scholarly journals Relevant categories and partial functions

2007 ◽  
pp. 17-23
Author(s):  
Kosta Dosen ◽  
Zoran Petric
Keyword(s):  
Natural Transformation ◽  
Relevant Logic ◽  
Partial Functions ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Relevant Category ◽  
Symmetric Monoidal Closed Category ◽  
Monoidal Closed Category ◽  
Cartesian Closed

A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination relevant comes from the connection with relevant logic. It is shown that the category of sets with partial functions, which is isomorphic to the category of pointed sets, is a category that is relevant, but not cartesian closed.

Isomorphic objects in symmetric monoidal closed categories

1997 ◽  
Vol 7 (6) ◽  
pp. 639-662 ◽  
Author(s):  
KOSTA DOšEN ◽  
ZORAN PETRIĆ
Keyword(s):  
Essential Role ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Symmetric Monoidal Closed Category ◽  
Monoidal Closed Category ◽  
Equational Calculus ◽  
Cartesian Closed ◽  
Arithmetical Interpretation

This paper presents a new and self-contained proof of a result characterizing objects isomorphic in the free symmetric monoidal closed category, i.e., objects isomorphic in every symmetric monoidal closed category. This characterization is given by a finitely axiomatizable and decidable equational calculus, which differs from the calculus that axiomatizes all arithmetical equalities in the language with 1, product and exponentiation by lacking 1c=1 and (a · b)c =ac · bc (the latter calculus characterizes objects isomorphic in the free cartesian closed category). Nevertheless, this calculus is complete for a certain arithmetical interpretation, and its arithmetical completeness plays an essential role in the proof given here of its completeness with respect to symmetric monoidal closed isomorphisms.

scholarly journals Proof of a Conjecture of S. Mac Lane

1996 ◽  
Vol 3 (61) ◽  
Author(s):  
Sergei Soloviev
Keyword(s):  
Linear Logic ◽  
Sufficient Conditions ◽  
Vector Spaces ◽  
Categorical Equivalence ◽  
Closed Category ◽  
Symmetric Monoidal Closed Category ◽  
Monoidal Closed Category ◽  
Coherence Theorem ◽  
Mac Lane

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (only this case was considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the "abstract coherence theorem" found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture proves also the correctness of the modified description.

An enrichment theorem for an axiomatisation of categories of domains and continuous functions

1997 ◽  
Vol 7 (5) ◽  
pp. 591-618 ◽  
Author(s):  
MARCELO P. FIORE
Keyword(s):  
Fixed Point ◽  
Continuous Functions ◽  
Domain Theory ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Cartesian Closed

Domain-theoretic categories are axiomatised by means of categorical non-order-theoretic requirements on a cartesian closed category equipped with a commutative monad. In this paper we prove an enrichment theorem showing that every axiomatic domain-theoretic category can be endowed with an intensional notion of approximation, the path relation, with respect to which the category Cpo-enriches.Our analysis suggests more liberal notions of domains. In particular, we present a category where the path order is not ω-complete, but in which the constructions of domain theory (such as, for example, the existence of uniform fixed-point operators and the solution of domain equations) are available.

Monad compositions II: Kleisli strength

2008 ◽  
Vol 18 (3) ◽  
pp. 613-643 ◽  
Author(s):  
ERNIE MANES ◽  
PHILIP MULRY
Keyword(s):  
Monoidal Category ◽  
Linear Equations ◽  
Large Collection ◽  
Data Types ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Symmetric Monoidal Category ◽  
Cartesian Closed

In this paper we introduce the concept of Kleisli strength for monads in an arbitrary symmetric monoidal category. This generalises the notion of commutative monad and gives us new examples, even in the cartesian-closed category of sets. We exploit the presence of Kleisli strength to derive methods for generating distributive laws. We also introduce linear equations to extend the results to certain quotient monads. Mechanisms are described for finding strengths that produce a large collection of new distributive laws, and consequently monad compositions, including the composition of monadic data types such as lists, trees, exceptions and state.

scholarly journals De Rham's theorem in a smooth topos

1984 ◽  
Vol 96 (1) ◽  
pp. 61-71 ◽  
Author(s):  
Ieke Moerdijk ◽  
Gonzalo E. Reyes
Keyword(s):  
Calculus Of Variations ◽  
Cartesian Closed Category ◽  
World Picture ◽  
Closed Category ◽  
Mathematical World ◽  
Cartesian Closed

It has been persuasively argued (e.g. by Lawvere[8]) that the mathematical world picture needed to develop the physics of continuous bodies and fields should involve a cartesian closed category of smooth morphisms between smooth spaces. As far as the foundations of the calculus of variations are concerned, the need for such a category was recognized by K. T. Chen(cf. [2]).

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