## Modules over Monads and Linearity

Citations: | 4 - 2 self |

### BibTeX

@MISC{Hirschowitz_modulesover,

author = {André Hirschowitz and Marco Maggesi},

title = {Modules over Monads and Linearity},

year = {}

}

### OpenURL

### Abstract

Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be

### Citations

216 |
Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics
- Lane
- 1998
(Show Context)
Citation Context ...onads and Algebras We briefly recall some standard material about monads and algebras. Experienced readers may want to skip this section or just use it as reference for our notations. Mac Lane’s book =-=[15]-=- can be used as reference on this material. 3 http://golem.ph.utexas.edu/string/archives/000715.htmlsLet C be a category. A monad over C is a monoid in the category C → C of endofunctors of C. In more... |

145 | A new approach to abstract syntax involving binders
- Gabbay, Pitts
- 1999
(Show Context)
Citation Context ...ducts). The monad LC is initial in Mon T0 . In fact, the previous theorem can be generalized as follows (interested readers may also want to look at other works on higher order abstract syntax, e.g., =-=[6,13,10]-=- see also our [12]). Let R be a monad over Set. We define an arity to be a list of nonnegative integers. We denote by N ∗ the set of arities. For each arity (a1, . . .,ar), and for any R-module M, we ... |

96 |
Semantical analysis of higher-order abstract syntax
- Hofmann
- 1999
(Show Context)
Citation Context ...ducts). The monad LC is initial in Mon T0 . In fact, the previous theorem can be generalized as follows (interested readers may also want to look at other works on higher order abstract syntax, e.g., =-=[6,13,10]-=- see also our [12]). Let R be a monad over Set. We define an arity to be a list of nonnegative integers. We denote by N ∗ the set of arities. For each arity (a1, . . .,ar), and for any R-module M, we ... |

77 | Monadic presentations of lambda terms using generalized inductive types
- Altenkirch, Reus
- 1999
(Show Context)
Citation Context ...by the singleton map ηX : x ∈ X ↦→ [x] ∈ L(X). Example 2 (Lambda Calculus). This example will be worked out with the necessary details in section 5.1, but let us give early some basic ideas (see also =-=[1]-=-). We denote by FV (M) the set of free variables of a λ-term M. For a fixed set X, consider the collection of λ-terms (modulo α-conversion) with free variables in X: LC(X) := {M| FV (M) ⊂ X}. Given a ... |

68 | R.: De Bruijn notation as a nested datatype
- Bird, Paterson
- 1999
(Show Context)
Citation Context ... that T possesses a least fixpoint that we denote by LC (LC standing for λ-calculus, cfr. the example in section 2). We consider LC(X) as the set of λ-terms with free variables taken from X (see also =-=[3]-=-). In fact, 5 There is a lexical conflict here with algebra of monads introduced in section 2, which is deeply rooted in the literature anyway. We hope that this will not lead to any confusion.sthe st... |

62 | Koszul duality of operads and homology of partition posets, from: “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic
- Fresse
(Show Context)
Citation Context ...er monads we have been able to find is on a blog by Urs Schreiber. 3 On the other hand, modules over operads have been introduced by M. Markl ([16, 17]) and are commonly used by topologists (see e.g. =-=[9,14,4]-=-). In [8], such modules over operads have been considered, under the name of actions, in the context of semantics. We think that the notions of module over a monad and linear transformations deserve m... |

46 | Semantics of name and value passing
- Fiore, Turi
- 2001
(Show Context)
Citation Context ...ave been able to find is on a blog by Urs Schreiber. 3 On the other hand, modules over operads have been introduced by M. Markl ([16, 17]) and are commonly used by topologists (see e.g. [9,14,4]). In =-=[8]-=-, such modules over operads have been considered, under the name of actions, in the context of semantics. We think that the notions of module over a monad and linear transformations deserve more atten... |

34 | Generalised folds for nested datatypes
- Bird, Paterson
- 1999
(Show Context)
Citation Context ...see also [12]). The idea that the notion of monad is suited for modelling substitution concerning syntax (and semantics) has been retained by many recent contributions concerned with syntax (see e.g. =-=[2, 11, 18]-=-) although some other settings have been considered. Notably in [6] the authors work within a setting roughly based on operads (although they do not write this word down; the definition of operad is o... |

21 | Abstract syntax and variable binding (extended abstract
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...panion definition of modules over an operad (c.f. e.g. [17,9]) follows easily from the observation [19] that operads are monoids in a suitable monoidal category. This monoidal structure is central in =-=[6]-=-. Given a left R-module M, we can introduce the mbind operator which, to each arrow f : X → RY , associates an arrow mbind f := MX → MY defined by mbindf := ρ · Mf. The axioms of left module are equiv... |

17 |
A compactification of the real configuration space as an operadic completion, preprint
- Markl
- 1996
(Show Context)
Citation Context ...he natural ideas involved, the only mention of modules over monads we have been able to find is on a blog by Urs Schreiber. 3 On the other hand, modules over operads have been introduced by M. Markl (=-=[16, 17]-=-) and are commonly used by topologists (see e.g. [9,14,4]). In [8], such modules over operads have been considered, under the name of actions, in the context of semantics. We think that the notions of... |

