## Coalgebraic Components in a Many-Sorted Microcosm

Citations: | 6 - 3 self |

### BibTeX

@MISC{Hasuo_coalgebraiccomponents,

author = {Ichiro Hasuo and Chris Heunen and Bart Jacobs and Ana Sokolova},

title = {Coalgebraic Components in a Many-Sorted Microcosm},

year = {}

}

### OpenURL

### Abstract

Abstract. The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a many-sorted setting. Then we can show that the coalgebraic component calculi of Barbosa are examples, with compositionality of behavior following from microcosm structure. The algebraic structure on these coalgebraic components corresponds to variants of Hughes’ notion of arrow, introduced to organize computations in functional programming. 1

### Citations

1310 | The essence of functional programming
- Wadler
(Show Context)
Citation Context ...s’ arrow [14] (specifically the axiomatization in [19]),13 the only gap being the one explained in Remark 5.1. The notion of arrow generalizes that of monad (modeling effects, i.e. structured output =-=[25, 31]-=-) and that of comonad (modeling structured input [30]); the notion of arrow models “structured computations” in general. See e.g. [19, §2.3]. Definition 5.2 An arrow is a set-theoretic model (Def. 3.2... |

921 |
Categories for the working mathematician
- Lane
- 1998
(Show Context)
Citation Context ...ng isomorphism (as part of the definition of a pseudo functor) is actually the identity.9 Remark 4.3 A standard way to avoid the complication with pseudo algebraic structure is by a coherence result =-=[20,24]-=-. For example: every monoidal category is equivalent to a strict one. This, however, only gives us a biequivalence (see e.g. [27]) between MonCAT and the 2-category of strict monoidal categories. Alth... |

733 | Notions of Computation and Monads
- Moggi
- 1991
(Show Context)
Citation Context ...(T(J × )) I : Set → Set, (1)2 where I is the set of possible input, and J that of output. The computational effect of the component is modeled by a monad T, as is customary in functional programming =-=[25]-=-. The monad T can capture features such as finite nondeterminism (T = Pω), possible non-termination or exceptions (T = 1 + ), probabilistic computation (T = D), global states (T = (S × ) S ), or combi... |

242 |
Braided tensor categories
- Joyal, Street
- 1983
(Show Context)
Citation Context ...ng isomorphism (as part of the definition of a pseudo functor) is actually the identity.9 Remark 4.3 A standard way to avoid the complication with pseudo algebraic structure is by a coherence result =-=[20,24]-=-. For example: every monoidal category is equivalent to a strict one. This, however, only gives us a biequivalence (see e.g. [27]) between MonCAT and the 2-category of strict monoidal categories. Alth... |

228 | A tutorial on (co)algebras and (co)induction
- Jacobs, Rutten
- 1997
(Show Context)
Citation Context ...n >>>. The symbol >>> is taken from that for (Hughes’) arrows, whose relevance is explained in §5.3 FX FZ c X beh(c) Z An important ingredient in the theory of coalgebra is “behavior-by-coinduction” =-=[18]-=-: when a state-based system is final ∼= viewed as an F-coalgebra, then a final F-coalgebra (which very often exists) consists of all the “behaviors” of systems of type F. Moreover, the morphism induce... |

189 | Toposes, Triples, and Theories
- Barr, Wells
- 1985
(Show Context)
Citation Context ...L is called a Lawvere theory (see e.g. [12,15,22]). In a many-sorted setting, such a category L—say a “many-sorted Lawvere theory”—is usually called a finite-product theory, or an FP-theory, see e.g. =-=[4,5]-=-. Definition 3.1 (FP-theory) An FP-theory is a category with finite products. The idea of such categorical presentation of algebraic structure originated in [22]. Significant about the approach is tha... |

160 | Generalising monads to arrows
- Hughes
- 2000
(Show Context)
Citation Context ... application of the many-sorted microcosm framework to component calculi, in §5. It turns out that components as FI,J-coalgebras carry algebraic structure that is a variant of Hughes’ notion of arrow =-=[14,19]-=-. 5 Arrows, generalizing monads, have been used to model structured computations in semantics of functional programming. In §5 we give a rigorous proof that components indeed carry such arrow-like str... |

