## An Algebraic Presentation of Term Graphs, via GS-Monoidal Categories (1999)

Venue: | Applied Categorical Structures |

Citations: | 38 - 25 self |

### BibTeX

@ARTICLE{Corradini99analgebraic,

author = {Andrea Corradini and Fabio Gadducci},

title = {An Algebraic Presentation of Term Graphs, via GS-Monoidal Categories},

journal = {Applied Categorical Structures},

year = {1999},

volume = {7},

pages = {7--299}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are one-to-one with the arrows of the free gs-monoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gs-monoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...

### Citations

1031 |
Categories for the Working Mathematician
- Lane, S
- 1971
(Show Context)
Citation Context ...e Section 4), our presentation is tailored over the need of our definitivo.tex; 30/11/1999; 16:52; p.11 12 main theorem. 3 For a full account on (symmetric) monoidal categories we refer the reader to =-=[44]-=-; gs-monoidal theories have been introduced in [21], under the name of s-monoidal theories. DEFINITION 10. (Symmetric monoidal categories). A monoidal categorysC is a triple C = hC 0 ;\Omega ; ei wher... |

586 | Term Rewriting Systems
- Klop
- 1992
(Show Context)
Citation Context ... for the theory of term rewriting is provided by the existence of three different but equivalent presentations, namely the operational , classical one (described in terms of redexes and substitutions =-=[39]-=-), the logical one (we think of Meseguer's rewriting logic [48]), and the categorical one, based on algebraic (Cartesian) 2-theories [53, 57]. While the operational presentation is the best suited for... |

518 |
Conditional rewriting logic as a unified model of concurrency
- Meseguer
- 1992
(Show Context)
Citation Context ...of three different but equivalent presentations, namely the operational , classical one (described in terms of redexes and substitutions [39]), the logical one (we think of Meseguer's rewriting logic =-=[48]-=-), and the categorical one, based on algebraic (Cartesian) 2-theories [53, 57]. While the operational presentation is the best suited for implementation purposes, both the logical and the categorical ... |

208 |
A New Implementation Technique for Applicative Languages
- Turner
- 1979
(Show Context)
Citation Context ...ng term graphs with cycles, one can represent in a finitary way certain structures that arise dealing with recursive definitions (as for the implementation of the fixed point combinator Y proposed in =-=[60]-=-). Therefore it is certainly interesting to consider the extension of our results to the possibly cyclic case. Preliminary results [12] show that this goal can be achieved by enriching gs-monoidal cat... |

196 |
Initial algebra semantics and continuous algebras
- Goguen, Thatcher, et al.
- 1977
(Show Context)
Citation Context ... lot of attention from the computer science community, due in large part to the seminal work of the New York--Stanford based ADJ group on the algebraic semantics for programming languages (see, e.g., =-=[23, 24]-=-). EXAMPLE 32. (Explicit construction of an algebraic theory). Let us consider again the signature \Sigma e . The basic arrows are a \Sigma e : 0 ! 1 and f \Sigma e : 2 ! 1. Now consider for example t... |

189 |
Functorial semantics of algebraic theories
- Lawvere
- 1963
(Show Context)
Citation Context ...g that the terms over a given signature \Sigma can be regarded as the arrows of a Cartesian category (called the algebraic theory of \Sigma) freely generated (in a suitable way) by \Sigma (see, e.g., =-=[40, 43]-=-). Such a category has (underlined) natural numbers as objects, and its generators are arrows like g : n ! 1, where g is an operator of rank n in \Sigma. It 1 Here `gs' stands for `graph substitution'... |

174 |
Traced monoidal categories
- Joyal, Street, et al.
- 1996
(Show Context)
Citation Context ...back structure (already studied in the algebraic specification community: See [2] for earlier references, and especially the work of Stefanescu [19, 18]) using the recently introduced notion of trace =-=[35]-=-, thus yielding traced gsmonoidalscategories. In this case, the correspondence with terms has to be reconsidered, because in general a cyclic term graph unravels to a rational term, i.e., a possibly i... |

