## MFPS 25 Preliminary Proceedings Two cotensors in one: Presentations of algebraic theories for local state and fresh names

### BibTeX

@MISC{Staton_mfps25,

author = {Sam Staton},

title = {MFPS 25 Preliminary Proceedings Two cotensors in one: Presentations of algebraic theories for local state and fresh names},

year = {}

}

### OpenURL

### Abstract

Various situations in computer science call for categories that support both cartesian closed and monoidal closed structure. Such situations include (i) models of local state, where the monoidal product describes disjointness of memory, and (ii) treatment of fresh names, as required in models of the π-calculus. I propose a technique to embed the two closed structures into one single structure. To demonstrate the technique, I show how previously studied theories of local state and fresh names can be understood formally as presentations of (enriched) algebraic theories. 1

### Citations

1064 | Bigraphs and mobile processes
- Jensen, Milner
- 2003
(Show Context)
Citation Context ...otensor’ structure, −•: as will be explained. [A → X] = [i(A) −• X] [A ⊸ X] = [iS(A) −• X] 1.1 Cartesian and monoidal structures in models of the π-calculus and local state Recall that the π-calculus =-=[16]-=- is a language that allows communication of channel names along the channels themselves. In the semantics of the π-calculus (e.g. [7,28]) the two closed structures both play crucial roles. The cartesi... |

766 | Notions of computation and monads
- Moggi
- 1991
(Show Context)
Citation Context ...trong (Prop. 3.1(ii)), and deduce that the monad T I respect to the tensor •; the result follows. (Σ,Eq) on I is strong with ✷ Proposition 3.4 is important because to use Moggi’s monadic metalanguage =-=[17]-=-, it is necessary for the monad to have a cartesian strength. Notice that the on A will not, in general, be strong with respect to the cartesian monad T A (Σ,Eq) structure of A, even if the restrictio... |

393 | Basic concepts of enriched category theory
- Kelly
- 2005
(Show Context)
Citation Context ...diments of algebraic theories and Lawvere theories in the enriched setting. The reader will find more detail and discussion, and more generality, in [14,22,26]; a reference for enriched categories is =-=[12]-=-. In Section 3.4, we consider the relevance of the theory to the categories I and A, considered above. 3.1 Preliminaries on enriched categories To recall the general definitions, we fix a cocomplete m... |

226 | A new approach to abstract syntax with variable binding
- Gabbay, Pitts
(Show Context)
Citation Context ...ense of [14]. In the process, we introduce a new way of understanding the two closed structures of I. This complements the well-established techniques of bunched implications [18,25] and nominal sets =-=[9]-=-. A next step is to better understand the kinds of reasoning permitted in models of Lawvere A-theories. As a first step, Power [24,23] provides an analysis of algebraic theories with a block operator ... |

101 |
On closed categories of functors
- Day
- 1970
(Show Context)
Citation Context ...tructures of I with a single closed structure in A (in Secs. 2.4–2.6). There is a summary of notation in Section 2.7. 2.1 Preliminaries: promonoidal categories To begin, recall the observation of Day =-=[3,4]-=-, that monoidal biclosed structures on (covariant) presheaf categories (Set C ) correspond to ‘promonoidal structures’ on small categories (C). (Day treats the general case of enriched functor categor... |

63 |
Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads
- Kelly, Power
- 1993
(Show Context)
Citation Context ...structure. In this paper, we take our illustrations from algebraic theories for local state and the π-calculus; there is a rich body of work on algebraic theories for monoidal closed categories (e.g. =-=[14,22,26]-=-). A crucial contribution of this paper is the observation that both kinds of function This is a preliminary version. The final version will be published in Electronic Notes in Theoretical Computer Sc... |

56 | Notions of computation determine monads
- Plotkin, Power
- 2002
(Show Context)
Citation Context ...y give ‘effects’, 1 deadlock −−−−−→ Pf(∅) and 1 choice −−−−→ Pf(2). In the literature, strong monads arise on a presheaf category I, for modelling local state and the π-calculus. As Plotkin and Power =-=[21]-=- and Stark [29] have observed, the algebras for these monads can be described in terms of operators — such as new and lookup above — subject to equations expressed as commuting diagrams. For instance,... |

