Deductive systems and categories III. Cartesian closed categories, intuitionist propositional calculus, and combinatory logic

1972 ◽  
pp. 57-82 ◽  
Author(s):  
Joachim Lambek
Keyword(s):  
Propositional Calculus ◽  
Combinatory Logic ◽  
Deductive Systems ◽  
Cartesian Closed Categories ◽  
Cartesian Closed

Least fixpoints of endofunctors of cartesian closed categories

1993 ◽  
Vol 3 (2) ◽  
pp. 229-257 ◽  
Author(s):  
J. Lambek
Keyword(s):  
Propositional Calculus ◽  
Second Order ◽  
Equivalence Relations ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Congruence Relations ◽  
Partial Recursive Functions ◽  
Cartesian Closed Categories ◽  
Partial Equivalence Relations ◽  
Cartesian Closed

Least fixpoints are constructed for finite coproducts of definable endofunctors of Cartesian closed categories that have weak polynomial products and joint equalizers of arbitrary families of pairs of parallel arrows. Both conditions hold in PER, the category whose objects are partial equivalence relations on N, and whose arrows are partial recursive functions. Weak polynomial products exist in any cartesian closed category with a finite number of objects as well as in any model of second order polymorphic lambda calculus: that is, in the proof theory of any second order positive intuitionistic propositional calculus, but such a category need not have equalizers. However, any finite coproduct of definable endofunctors of a cartesian closed category with weak polynomial products will have a least fixpoint in a larger category with equalizers whose objects are right ideals (or sieves) of modulo certain congruence relations, and whose arrows are induced from .

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

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 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.

scholarly journals Partial map classifiers and partial cartesian closed categories

1994 ◽  
Vol 136 (1) ◽  
pp. 109-123 ◽  
Author(s):  
Philip S. Mulry
Keyword(s):  
Cartesian Closed Categories ◽  
Cartesian Closed

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.

Sign in / Sign up

Export Citation Format

Share Document