160 | Rewriting Logic as a Logical and Semantic Framework,” Handbook of philosophical logic
- Marti-Oliet, Meseguer
- 1993
(Show Context)
Citation Context ...ons, definitivo.tex; 30/11/1999; 16:52; p.32 33 on which most of the theory is based; the categorical , discussed to some extent in the previous subsection; and the logical one, for which we refer to =-=[47, 48]-=-, but also [29, 55]. It is worth stressing that the logical and the categorical views are very strongly related, the former providing often a more syntactical presentation of the latter. Various paper... |

137 |
Functional Programming and Parallel Graph Rewriting
- Plasmeijer, Eekelen
- 1993
(Show Context)
Citation Context ...c fl 1999 Kluwer Academic Publishers. Printed in the Netherlands. definitivo.tex; 30/11/1999; 16:52; p.1 2 term graph rewriting is often used in the implementation of functional programming languages =-=[52]-=-. In our opinion, however, there is an unsatisfactory gap between the achievements of the theories of term and of term graph rewriting, respectively. In particular, a solid ground for the theory of te... |

107 | Premonoidal categories and notions of computation
- Power, Robinson
- 1997
(Show Context)
Citation Context ...those by Stefanescu, in their framework a traced structure is not explicitly needed but it is derivable. Finally, also related to our work are premonoidal categories, introduced by Power and Robinson =-=[54]-=-. These categories are extensively used as a semantical model in the community of action calculi, since they provide a categorical characterisation of some sort of data flow structures (see Hasegawa [... |

77 |
Graph expressions and graph rewritings
- Bauderon, Courcelle
- 1987
(Show Context)
Citation Context ...cond one, and then by eliminating them. For example, term graph G 1 ; G 2 is the composition of G 1 and of G 2 of rank (1; 4). The last term graph is G 1 \Phi G 2 , the union of G 1 and G 2 , of rank =-=(5; 6)-=-. We are now ready to show the main result of this section, that is, that every term graph can be constructed, using composition and union, from a small set of atomic term graphs. definitivo.tex; 30/1... |

76 | Equational term graph rewriting
- Ariola, Klop
- 1996
(Show Context)
Citation Context ...ould be defined. In fact, they have been represented as directed graphs satisfying a number of constraints [4], as suitable labeled hyper-graphs called jungles [32], or as sets of recursive equations =-=[1]-=-, among others. This paper proposes a first contribution towards a categorical theory of term graph rewriting, which parallels the categorical description of term rewriting. In fact, we present a char... |

69 | The tile model
- Gadducci, Montanari
- 2000
(Show Context)
Citation Context ...e topic or intuition. As far as our work is concerned, let us summarise the motivations that led us to consider a categorical presentation of term graphs. When introducing gs-monoidal categories (see =-=[21, 22]-=-: We called them smonoidal, since we were not yet aware of Pfender's work cited below), definitivo.tex; 30/11/1999; 16:52; p.33 34 our intention was to enrich symmetric monoidal categories in order to... |

69 |
Semantics of weakening and contraction
- Jacobs
- 1994
(Show Context)
Citation Context ...mations in enforced, thus providing an easy, equational characterisation of the notion of finite products. This is shown by the following result, which has become folklore in recent years (see, e.g., =-=[33, 41, 56]-=-). PROPOSITION 33. (Algebraic as gs-monoidal plus naturality). For a given signature \Sigma, the category C-Th(\Sigma) obtained by quotienting the associated gs-monoidal theory GS-Th(\Sigma) with the ... |

59 |
Categories of partial maps
- Robinson, Roslini
- 1988
(Show Context)
Citation Context ...mations in enforced, thus providing an easy, equational characterisation of the notion of finite products. This is shown by the following result, which has become folklore in recent years (see, e.g., =-=[33, 41, 56]-=-). PROPOSITION 33. (Algebraic as gs-monoidal plus naturality). For a given signature \Sigma, the category C-Th(\Sigma) obtained by quotienting the associated gs-monoidal theory GS-Th(\Sigma) with the ... |

