## Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic (Extended Abstract)

Citations: | 6 - 0 self |

### BibTeX

@MISC{Fiore_differentialstructure,

author = {Marcelo P. Fiore},

title = {Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic (Extended Abstract)},

year = {}

}

### OpenURL

### Abstract

Abstract. In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigating differential structure in the context of linear logic. Consequently, within this setting, I introduce a notion of creation operator (as considered by physicists for bosonic Fock space in the context of quantum field theory), provide an equivalent description of creation operators in terms of creation maps, and show that they induce a differential operator satisfying all the basic laws of differentiation (the product and chain rules, the commutation relations, etc.). 1

### Citations

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- 1989
(Show Context)
Citation Context ...c Models of multiplicative intuitionistic linear logic. I recall the definition of categorical model of multiplicative intuitionistic linear logic as it has been developed in the literature, see e.g. =-=[17, 20, 2, 3, 1, 18, 19]-=-. Definition 3.1. An L ! ⊗-model is given by a category equipped with 1. a symmetric monoidal structure (I, ⊗); 2. a symmetric monoidal endofunctor � !, ϕI : I 3. a monoidal comonad structure A ɛ �� !... |

96 | What is a categorical model of intuitionistic linear logic
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Citation Context ...c Models of multiplicative intuitionistic linear logic. I recall the definition of categorical model of multiplicative intuitionistic linear logic as it has been developed in the literature, see e.g. =-=[17, 20, 2, 3, 1, 18, 19]-=-. Definition 3.1. An L ! ⊗-model is given by a category equipped with 1. a symmetric monoidal structure (I, ⊗); 2. a symmetric monoidal endofunctor � !, ϕI : I 3. a monoidal comonad structure A ɛ �� !... |

93 | A mixed linear non-linear logic: proofs, terms and models
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- 1995
(Show Context)
Citation Context ...c Models of multiplicative intuitionistic linear logic. I recall the definition of categorical model of multiplicative intuitionistic linear logic as it has been developed in the literature, see e.g. =-=[17, 20, 2, 3, 1, 18, 19]-=-. Definition 3.1. An L ! ⊗-model is given by a category equipped with 1. a symmetric monoidal structure (I, ⊗); 2. a symmetric monoidal endofunctor � !, ϕI : I 3. a monoidal comonad structure A ɛ �� !... |

48 | Finiteness spaces
- Ehrhard
(Show Context)
Citation Context ...basic laws of differentiation (the product and chain rules, the commutation relations, etc.). 1 Introduction Recent developments in the model theory of linear logic, starting with the work of Ehrhard =-=[6, 7]-=-, have uncovered a variety of models with differential structure. Examples include Köthe sequence spaces [6], finiteness spaces [7], the relational model, generalised species of structures [11, 12], i... |

44 | The differential lambda-calculus
- Ehrhard, Regnier
- 2003
(Show Context)
Citation Context ...here. From the abstract theoretical viewpoint, the consideration of L !,⊸ ⊗,+× -models equipped with differential structure as categorical models of the differential λ-calculus of Ehrhard and Regnier =-=[9]-=- will be considered in the full version of the paper. A more important next step, however, is to work out the type and proof theory of L !,⊸ ⊗,+× -models, both as a term assignment system and a graphi... |

40 |
Linear Spaces and Differentiation Theory
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Citation Context ...els mentioned at the beginning of the Introduction are examples will be covered. More interestingly, I conjecture that the category of convenient vector spaces and linear maps of Frölicher and Kriegl =-=[13]-=- provides yet another example; as so may be the case, indicated to me by Anders Kock in correspondence, with the category of modules for the ring object of line type in some models of Synthetic Differ... |

31 |
Differential interaction nets
- Ehrhard, Regnier
- 2006
(Show Context)
Citation Context ...operator D provides a linear approximation D[f]x : A ⊸ B for every function f : A �� B at any point x : A. The algebra underlying these models has also been investigated recently. Ehrhard and Regnier =-=[8]-=-, isolated local-additive and commutative bialgebraicexponential structure and explained, amongst other things, how they support Addendum: The Strength Law (14) in Definition 4.2(1) is redundant. 1s2 ... |