40 | Syntactic control of interference revisited
- O’Hearn, Power, et al.
- 1999
(Show Context)
Citation Context ...volving an atom which is known to be distinct from all the others. Other examples. Various authors have used more sophisticated monoidal structures on presheaf categories. The structure considered in =-=[19]-=-, for modelling syntactic control of interference, appears to be an example of a compatibility structure. It would be interesting to investigate a process of adjoining ‘indeterminates’ to compatibilit... |

34 |
Structures Defined by Finite Limits in the Enriched Context I, Cahiers de Topologie et Géométrie Différentielle 23
- Kelly
- 1982
(Show Context)
Citation Context ...and every finitely presentable n in V. Note that V has tensors given by monoidal product, and cotensors given by the left closed structure, and that it is locally finitely presentable as a V-category =-=[13]-=-. 3.2 Presentations of algebraic theories A V-signature is a family (Σn ∈ V) n∈|Vf| of objects of V. In other words, a V-signature is a functor |Vf| → V. For each finitely presentable n in V, the obje... |

33 | On Bunched Typing
- O'Hearn
- 2003
(Show Context)
Citation Context ...semantics of which is determined by a morphism new : [A ⊸ X] → X making a denotation out of an element requiring a fresh name. In giving the semantics of a programming language with local state (e.g. =-=[18]-=-), both closed structures are used again. The cartesian closed structure is used for an operator lookup : A × [V → X] → X, giving semantics to commands such as let v := !a in M. The monoidal closed st... |

31 | Computational Effects and Operations: an Overview
- Plotkin, Power
- 2004
(Show Context)
Citation Context ...A ⊸ X] → X, giving semantics to commands such as block a := v in M, that introduce local state. (Here V is a set of values.) 1.2 Algebraic operations and generic effects In a sequence of papers (e.g. =-=[20]-=-), Plotkin, Power and collaborators have outlined a program to understand Moggi’s monadic semantics from a more axiomatic perspective: the monads can be presented in terms of operators and equations. ... |

30 |
Nominal equational logic
- Clouston, Pitts
(Show Context)
Citation Context ...echnique for synthesizing an equational logic from an enriched Lawvere theory. It will be interesting to evaluate this technique for the 3Staton theories in Section 4. Nominal equational logic (NEL) =-=[2]-=- and nominal algebra [8] are reasoning systems for theories involving freshness and binding. Unfortunately the theories considered here, for the π-calculus and local state, do not seem to be ‘nominal ... |

16 | Enriched Lawvere theories
- Power
- 1999
(Show Context)
Citation Context ... theories and their presentations We survey some rudiments of algebraic theories and Lawvere theories in the enriched setting. The reader will find more detail and discussion, and more generality, in =-=[14,22,26]-=-; a reference for enriched categories is [12]. In Section 3.4, we consider the relevance of the theory to the categories I and A, considered above. 3.1 Preliminaries on enriched categories To recall t... |

15 |
A fully abstract model for the π-calculus
- Fiore, Moggi, et al.
- 2002
(Show Context)
Citation Context ...dels of the π-calculus and local state Recall that the π-calculus [16] is a language that allows communication of channel names along the channels themselves. In the semantics of the π-calculus (e.g. =-=[7,28]-=-) the two closed structures both play crucial roles. The cartesian structure is used for an operator input : A × [A → X] → X, giving semantics to the input behaviour of the π-calculus; informally, inp... |

13 | A formal calculus for informal equality with binding
- Gabbay, Mathijssen
- 2007
(Show Context)
Citation Context ...g an equational logic from an enriched Lawvere theory. It will be interesting to evaluate this technique for the 3Staton theories in Section 4. Nominal equational logic (NEL) [2] and nominal algebra =-=[8]-=- are reasoning systems for theories involving freshness and binding. Unfortunately the theories considered here, for the π-calculus and local state, do not seem to be ‘nominal theories’, although, con... |

13 | A unifying model of variables and names
- Miculan, Yemane
- 2005
(Show Context)
Citation Context ... Models for the π-calculus with distinctions. In some aspects of the π-calculus, particularly open bisimulation [27], it is important to keep track of which names are distinct. To this end, following =-=[10,15,30]-=-, let D be the category whose objects (A, dA) are pairs of a finite set A together with an irreflexive symmetric binary relation dA on A; a morphism f : (A, dA) → (B, dB) is given by a function f : A ... |