48 |
Computational semantics of term rewriting systems
- Boudol
- 1985
(Show Context)
Citation Context ...ts. Vertical composition acts instead as sequential composition. Furthermore, the generated rewriting sequences are subject to an equivalence that coincides with the so-called permutation equivalence =-=[7]-=-, due to the axioms of 2-categories [21, 42]. A similar construction can be followed for term graph rewriting as well, as it is shown in [10]. The idea is to add to the gs-monoidal category generated ... |

46 | Recursion from cyclic sharing: traced monoidal categories and models of cyclic lambda-calculi
- Hasegawa
(Show Context)
Citation Context ...iding often a more syntactical presentation of the latter. Various papers in the literature discuss term graph rewriting using a logical presentation in terms of inference rules and axioms, including =-=[26, 27, 49]-=-. As far as the topic of the present paper is concerned, we just want to recall the inference rule that defines term graphs as an extension of standard first order terms. Roughly, the usual syntax is ... |

44 | Bicategories of processes
- Katis, Sabadini, et al.
- 1997
(Show Context)
Citation Context ... r to be natural. Also related to ours are the copy-categories introduced in [28]. Walters et alii investigate the algebraic structure of relations, regarded as arrows of suitable categories of spans =-=[36, 37]-=-. They have on the operational side an interpretation in terms of processes, and on the categorical side suitable bicategories. Interestingly, the axiomatisation of relations they propose makes use of... |

41 |
From Petri nets to linear logic through categories: a survey, Znternat
- Marti-Oliet, Meseguer
- 1991
(Show Context)
Citation Context ...: Our starting points were the already mentioned [33, 41]. Next, in [14], when analysing two categorical models for term rewriting, we observed that neither the standard 2-categorical presentation of =-=[46, 48]-=- (see Section 6.4), nor the formulation based on the weaker notion of sesqui-category by Stell [57, 59] were completely satisfactory from the perspective of a truly concurrent semantics. In fact, from... |

36 | A 2-categorical presentation of term graph rewriting
- Corradini, Gadducci
- 1997
(Show Context)
Citation Context ...e that coincides with the so-called permutation equivalence [7], due to the axioms of 2-categories [21, 42]. A similar construction can be followed for term graph rewriting as well, as it is shown in =-=[10]-=-. The idea is to add to the gs-monoidal category generated by \Sigma cells representing the rules of a term graph rewriting system, and to consider the (gs-monoidal) 2-category freely generated by suc... |

36 |
First-order categorical logic
- Makkai, Reyes
- 1977
(Show Context)
Citation Context ...g that the terms over a given signature \Sigma can be regarded as the arrows of a Cartesian category (called the algebraic theory of \Sigma) freely generated (in a suitable way) by \Sigma (see, e.g., =-=[40, 43]-=-). Such a category has (underlined) natural numbers as objects, and its generators are arrows like g : n ! 1, where g is an operator of rank n in \Sigma. It 1 Here `gs' stands for `graph substitution'... |

33 |
Hyperedge Replacement: Grammars and Languages, volume 643
- Habel
- 1992
(Show Context)
Citation Context ... `up to isomorphism'. Thirdly, our term graphs are ranked. The idea of equipping graphs with lists of distinguished nodes in order to define composition operations on them is not new (see for example =-=[5, 25]-=-), but for the first time, to our knowledge, it is applied here to the class of term graphs. In [4] this technique is not used simply because it is not needed for the goals of the paper, and the singl... |

28 |
Span(Graph): A categorical algebra of transition systems
- Katis, Sabadini, et al.
- 1997
(Show Context)
Citation Context ... r to be natural. Also related to ours are the copy-categories introduced in [28]. Walters et alii investigate the algebraic structure of relations, regarded as arrows of suitable categories of spans =-=[36, 37]-=-. They have on the operational side an interpretation in terms of processes, and on the categorical side suitable bicategories. Interestingly, the axiomatisation of relations they propose makes use of... |

