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.

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.

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]).

scholarly journals A Cartesian closed category of event structures with quotients

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Samy Abbes
Keyword(s):  
Topological Space ◽  
Event Structure ◽  
Cartesian Closed Category ◽  
New Class ◽  
Closed Category ◽  
Event Structures ◽  
Natural Notion ◽  
International Audience ◽  
Cartesian Closed

International audience We introduce a new class of morphisms for event structures. The category obtained is cartesian closed, and a natural notion of quotient event structure is defined within it. We study in particular the topological space of maximal configurations of quotient event structures. We introduce the compression of event structures as an example of quotient: the compression of an event structure E is a minimal event structure with the same space of maximal configurations as E.

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