11 | Relationally staged computations in calculi of mobile processes
- Ghani, Yemane, et al.
(Show Context)
Citation Context ... Models for the π-calculus with distinctions. In some aspects of the π-calculus, particularly open bisimulation [27], it is important to keep track of which names are distinct. To this end, following =-=[10,15,30]-=-, let D be the category whose objects (A, dA) are pairs of a finite set A together with an irreflexive symmetric binary relation dA on A; a morphism f : (A, dA) → (B, dB) is given by a function f : A ... |

10 |
Term equational systems and logics
- Fiore, Hur
- 2008
(Show Context)
Citation Context ...in models of Lawvere A-theories. As a first step, Power [24,23] provides an analysis of algebraic theories with a block operator — a generalization of the theory of local state in [21]. Fiore and Hur =-=[6]-=- propose a technique for synthesizing an equational logic from an enriched Lawvere theory. It will be interesting to evaluate this technique for the 3Staton theories in Section 4. Nominal equational ... |

7 |
Equational systems and free constructions
- Fiore, Hur
- 2007
(Show Context)
Citation Context ...egler, Miller and Palamidessi [32]. I hope that the constructions in this paper will help to explain that work from a more model theoretic perspective. Acknowledgements It has been helpful to discuss =-=[5,21,29,32]-=- with their authors. I’ve had helpful discussions with Martin Hyland and Paul Levy too. Referees’ comments were helpful. Research supported by EPSRC fellowship EP/E042414/1. 2 Two cotensors, in one In... |

5 |
Construction of Biclosed Categories
- Day
- 1970
(Show Context)
Citation Context ...tructures of I with a single closed structure in A (in Secs. 2.4–2.6). There is a summary of notation in Section 2.7. 2.1 Preliminaries: promonoidal categories To begin, recall the observation of Day =-=[3,4]-=-, that monoidal biclosed structures on (covariant) presheaf categories (Set C ) correspond to ‘promonoidal structures’ on small categories (C). (Day treats the general case of enriched functor categor... |

5 | Semantics for local computational effects
- Power
- 2006
(Show Context)
Citation Context ...tablished techniques of bunched implications [18,25] and nominal sets [9]. A next step is to better understand the kinds of reasoning permitted in models of Lawvere A-theories. As a first step, Power =-=[24,23]-=- provides an analysis of algebraic theories with a block operator — a generalization of the theory of local state in [21]. Fiore and Hur [6] propose a technique for synthesizing an equational logic fr... |

1 |
Nominal Lawvere theories. Unpublished manuscript
- Clouston
- 2009
(Show Context)
Citation Context ...e π-calculus and local state, do not seem to be ‘nominal theories’, although, conversely, it seems that every NEL theory can be understood as an Lawvere A-theory (in the sense of Sec. 3.4). (Clouston =-=[1]-=- recently introduced a bespoke notion of ‘nominal Lawvere theory’ for NEL.) The idea of having two types of atoms to encode two function spaces seems to be implicit in the proof theoretic work of Zieg... |

1 | Monoidal indeterminates and categories of possible worlds
- Hermida, Tennent
- 2009
(Show Context)
Citation Context ...c control of interference, appears to be an example of a compatibility structure. It would be interesting to investigate a process of adjoining ‘indeterminates’ to compatibility structures, following =-=[11]-=-. 16�� � �� �� � �� Staton 5.3 Compatibility structure on categories of monomorphisms Let (C, M, ⌣) be a compatibility structure. Write C[M] for the category whose objects are monomorphisms (SA ↣ A) ... |

1 | Semantics for computational effects: from global to local
- Power
- 2006
(Show Context)
Citation Context ...tablished techniques of bunched implications [18,25] and nominal sets [9]. A next step is to better understand the kinds of reasoning permitted in models of Lawvere A-theories. As a first step, Power =-=[24,23]-=- provides an analysis of algebraic theories with a block operator — a generalization of the theory of local state in [21]. Fiore and Hur [6] propose a technique for synthesizing an equational logic fr... |