## On the construction of free algebras for equational systems (2009)

Venue: | IN: SPECIAL ISSUE FOR AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007). VOLUME 410 OF THEORETICAL COMPUTER SCIENCE |

Citations: | 5 - 4 self |

### BibTeX

@INPROCEEDINGS{Fiore09onthe,

author = {Marcelo Fiore and Chung-Kil Hur},

title = {On the construction of free algebras for equational systems},

booktitle = {IN: SPECIAL ISSUE FOR AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007). VOLUME 410 OF THEORETICAL COMPUTER SCIENCE},

year = {2009},

pages = {1704--1729},

publisher = {Elsevier}

}

### OpenURL

### Abstract

The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.

### Citations

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Citation Context ...equational systems. Proposition 6.4 Let S be an equational system. If the forgetful functor US : S-Alg→ C has a left adjoint, then it is monadic. PROOF. To show the monadicity of US by Beck’s theorem =-=[25]-=-, it is enough to show that US creates coequalizers of parallel pairs f, g : (X, r) → (Y, s) in S-Alg for which f, g : X → Y has an absolute coequalizer, say e : Y Z, in C . In this case, then, Σe i... |

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Citation Context ...nt can be described in two stages as the composition of a left adjoint to UΣ followed by a left adjoint to JS. Conditions for the existence of the former have been studied in the literature (see e.g. =-=[3,4]-=-). Thus, we concentrate here on obtaining a reflection to the embedding of S-Alg into Σ-Alg. 4.1 Algebraic coequalizers The construction of free algebras for an equational system explained in Section ... |

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Citation Context ...re recent applications have had to be based on the more general categorical algebra. Examples include models of name-passing process calculi [12,32], theories of abstract syntax with variable binding =-=[13,16]-=-, and the algebraic treatment of computational effects [27,28]. In the above and most other applications of equational theories, the existence and construction of initial and/or free algebras, and con... |

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Citation Context ...an affine doubly closed structure (see [30]) given by: • the cartesian closed structure ( 1,×, (=)(−) ) , and • the symmetric monoidal closed structure ( 1,⊗, (−)( (=) ) induced by Day’s construction =-=[9]-=- from the symmetric monoidal structure (∅,unionmulti) on Iop given by the empty set ∅ and the disjoint-union tensor unionmulti. Note that, as the tensor unit is terminal, the tensor product comes equi... |

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Citation Context ...nce of free models. We need consider the presheaf category SetI, for I the (essentially small) category of finite sets and injections. The category SetI carries an affine doubly closed structure (see =-=[30]-=-) given by: • the cartesian closed structure ( 1,×, (=)(−) ) , and • the symmetric monoidal closed structure ( 1,⊗, (−)( (=) ) induced by Day’s construction [9] from the symmetric monoidal structure (... |

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Citation Context ...guages and the specification of abstract data types pioneered by the ADJ group [19], and the abstract description of powerdomain constructions as free algebras of non-determinism advocated by Plotkin =-=[21,26]-=- (see also [1]). While these developments essentially belong to the realm of universal algebra, more recent applications have had to be based on the more general categorical algebra. Examples include ... |

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Citation Context ...xamples of 1 Research supported by a Samsung Scholarship from the Samsung Foundation of Culture. Preprint submitted to Elsevier 27 November 2008 equational systems include enriched algebraic theories =-=[23,31]-=-, algebras for a monad, monoids in a monoidal category, etc. (see Section 3). The original motivation for the development of the theory of equational systems arose from the need of a mathematical theo... |

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Citation Context ...nt can be described in two stages as the composition of a left adjoint to UΣ followed by a left adjoint to JS. Conditions for the existence of the former have been studied in the literature (see e.g. =-=[3,4]-=-). Thus, we concentrate here on obtaining a reflection to the embedding of S-Alg into Σ-Alg. 4.1 Algebraic coequalizers The construction of free algebras for an equational system explained in Section ... |