11 |
Homotopy theory of coalgebras
- Smirnov
(Show Context)
Citation Context ... refer to the category D as the range of M. M · R ρ �� Mη 1M �� ����������� M M · I Remark 1. The companion definition of modules over an operad (c.f. e.g. [17,9]) follows easily from the observation =-=[19]-=- that operads are monoids in a suitable monoidal category. This monoidal structure is central in [6]. Given a left R-module M, we can introduce the mbind operator which, to each arrow f : X → RY , ass... |

7 | From left modules to algebras over an operad: application to combinatorial Hopf algebras
- Livernet
- 2006
(Show Context)
Citation Context ...er monads we have been able to find is on a blog by Urs Schreiber. 3 On the other hand, modules over operads have been introduced by M. Markl ([16, 17]) and are commonly used by topologists (see e.g. =-=[9,14,4]-=-). In [8], such modules over operads have been considered, under the name of actions, in the context of semantics. We think that the notions of module over a monad and linear transformations deserve m... |

7 |
Pseudo-distributive laws and axiomatics for variable binding
- Tanaka, Power
- 2006
(Show Context)
Citation Context ...ad is on Wikipedia; operads and monads are not too far from each other). As they mention, their approach is, to some extent, equivalent to an approach through monads. It has been both applied e.g. in =-=[21]-=- and generalized e.g. in [20]. Another approach to syntax with bindings, initiated by Gabbay and Pitts [10], relies on a systematic consideration of freshness, an issue which is definitely ignored in ... |

6 |
and Tarmo Uustalu, Substitution in non-wellfounded syntax with variable binding, Theor
- Matthes
(Show Context)
Citation Context ...see also [12]). The idea that the notion of monad is suited for modelling substitution concerning syntax (and semantics) has been retained by many recent contributions concerned with syntax (see e.g. =-=[2, 11, 18]-=-) although some other settings have been considered. Notably in [6] the authors work within a setting roughly based on operads (although they do not write this word down; the definition of operad is o... |

5 | Explicit substitutions and higher-order syntax, MERLIN ’03
- Ghani, Uustalu
- 2003
(Show Context)
Citation Context ...see also [12]). The idea that the notion of monad is suited for modelling substitution concerning syntax (and semantics) has been retained by many recent contributions concerned with syntax (see e.g. =-=[2, 11, 18]-=-) although some other settings have been considered. Notably in [6] the authors work within a setting roughly based on operads (although they do not write this word down; the definition of operad is o... |

4 |
constructions for topological operads and the Goodwillie derivatives of the identity
- Bar
(Show Context)
Citation Context ...er monads we have been able to find is on a blog by Urs Schreiber. 3 On the other hand, modules over operads have been introduced by M. Markl ([16, 17]) and are commonly used by topologists (see e.g. =-=[9,14,4]-=-). In [8], such modules over operads have been considered, under the name of actions, in the context of semantics. We think that the notions of module over a monad and linear transformations deserve m... |

4 |
A unified categorytheoretic formulation of typed binding signatures
- Tanaka, Power
- 2005
(Show Context)
Citation Context ...nd monads are not too far from each other). As they mention, their approach is, to some extent, equivalent to an approach through monads. It has been both applied e.g. in [21] and generalized e.g. in =-=[20]-=-. Another approach to syntax with bindings, initiated by Gabbay and Pitts [10], relies on a systematic consideration of freshness, an issue which is definitely ignored in the monadic or operadic appro... |

2 |
On the structure of substitution. Invited address for the 22nd
- Fiore
- 2006
(Show Context)
Citation Context ...show examples concerning in particular abstract syntax and lambda-calculus. 1 Introduction Substitution is a major operation. Its relevance to computer sciences has been stressed constantly (see e.g. =-=[7]-=-). Mathematicians of the last century have coined two strongly related notions which capture the formal properties of this operation. The first one is the notion of monad, while the second one is the ... |

2 | M.: The algebraicity of the lambda-calculus
- Hirschowitz, Maggesi
(Show Context)
Citation Context ... initial in Mon T0 . In fact, the previous theorem can be generalized as follows (interested readers may also want to look at other works on higher order abstract syntax, e.g., [6,13,10] see also our =-=[12]-=-). Let R be a monad over Set. We define an arity to be a list of nonnegative integers. We denote by N ∗ the set of arities. For each arity (a1, . . .,ar), and for any R-module M, we define the R-modul... |

1 |
Models for operads. arXiv:hep-th/9411208v1
- Markl
- 1994
(Show Context)
Citation Context ...he natural ideas involved, the only mention of modules over monads we have been able to find is on a blog by Urs Schreiber. 3 On the other hand, modules over operads have been introduced by M. Markl (=-=[16, 17]-=-) and are commonly used by topologists (see e.g. [9,14,4]). In [8], such modules over operads have been considered, under the name of actions, in the context of semantics. We think that the notions of... |

1 |
lambda calcul vu comme monade initiale. Master’s thesis, Université de Nice
- Le
(Show Context)
Citation Context ... are easily checked to be L-linear. 6.2 Simply Typed Lambda Calculus Our second example of typed monad is the simply-typed λ-calculus. We denote by τ the set of simple types τ := ∗ | τ → τ. Following =-=[22]-=-, we consider the syntactic typed λ-calculus as an assignment V ↦→ LCτ(V ), where V = � t∈τ (Vt) is a (variable) set (of typed variables) while LCτ(V ) = � (LCτ(V ))t t∈τ is the set of typed λ-terms (... |