A cartesian closed topological category of sequential spaces

1990 ◽  
Vol 21 (2) ◽  
pp. 109-112 ◽  
Author(s):  
J. Šlapal
Keyword(s):  
Topological Category ◽  
Cartesian Closed ◽  
Sequential Spaces

Reflexive objects in topological categories

1996 ◽  
Vol 6 (4) ◽  
pp. 375-386
Author(s):  
Michael D. Rice
Keyword(s):  
Geometric Structure ◽  
Unit Interval ◽  
Continuous Image ◽  
Topological Category ◽  
Hausdorff Spaces ◽  
Tychonoff Space ◽  
Countably Compact ◽  
Discrete Subspace ◽  
Cartesian Closed

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

scholarly journals A categorical approach to convergence

Filomat ◽  
2016 ◽  
Vol 30 (12) ◽  
pp. 3329-3338
Author(s):  
Josef Slapal
Keyword(s):  
Closure Operator ◽  
Sufficient Conditions ◽  
Topological Category ◽  
Categorical Approach ◽  
Cartesian Closed ◽  
Natural Way

We define the concept of a convergence class on an object of a given category by using certain generalized nets for expressing the convergence. The resulting topological category, whose objects are the pairs consisting of objects of the original category and convergence classes on them, is then investigated. We study the full subcategories of this category which are obtained by imposing on it some natural convergence axioms. In particular, we find sufficient conditions for the subcategories to be cartesian closed. We also investigate the behavior of the closure operator associated with the convergence in a natural way.

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

2002 ◽  
Vol 12 (6) ◽  
pp. 739-770 ◽  
Author(s):  
MATÍAS MENNI ◽  
ALEX SIMPSON
Keyword(s):  
Large Class ◽  
Full Subcategory ◽  
The Other ◽  
Topological Spaces ◽  
Limit Space ◽  
Quotient Spaces ◽  
Cartesian Closed Categories ◽  
The One ◽  
Cartesian Closed ◽  
Sequential 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.

scholarly journals Dynamic game semantics

2020 ◽  
pp. 1-60
Author(s):  
Norihiro Yamada ◽  
Samson Abramsky
Keyword(s):  
Programming Languages ◽  
Functional Programming ◽  
Operational Semantics ◽  
Dynamic Game ◽  
Standard Interpretation ◽  
Functional Programming Languages ◽  
Game Semantics ◽  
Wide Range ◽  
Cartesian Closed Categories ◽  
Cartesian Closed

Abstract The present work achieves a mathematical, in particular syntax-independent, formulation of dynamics and intensionality of computation in terms of games and strategies. Specifically, we give game semantics of a higher-order programming language that distinguishes programmes with the same value yet different algorithms (or intensionality) and the hiding operation on strategies that precisely corresponds to the (small-step) operational semantics (or dynamics) of the language. Categorically, our games and strategies give rise to a cartesian closed bicategory, and our game semantics forms an instance of a bicategorical generalisation of the standard interpretation of functional programming languages in cartesian closed categories. This work is intended to be a step towards a mathematical foundation of intensional and dynamic aspects of logic and computation; it should be applicable to a wide range of logics and computations.

scholarly journals All cartesian closed categories of quasicontinuous domains consist of domains

2015 ◽  
Vol 594 ◽  
pp. 143-150 ◽  
Author(s):  
Xiaodong Jia ◽  
Achim Jung ◽  
Hui Kou ◽  
Qingguo Li ◽  
Haoran Zhao
Keyword(s):  
Cartesian Closed Categories ◽  
Cartesian Closed

scholarly journals Comparing Cartesian closed categories of (core) compactly generated spaces

2004 ◽  
Vol 143 (1-3) ◽  
pp. 105-145 ◽  
Author(s):  
Martín Escardó ◽  
Jimmie Lawson ◽  
Alex Simpson
Keyword(s):  
Cartesian Closed Categories ◽  
Cartesian Closed ◽  
Compactly Generated

scholarly journals On countable choice and sequential spaces

2008 ◽  
Vol 54 (2) ◽  
pp. 145-152 ◽  
Author(s):  
Gonçalo Gutierres
Keyword(s):  
Sequential Spaces
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