138 |
Handbook of Categorical Algebra
- Borceux
- 1994
(Show Context)
Citation Context ...ition. In short, we get equations satisfied up-to isomorphism, by weakening a functor into a pseudo functor; the latter preserves identities and composition only up-to coherent isomorphisms (see e.g. =-=[8]-=-). The delicate question here is what it means for a pseudo functor to be FP-preserving. The conditions below are chosen so that Prop. 4.2 holds (see also Rem. 4.3 later). Definition 4.1 (L-category) ... |

112 |
Categorical Logic and Type Theory
- Jacobs
- 1999
(Show Context)
Citation Context ...× Sm, and its out-arity outar(σ) that is some sort S ∈ S; – and a set E of equations. A straightforward presentation of such is as a tuple (S,Σ,E) which is called an algebraic specification (see e.g. =-=[17]-=-). In this paper we prefer different, categorical presentation of algebraic structure. The idea is that algebraic structure can be presented by a category L with: – all the finite sequences of sorts S... |

107 |
Coherence for tricategories
- Gordon, Power, et al.
- 1995
(Show Context)
Citation Context ...—a commutative diagram in L—must now be carried to a diagram which is “commutative up-to isomorphism,” i.e. a 9 Equivalence of 0-cells is the right “equality” in a 2-category; biequivalence (see e.g. =-=[27]-=-) is the one in a 3-category; and so on.8 diagram filled in with an iso-2-cell. Using the (one-sorted) example (6): in L 3 m×id id×m = 2 m 1 2 m in Set �m×id�X X 2 X 3 �id×m�X X 2 = �m� X X �m� X x1 ... |

99 | Premonoidal categories and notions of computation
- Power, Robinson
- 1997
(Show Context)
Citation Context ...remonoidal category K together with a Cartesian category B embedded via an identity-on-object strict premonoidal functor B → K, subject to a condition on center morphisms. The notion is introduced in =-=[28]-=-, and it is named as such later in [23]. In this way one can look at ArrTh as an axiomatization of the notion of Freyd category. There is a similar corresponding structure for the (stronger) FP-theory... |

97 |
On closed categories of functors
- Day
- 1974
(Show Context)
Citation Context ...r characterization of this structure, discovered in [19], is as a monoid object in the monoidal category of bifunctors [B op × B,Set] where the monoidal products are given by the so-called Day tensor =-=[9]-=-. Equivalently, it is a monad on B in the bicategory of profunctors (also called distributors or bimodules, see e.g. [6,8]). 5.3 PLTh, ArrTh and MArrTh as component calculi We now show that our coalge... |

74 | Higher-dimensional algebra III: n-categories and the algebra of opetopes
- Baez, Dolan
- 1998
(Show Context)
Citation Context ... behavior is compositional. This is easier said than done, especially in the presence of concurrency, that is, when systems can be composed in parallel as well as in sequence. The microcosm principle =-=[1,12]-=- brings some order to the situation. Roughly speaking, compositionality means that the behavior of a compound system is the composition of the components’ behaviors. The microcosm principle then obser... |

73 |
The Definition Of Conformal Field Theory
- Segal
- 2004
(Show Context)
Citation Context ...s 3–5 means that the corresponding mediating isomorphism (as part of the definition of a pseudo functor) is actually the identity.The same idea as ours has been pursued by a few other authors. Segal =-=[29]-=- defines pseudo algebras as pseudo functors, for monoidal theories (as opposed to cartesian theories in our case), with applications to conformal field theory. Fiore’s definition [11] is equivalent to... |

67 | Semantics of weakening and contraction - Jacobs - 1994 |

20 |
Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories
- Lawvere
- 1963
(Show Context)
Citation Context ...in one way, from (S,Σ,E) to L.5 In a one-sorted setting—where arities (objects of L) are identified with natural numbers by taking their length—such a category L is called a Lawvere theory (see e.g. =-=[12,15,22]-=-). In a many-sorted setting, such a category L—say a “many-sorted Lawvere theory”—is usually called a finite-product theory, or an FP-theory, see e.g. [4,5]. Definition 3.1 (FP-theory) An FP-theory is... |

19 | Algebraic specification and coalgebraic synthesis of mealy automata, in: L.B. Zhiming Liu (Ed
- Rutten
(Show Context)
Citation Context ...eed it with an input stream. Let us denote the final FI,J-coalgebra by ζI,J : ZI,J ∼ = −→ FI,J(ZI,J) , that is, ZI,J = {t : I ω → J ω | t is causal} . The structure map ζI,J is described in detail in =-=[29]-=-. Then there naturally arises a “sequential composition” operation that is different from (2): it acts on behaviors of components, simply composing two behaviors of matching types. >>>I,J,K : ZI,J × Z... |