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Citation Context ...nature. Under this view, thus, we obtain an abstract notion of term as a generalized element U → TΣV . Assume now that C is symmetric monoidal closed (with structure I, ⊗, [−,=]) and that Σ is strong =-=[24]-=-, with strength stX,V : X ⊗ Σ(V )→ Σ(X ⊗ V ). It follows that TΣ is strong, say with strength stX,V : X ⊗ TΣ(V ) → TΣ(X ⊗ V ) providing a means to distribute parameters within terms as specified by st... |

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Citation Context ...is is also the case for its colimiting cone { cn : Zn → Z∞ }n≥0. 2 10 4.2 Finitary free Σ-algebras The following result describes a well-known condition for the existence of free Σ-algebras (see e.g. =-=[2]-=-). Theorem 4.6 Let Σ be an endofunctor on a category C with finite coproducts. If C has colimits of ω-chains and Σ preserves them, then the forgetful functor UΣ : Σ-Alg→ C has a left adjoint. The free... |

34 | Algebraic operations and generic effects
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Citation Context ...l categorical algebra. Examples include models of name-passing process calculi [12,32], theories of abstract syntax with variable binding [13,16], and the algebraic treatment of computational effects =-=[27,28]-=-. In the above and most other applications of equational theories, the existence and construction of initial and/or free algebras, and consequently of monads, plays a central role; as so does the stud... |

31 | Computational Effects and Operations: an Overview
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Citation Context ...l categorical algebra. Examples include models of name-passing process calculi [12,32], theories of abstract syntax with variable binding [13,16], and the algebraic treatment of computational effects =-=[27,28]-=-. In the above and most other applications of equational theories, the existence and construction of initial and/or free algebras, and consequently of monads, plays a central role; as so does the stud... |

30 |
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(Show Context)
Citation Context ... examples of this situation, we consider algebraic models of the untyped λ-calculus up to βη identities (see Section 7.2) and the recently introduced nominal equational theories of Clouston and Pitts =-=[7]-=- (see Section 7.3). Monads and categories of algebras for equational systems are discussed in Section 6. In the vein of the above results, we provide conditions under which the monadicity and cocomple... |

26 | Variations on algebra: monadicity and generalisations of equational theories
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Citation Context ...xamples of 1 Research supported by a Samsung Scholarship from the Samsung Foundation of Culture. Preprint submitted to Elsevier 27 November 2008 equational systems include enriched algebraic theories =-=[23,31]-=-, algebras for a monad, monoids in a monoidal category, etc. (see Section 3). The original motivation for the development of the theory of equational systems arose from the need of a mathematical theo... |

23 |
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Citation Context ...[6], and it can be related to that of [18] as follows. The proof of the dual of Corollary 5.9 (3) together with the construction of cofree coalgebras for endofunctors by terminal sequences of Worrell =-=[33]-=-, gives a construction of cofree coalgebras for equational cosystems on a locally presentable base category with an accessible functorial signature that preserves monomorphisms. This is a variation of... |

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Citation Context ... the concept of equational system, and its dual of equational cosystem (viz., an equational system on an opposite base category), had already been considered in the literature. For instance, Fokkinga =-=[14]-=- introduces the more general concept of law between so-called transformers, but only studies initial algebras for the laws that are equational systems; Ĉırstea [6] introduces the concept of coequatio... |

18 |
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Citation Context ... are algebras for an equational cosystem). On the other hand, the price paid for all this generality is that the important connection between enriched algebraic theories and enriched Lawvere theories =-=[29]-=- is lost for equational systems. An equational system S = (C : Σ B Γ ` L = R) is defined as a parallel pair L,R : Σ-Alg→ Γ-Alg of functors between categories of algebras over a base category C . In th... |

15 | A fully abstract model for the π-calculus - Fiore, Moggi, et al. - 2002 |

14 | Presheaf models for the pi-calculus
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Citation Context ...ts essentially belong to the realm of universal algebra, more recent applications have had to be based on the more general categorical algebra. Examples include models of name-passing process calculi =-=[12,32]-=-, theories of abstract syntax with variable binding [13,16], and the algebraic treatment of computational effects [27,28]. In the above and most other applications of equational theories, the existenc... |