26 | On Köthe sequence spaces and linear logic
- Ehrhard
- 2002
(Show Context)
Citation Context ...basic laws of differentiation (the product and chain rules, the commutation relations, etc.). 1 Introduction Recent developments in the model theory of linear logic, starting with the work of Ehrhard =-=[6, 7]-=-, have uncovered a variety of models with differential structure. Examples include Köthe sequence spaces [6], finiteness spaces [7], the relational model, generalised species of structures [11, 12], i... |

23 | Categorical models of linear logic revisited. Available at http://www.pps.jussieu.fr/mellies/papers/catmodels.ps.gz
- Mellies
- 2002
(Show Context)
Citation Context |

10 | Differential categories
- Blute, Cockett, et al.
- 2005
(Show Context)
Citation Context ...ial structure. Examples include Köthe sequence spaces [6], finiteness spaces [7], the relational model, generalised species of structures [11, 12], interaction systems [15], and complete semilattices =-=[4]-=-. This differential structure manifests itself as differential operators. In this context, a differential operator is a natural linear map !A ⊸ B that, when embedded as a map A �� B �� !A ⊸ A ⊸ B (1) ... |

10 |
catégories et machines
- Logiques
- 1988
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Citation Context |

9 | Mathematical models of computational and combinatorial structures. Invited address for
- Fiore
- 2005
(Show Context)
Citation Context ...hard [6, 7], have uncovered a variety of models with differential structure. Examples include Köthe sequence spaces [6], finiteness spaces [7], the relational model, generalised species of structures =-=[11, 12]-=-, interaction systems [15], and complete semilattices [4]. This differential structure manifests itself as differential operators. In this context, a differential operator is a natural linear map !A ⊸... |

7 | The cartesian closed bicategory of generalised species of structures
- Fiore, Gambino, et al.
- 2008
(Show Context)
Citation Context ...hard [6, 7], have uncovered a variety of models with differential structure. Examples include Köthe sequence spaces [6], finiteness spaces [7], the relational model, generalised species of structures =-=[11, 12]-=-, interaction systems [15], and complete semilattices [4]. This differential structure manifests itself as differential operators. In this context, a differential operator is a natural linear map !A ⊸... |

5 |
Linear λ-calculus and categorical models revisited
- Benton, Bierman, et al.
- 1993
(Show Context)
Citation Context |

3 | Fock space: A model of linear exponential types
- Blute, Panangaden, et al.
- 1994
(Show Context)
Citation Context ...y, my analysis of differential structure starts with the consideration of operators as in (3). These I call creation operators; as, interpreting the exponential as the bosonic Fock space construction =-=[5]-=-, that models quantum systems of many identical non-interacting particles, they intuitively correspond to operators modelling particle creation. Indeed, categorical models of multiplicative intuitioni... |

3 |
Synthetic Differential Geometry. Number 333
- Kock
- 2006
(Show Context)
Citation Context ...1⊗e �� � �� ��� A ⊗ I 3. Multiplication. A ⊗ !A 1⊗d �� A ⊗ !A ⊗ !A ∂ �� !A ∂⊗δ A ⊗ !A ⊗ !A ∂⊗1 �� � !A ⊗ !A ���� ��� 1⊗m ����� m �� !A ��� �� �� �� ∂ �� A ⊗ !A δ �� !!A� � ∂! �� !A ⊗ !!A 11 (14) (15) =-=(16)-=- The above form for creation and annihilation operators is non-standard. More commonly, see e.g. [14], the literature deals with creation operators ∂v : !A �� !A for vectors v : I �� A and annihilatio... |

2 |
Eric Jul and Bjarne Steensgaard. Implementation of distributed objects in Emerald
- Fiore
- 2004
(Show Context)
Citation Context ...e exponential. The present axiomatisation of creation maps has been directly influenced by and developed through a thorough analysis of the differential structure of generalised species of structures =-=[10, 11]-=-, which is a bicategorical generalisation of that of the relational model of linear logic. Organisation and contribution of the paper. Section 2 provides basic background on biproduct structure. The e... |

2 |
A logical investigation of interaction systems
- Hyvernat
- 2005
(Show Context)
Citation Context ...variety of models with differential structure. Examples include Köthe sequence spaces [6], finiteness spaces [7], the relational model, generalised species of structures [11, 12], interaction systems =-=[15]-=-, and complete semilattices [4]. This differential structure manifests itself as differential operators. In this context, a differential operator is a natural linear map !A ⊸ B that, when embedded as ... |

1 |
What is a categorical model of linear logic? Notes for research students
- Schalk
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Citation Context |