that every (non-degenerate) locale H is the set of truth values of a complete and co-complete quasi-topos into which the category of topological spaces embeds and the topos of sheaves over H reflectively embeds.

A model of guarded recursion via generalised equilogical spaces

We present a new model, called GuardedEqu, of guarded dependent type theory using generalised equilogical spaces. GuardedEqu models guarded recursive types, which can be used to program with coinductive types and we prove that GuardedEqu ensures that all definable functions on coinductive types, e.g., streams, are continuous with respect to the natural topology. We present a direct, elementary, construction of the new model, which, importantly, is coherent (split) by construction.

Equilogical spaces and algebras for a double-power monad

We investigate the algebras for the double-power monad on the Sierpisnki space in the category $\mathcal{Equ}$ of equilogical spaces, a cartesian closed extension of $\mathcal{Top}_0$ introduced by Scott, and the relationship of such algebras with frames. In particular, we focus our attention on interesting subcategories of $\mathcal{Equ}$. We prove uniqueness of the algebraic structure for a large class of equilogical spaces, and we characterize the algebras for the double-power monad in the category of algebraic lattices and in the category of continuous lattices, seen as full subcategories of $\mathcal{Equ}$. We also analyse the case of algebras in the category $\mathcal{Top}_0$ of $\mathrm{T}_0$-spaces, again seen as a full subcategoy of $\mathcal{Equ}$, proving that each algebra for the double-power monad in $\mathcal{Top}_0$ has an underlying sober, compact, connected space.

The category of equilogical spaces and the effective topos as homotopical quotients

We show that the two models of an extensional version of Martin-Lof type theory, those given by the category of equilogical spaces and by the effective topos, are homotopical quotients of appropriate categories of 2-groupoids.

Sobriety for equilogical spaces

The category of equilogical spaces, as well as the exact completions of the category of T0-spaces and of the category of topological spaces, offers locally cartesian closed extensions of the category of topological spaces. Hence in any one of such categories, it is straightforward to consider spaces of continuous functions without bothering about ensuring that they be topological spaces.We test this fact with the notion of sober topological space, producing a synthetic characterization of those topological spaces which are sober in terms of a construction on equilogical spaces of functions.

Topological objects in the category EQU

We give a criterion for equivalence between equilogical and topological spaces. This enables us to prove that some interesting categories of topological spaces are complete subcartesian closed subcategories of equilogical spaces.

Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions

We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably interpreted in realizability models. We work with general realizability models rather than with a particular model of computation. Consequently, all the results are applicable in various established schools of computability, such as type 1 and type 2 effectivity, domain representations, equilogical spaces, and others.

Injective Convergence Spaces and Equilogical Spaces via Pretopological Spaces

Sierpinski space Ω is injective in the category Top of topological spaces, but not in any of the larger cartesian closed categories Conv of convergence spaces and Equ of equilogical spaces. We show that this negative result extends to all sub-cccs of Equ and Conv that are closed under subspaces and contain Top. On the other hand, we study the category PrTop of pretopological spaces that lies in-between Top and Conv/Equ, identify its injective spaces, and show that they are also injective in Conv and Equ.

Synthetic Topology: of Data Types and Classical Spaces

Synthetic topology as conceived in this monograph has three fundamental aspects: 1. to explain what has been done in classical topology in conceptual terms, 2. to provide one-line, enlightening proofs of the theorems that constitute the core of the theory, and 3. to smoothly export topological concepts and theorems to unintended situations, keeping the synthetic proofs unmodified. The unintended situation that we focus on is the theory of computation, in particular regarding programming languages from both operational (Part I) and denotational (Part III) points of view, with emphasis on sequential computation. We are aware of other applications of synthetic topology, e.g. to locales, convergence spaces, sequential spaces, equilogical spaces, and some sheaf and realizability toposes, but this will be reported elsewhere. Aspects 1 and 2 are the subject of Part II. However, it turns out that it is possible to tackle aspect 3 without previous reference to 1 or 2. In fact, we start by developing synthetic topology of programming-language data types in Part I, without assuming any background in classical topology and without introducing any. Part III combines ideas from Parts I and II, developing non-trivial computational applications. The main new result is a computational version of the Tychonoff theorem. We also review previously known applications and explain how topology and semantics interact in program-correctness proofs. Although computers are finite, infinity shows up in a number of important situations in the theory of computation, e.g. infinity in syntax : loops, recursion; infinity in time: non-terminating computations; infinity of data: stream computation and higher-type computation; infinity in precision: real-number computation; infinity through abstraction: probabilistic descriptions. The first few chapters of Part I explore how the fundamental topological notions of continuous map, open set, closed set, compact set, Hausdorff space, and discrete space reconcile the finite character of computers with the infinite nature of the entities one wishes to calculate with. One of the main contributions of this monograph is to explain the computational nature of the the notion of compactness. Roughly speaking, a set is compact if and only if, given any semidecidable property, one can semidecide whether it holds for all elements of the set in finite time. Surprisingly, there are infinite computationally compact sets, for example that of infinite streams of binary digits.