18 | Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
- Fiore
(Show Context)
Citation Context ...r authors. Segal [29] defines pseudo algebras as pseudo functors, for monoidal theories (as opposed to cartesian theories in our case), with applications to conformal field theory. Fiore’s definition =-=[11]-=- is equivalent to ours, but its aspect as a pseudo functor is not emphasized there. 4.2 Inner model: L-object Once we have an outer model C of L, we can define the notion of inner model in C. It is a ... |

17 | Towards a Calculus of State-based Software Components
- Barbosa
- 2003
(Show Context)
Citation Context ...sic link between the two. The present article gives a rigorous analysis of compositionality of components as sketched above. Considering models as coalgebras, we study Barbosa’s calculi of components =-=[2,3]-=- as coalgebras with specified input and output interfaces. Explicitly, a component is a coalgebra for the endofunctor FI,J = (T(J × )) I : Set → Set, (1)2 where I is the set of possible input, and J ... |

16 |
Components as Coalgebras
- Barbosa
- 2001
(Show Context)
Citation Context ...sic link between the two. The present article gives a rigorous analysis of compositionality of components as sketched above. Considering models as coalgebras, we study Barbosa’s calculi of components =-=[2,3]-=- as coalgebras with specified input and output interfaces. Explicitly, a component is a coalgebra for the endofunctor FI,J = (T(J × )) I : Set → Set, (1)2 where I is the set of possible input, and J ... |

14 | The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
- Hyland, Power
- 2007
(Show Context)
Citation Context ...in one way, from (S,Σ,E) to L.5 In a one-sorted setting—where arities (objects of L) are identified with natural numbers by taking their length—such a category L is called a Lawvere theory (see e.g. =-=[12,15,22]-=-). In a many-sorted setting, such a category L—say a “many-sorted Lawvere theory”—is usually called a finite-product theory, or an FP-theory, see e.g. [4,5]. Definition 3.1 (FP-theory) An FP-theory is... |

14 | Modelling environments in call-by-value programming languages
- Levy, Power, et al.
(Show Context)
Citation Context ...The latter two are equipped with different “parallel composition” operations. Notably the algebraic structure expressed by ArrTh is that of (Hughes’) arrow [14], equivalently that of Freyd categories =-=[23]-=-, the notions introduced for modeling structured computations in functional programming. The main result in this section is that the categories {Coalg(FI,J)}I,J— modeling components with FI,J = (T(J ×... |

11 | The microcosm principle and concurrency in coalgebras, 2007. preprint, available from http://www.cs.ru.nl/ ichiro/papers
- Hasuo, Jacobs, et al.
(Show Context)
Citation Context ... behavior is compositional. This is easier said than done, especially in the presence of concurrency, that is, when systems can be composed in parallel as well as in sequence. The microcosm principle =-=[1,12]-=- brings some order to the situation. Roughly speaking, compositionality means that the behavior of a compound system is the composition of the components’ behaviors. The microcosm principle then obser... |

9 |
Category Theory for Computing Science. Centre de Recherches Mathématiques, Université de Montreal, third edition
- Barr, Wells
- 1999
(Show Context)
Citation Context ...L is called a Lawvere theory (see e.g. [12,15,22]). In a many-sorted setting, such a category L—say a “many-sorted Lawvere theory”—is usually called a finite-product theory, or an FP-theory, see e.g. =-=[4,5]-=-. Definition 3.1 (FP-theory) An FP-theory is a category with finite products. The idea of such categorical presentation of algebraic structure originated in [22]. Significant about the approach is tha... |

9 | Coalgebraic logic and synthesis of Mealy machines
- Bonsangue, Rutten, et al.
- 2008
(Show Context)
Citation Context ...nt calculi that we are interested in. For simplicity let us assume that we have no effect in components (i.e. T = Id,FI,J = (J × ) I ). Coalgebras for this functor are called Mealy machines, see e.g. =-=[7]-=-. A prominent operation in component calculi is sequential composition, or pipeline. It attaches two components with matching I/O interfaces, one after another: ( I c J , J d K ) >>>I,J,K ↦−→ I c J d ... |

9 |
Comonadic notions of computation, in
- Uustalu, Vene
(Show Context)
Citation Context ...13 the only gap being the one explained in Remark 5.1. The notion of arrow generalizes that of monad (modeling effects, i.e. structured output [25, 31]) and that of comonad (modeling structured input =-=[30]-=-); the notion of arrow models “structured computations” in general. See e.g. [19, §2.3]. Definition 5.2 An arrow is a set-theoretic model (Def. 3.2) of ArrTh. It had been folklore, and was proved in [... |

