Coherence in cartesian closed categories and the generality of proofs

Studia Logica ◽  
1989 ◽  
Vol 48 (3) ◽  
pp. 285-297
Author(s):  
M. E. Szabo
Keyword(s):  
Cartesian Closed Categories ◽  
Cartesian Closed

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 Polynomial functors and polynomial monads

2012 ◽  
Vol 154 (1) ◽  
pp. 153-192 ◽  
Author(s):  
NICOLA GAMBINO ◽  
JOACHIM KOCK
Keyword(s):  
Basic Theory ◽  
Polynomial Functors ◽  
Cartesian Closed Categories ◽  
Cartesian Closed ◽  
The Relationship

AbstractWe study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.

The sequential topology on is not regular

2009 ◽  
Vol 19 (5) ◽  
pp. 943-957 ◽  
Author(s):  
MATTHIAS SCHRÖDER
Keyword(s):  
Open Subset ◽  
Open Problem ◽  
Polish Space ◽  
Topological Spaces ◽  
Computable Analysis ◽  
Topological Properties ◽  
Compact Open Topology ◽  
Cartesian Closed Categories ◽  
Cartesian Closed ◽  
Continuous Functionals

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.

A Counter-Example to Coherence in Cartesian Closed Categories

1975 ◽  
Vol 18 (1) ◽  
pp. 111-114 ◽  
Author(s):  
M. E. Szabo
Keyword(s):  
Counter Example ◽  
Natural Transformations ◽  
Cartesian Closed Categories ◽  
Cartesian Closed

It follows from [3] that all morphisms of free closed categories on finite discrete categories are components of natural or “generalized” natural transformations, and from [8] that all hom-sets of such categories are finite. The purpose of this paper is to show that neither statement remains true if the categories are also assumed to be cartesian.

A General Notion of Realizability

2002 ◽  
Vol 8 (2) ◽  
pp. 266-282
Author(s):  
Lars Birkedal
Keyword(s):  
Predicate Logic ◽  
Higher Order ◽  
Order Logic ◽  
General Notion ◽  
Higher Order Logic ◽  
Cartesian Closed Categories ◽  
Cartesian Closed

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.

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