13 | A formal calculus for informal equality with binding
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Citation Context ...ion. For brevity, we only consider the single-sorted case; the multi-sorted one being treated analogously. All these results can be also seen to apply to the nominal algebras of Gabbay and Mathijssen =-=[15]-=-. However, we do not dwell on this here. 35 7.3.1 Nominal sets For a fixed countably infinite set A of atoms, the group S0(A) of finite permutations of atoms consists of the bijections on A that fix a... |

12 |
Dualising initial algebras
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Citation Context ...introduces the concept of coequation between abstract cosignatures, which is equivalent to our notion of equational cosystem, and studies final coalgebras for them; Ghani, Lüth, De Marchi, and Power =-=[18]-=- introduce the concept of functorial coequational presentations, which is equivalent to our notion of 45 equational cosystem on a locally presentable base category with an accessible functorial signat... |

11 |
An algebra-coalgebra framework for system specification
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(Show Context)
Citation Context ...literature. For instance, Fokkinga [14] introduces the more general concept of law between so-called transformers, but only studies initial algebras for the laws that are equational systems; Ĉırstea =-=[6]-=- introduces the concept of coequation between abstract cosignatures, which is equivalent to our notion of equational cosystem, and studies final coalgebras for them; Ghani, Lüth, De Marchi, and Power... |

10 |
Second-order and dependently-sorted abstract syntax
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(Show Context)
Citation Context ...lculus [32], and (ii) Σ-monoids (see Section 7.2), which are needed for the initial algebra approach to the semantics of languages with variable binding and capture-avoiding simultaneous substitution =-=[13,10]-=-. Indeed, these two examples respectively highlight two inadequacies of enriched algebraic theories in applications: (i) models may require a theory based on more than one enrichment, as it is the cas... |

9 | Free-algebra models for the π-calculus - Stark |

7 | Equational systems and free constructions - Fiore, Hur - 2007 |

5 |
Rewriting via coinserters
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(Show Context)
Citation Context ...ries of Kelly and Power [23], which we have exhibited as equational systems in Section 3.4 (2), one may also consider the categorical presentation of term rewriting via coinserters of Ghani and Lüth =-=[17]-=- in the setting of algebraic theories on the category of preorders. In this vein, we have developed a theory of free constructions for inequational systems in an abstract-rewriting enriched setting to... |

5 |
Σ-monoids: A higher-order syntax with metavariables
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(Show Context)
Citation Context ...quivalence classes of λ-terms with variables from V , and thus provides an abstract notion of syntax for the λ-calculus (see [13]). The syntactic description of free Σλ-monoids has been considered in =-=[20,10]-=-. The βη identities for a λ-prealgebra on X are expressed, in the internal language, by the following equations (β) f : XV , x : X ` app(abs(f), x) = sub(f〈x〉) : X (η) x : X ` abs ( λv : V. app(x, var... |

5 |
A note on actions of a monoidal category. Theory and Applications of Categories
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(Show Context)
Citation Context ...C and (B, b : I → B), (C, c : I → C) ∈ I/C . Remark The notion of pointed strength arises as a special case of that of a strength for an action of a monoidal category on a category (see [10] and also =-=[22]-=-). For an endofunctor Σ with a pointed strength st on a monoidal category C , the category of Σ-monoids Σ-Mon(C ) has objects given by quadruples (X, s,m, e) where (X, s : ΣX → X) is a Σ-algebra and (... |

4 |
A fully abstract model for the pi-calculus
- Fiore, Moggi, et al.
- 2002
(Show Context)
Citation Context ...ts essentially belong to the realm of universal algebra, more recent applications have had to be based on the more general categorical algebra. Examples include models of name-passing process calculi =-=[12,32]-=-, theories of abstract syntax with variable binding [13,16], and the algebraic treatment of computational effects [27,28]. In the above and most other applications of equational theories, the existenc... |