Equilogical spaces

It is well known that one can build models of full higher-order dependent-type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra. But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily define the category of ERs and equivalence-preserving continuous mappings over the standard category Top 0 of topological T 0-spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this category—in contradistinction to Top 0 —is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein), a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the Kleene–Kreisel hierarchy of countable functionals of finite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models.

Topological and limit-space subcategories of countably-based equilogical spaces

There are two main approaches to obtaining ‘topological’ cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed – for example, the category of sequential spaces. Under the other, one generalises the notion of space – for example, to Scott's notion of equilogical space. In this paper, we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countably based equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. In fact, this category turns out to be equivalent to the category of all quotient spaces of countably based topological spaces. We show that the category is bicartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the category of equilogical spaces. We also show that the category of countably based equilogical spaces has a larger full subcategory that can be simultaneously viewed as a full subcategory of limit spaces. This full subcategory is locally cartesian closed and the embeddings into limit spaces and countably based equilogical spaces preserve this structure. We observe that it seems essential to go beyond the realm of topological spaces to achieve this result.

A Relationship between Equilogical Spaces and Type Two Effectivity

In this paper I compare two well studied approaches to topological semantics — the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ, and Type Two Effectivity, exemplified by the category of Baire space representations, Rep( B ). These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is how they are related. First, we show that Rep( B ) is equivalent to a full coreflective subcategory of Equ, consisting of the so-called 0-equilogical spaces. This establishes a pair of adjoint functors between Rep( B ) and Equ. The inclusion Rep( B ) → Equ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of Rep( B ) and Equ are essentially different. However, in a second comparison we show that Rep( B ) and Equ do share a common cartesian closed subcategory that contains all countably based T0-spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based T0-spaces. We consider several examples involving the natural numbers and the real numbers to demonstrate how these comparisons make it possible to transfer results from one setting to another.

A General Notion of Realizability

AbstractWe present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We show how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives rise to a topos, i.e., a model of impredicative intuitionistic higher-order logic.

A General Notion of Realizability

We present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We show how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives rise to a topos, i.e., a model of impredicative intuitionistic higher-order logic.

A Relationship between Equilogical Spaces and Type Two Effectivity

In this paper I compare two well studied approaches to topological semantics — the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ, and Type Two Effectivity, exemplified by the category of Baire space representations, Rep( B ). These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is how they are related.First, we show that Rep( B ) is equivalent to a full coreflective subcategory of Equ, consisting of the so-called 0-equilogical spaces. This establishes a pair of adjoint functors between Rep( B ) and Equ. The inclusion Rep( B ) → Equ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of Rep( B ) and Equ are essentially different. However, in a second comparison we show that Rep( B ) and Equ do share a common cartesian closed subcategory that contains all countably based T0-spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based T0-spaces. We consider several examples involving the natural numbers and the real numbers to demonstrate how these comparisons make it possible to transfer results from one setting to another.

Equilogical spaces and domains with totality

Equilogical spaces and domains with totality

Effective versions of equilogical spaces

Effective versions of equilogical spaces

Equilogical spaces and domains with totality

I discuss connections between equilogical spaces and total and dense objects in domain theory. The category DPER(DOM) of dense partial equivalence relations on Scott domains will be shown to be equivalent to the category EQU of equilogical spaces. Total and dense objects form a sub-ccc of DPER(DOM), which means that equilogical spaces can be viewed as a generalization of a domain-theoretic notion of totality. A consequence of this (based on work of Ershov, Berger, and others) is that the Kleene-Kreisel countable functionals of finite types can be naturally constructed in EQU from the natural numbers object N by repeated use of exponentiation and binary products. I will also comment on the type hierarchy over the reals R.