6 |
Tracing Anonymity with Coalgebras
- Hasuo
- 2010
(Show Context)
Citation Context ...ten denote C’s action C(a) on a morphism a by �a�C. One consequence from the definition is that C also preserves composition of the form δ ◦ a, where δ : A → A × A is a diagonal. It is illustrated in =-=[11,12]-=- how pseudo functoriality induces isomorphisms up-to which equations are satisfied. The definition is justified by the following fact. Its proof, as well as its generalization to other algebraic struc... |

5 |
Categorical semantics for arrows
- Jacobs, Heunen, et al.
(Show Context)
Citation Context ... application of the many-sorted microcosm framework to component calculi, in §5. It turns out that components as FI,J-coalgebras carry algebraic structure that is a variant of Hughes’ notion of arrow =-=[14,19]-=-. 5 Arrows, generalizing monads, have been used to model structured computations in semantics of functional programming. In §5 we give a rigorous proof that components indeed carry such arrow-like str... |

3 |
An introduction to the theory of coalgebras. Course notes for NASSLLI
- Pattinson
- 2003
(Show Context)
Citation Context ...here; finality immediately yields a positive answer. 7 This is how they are formalized in [29]. Equivalent formulations are: as string functions I ∗ → J ∗ that are length-preserving and prefix-closed =-=[26]-=-; and as functions I + → J where I + is the set of strings of length ≥ 1.4 In fact, the microcosm principle is the mathematical structure that has been behind the story. It refers to the phenomenon t... |

1 |
Distributors at work. Lecture notes by Thomas Streicher
- Bénabou
- 2000
(Show Context)
Citation Context ...B op × B,Set] where the monoidal products are given by the so-called Day tensor [9]. Equivalently, it is a monad on B in the bicategory of profunctors (also called distributors or bimodules, see e.g. =-=[6,8]-=-). 5.3 PLTh, ArrTh and MArrTh as component calculi We now show that our coalgebraic modeling of components indeed models the calculi PLTh, ArrTh and MArrTh, according to the choice of an effect monad ... |

1 |
A roadmap to the unification of weak categorical structures: transformations and equivalences among the various notions of pseudo-algebra
- Hermida
- 2004
(Show Context)
Citation Context ...nd obtain a more general notion of pseudo L-model. Such generality is not needed in this paper. 10 Later we came to know that the idea is folklore at least for “monoidal” theories. It is mentioned in =-=[13]-=- as Segalic presentation of monoidal categories. 11 To be precise, each of the conditions 3–5 means that the corresponding mediating isomorphism (as part of the definition of a pseudo functor) is actu... |

1 |
Lawvere 2-theories. Presented at CT2007
- Lack, Power
- 2007
(Show Context)
Citation Context ...spondence results combined only yield biequivalence. In contrast, Prop. 4.2 realizes equivalence between MonCAT and the 2-category of MonTh-categories, by fine-tuning the latter notion. Remark 4.4 In =-=[21]-=- a different approach for modeling pseudo algebraic structure is presented. There a “Lawvere theory” for monoidal categories is a 2category with all the coherent isomorphisms α,λ,ρ explicit as 2-cells... |

1 |
Two-dimensional monad theory. Journ. of Pure
- Blackwell, Kelly, et al.
(Show Context)
Citation Context ... and see [13, §5.3.3] how pseudo functoriality yields the mediating iso 2-cell in (7). Remark 4.3 There have been different approaches to formalization of “pseudo algebra.” A traditional one (e.g. in =-=[7]-=-) is to find a suitable 2-monad which already takes pseudo satisfaction of equations into account. Another one is by a Lawvere 2-theory, which also includes explicitly the isomorphism up-to which equa... |

1 |
Pseudo functorial semantics. Preprint, available at www.kurims.kyoto-u.ac.jp/˜ichiro
- Hasuo
(Show Context)
Citation Context ...strong monoidal functors and monoidal transformations is equivalent to the 2-category of MonTh-categories with suitable 1and 2-cells. Proof. The proof involves overwhelming details. It is deferred to =-=[12]-=-, where a general result—not only for the specification for monoids but for any algebraic specification— is proved. ⊓⊔ What the last proposition asserts is stronger than merely establishing a biequiva... |