24 | B.: Sesqui-pushout rewriting
- Corradini, Heindel, et al.
(Show Context)
Citation Context ...nravels to a rational term, i.e., a possibly infinite term with a finite number of distinct sub-terms. The relationship between rational terms and cyclic term graphs is addressed in [9, 38], while in =-=[11]-=- rational terms are related to -terms (which are essentially a subclass of term graphs): None of these papers, however, phrases this relationship categorically as a functor between suitable theories, ... |

24 | On the Algebraic Approach to Concurrent Term Rewriting
- Gadducci
- 1996
(Show Context)
Citation Context ... need of our definitivo.tex; 30/11/1999; 16:52; p.11 12 main theorem. 3 For a full account on (symmetric) monoidal categories we refer the reader to [44]; gs-monoidal theories have been introduced in =-=[21]-=-, under the name of s-monoidal theories. DEFINITION 10. (Symmetric monoidal categories). A monoidal categorysC is a triple C = hC 0 ;\Omega ; ei where C 0 is a category, e 2 C 0 is a distinguished obj... |

24 | Algebraic approaches to nondeterminism: An overview - Walicki, Meldal - 1997 |

23 |
Towards a new algebraic foundation of flowchart scheme theory
- Stefanescu
- 1990
(Show Context)
Citation Context ...achieved by enriching gs-monoidal categories with a feedback structure (already studied in the algebraic specification community: See [2] for earlier references, and especially the work of Stefanescu =-=[19, 18]-=-) using the recently introduced notion of trace [35], thus yielding traced gsmonoidalscategories. In this case, the correspondence with terms has to be reconsidered, because in general a cyclic term g... |

23 |
de Vries. On the adequacy of graph rewriting for simulating term rewriting
- Kennaway, Klop, et al.
- 1994
(Show Context)
Citation Context ...yclic term graph unravels to a rational term, i.e., a possibly infinite term with a finite number of distinct sub-terms. The relationship between rational terms and cyclic term graphs is addressed in =-=[9, 38]-=-, while in [11] rational terms are related to -terms (which are essentially a subclass of term graphs): None of these papers, however, phrases this relationship categorically as a functor between suit... |

22 |
Premonoidal categories and a graphical view of programs
- Jeffrey
- 1997
(Show Context)
Citation Context ...es are extensively used as a semantical model in the community of action calculi, since they provide a categorical characterisation of some sort of data flow structures (see Hasegawa [26] and Jeffrey =-=[34]-=-): we refer to Pavlovic [50] and to Barber et alii [3] for an overview. A combination of their approach and ours, with a specific application to term rewriting, is due to Miyoshi [49]. 10 A multi-alge... |

22 | Categorical logic of names and abstraction in action calculi
- Pavlović
- 1997
(Show Context)
Citation Context ... semantical model in the community of action calculi, since they provide a categorical characterisation of some sort of data flow structures (see Hasegawa [26] and Jeffrey [34]): we refer to Pavlovic =-=[50]-=- and to Barber et alii [3] for an overview. A combination of their approach and ours, with a specific application to term rewriting, is due to Miyoshi [49]. 10 A multi-algebra A over a one-sorted sign... |

21 | Cyclic) term graph rewriting is adequate for rational parallel term rewriting
- Corradini, Drewes
- 1997
(Show Context)
Citation Context ...yclic term graph unravels to a rational term, i.e., a possibly infinite term with a finite number of distinct sub-terms. The relationship between rational terms and cyclic term graphs is addressed in =-=[9, 38]-=-, while in [11] rational terms are related to -terms (which are essentially a subclass of term graphs): None of these papers, however, phrases this relationship categorically as a functor between suit... |

21 |
Hyperedge replacement jungle rewriting for term-rewriting systems and logic programming
- Corradini, Rossi
- 1993
(Show Context)
Citation Context ... pushout of two arrows in Dag \Sigma does not always exist: It does exist however in the case we are interested in, since morphism v : j ! d is injective and has only empty nodes in the codomain. See =-=[16]-=- for necessary and sufficient conditions for the existence of pushouts in the equivalent category of jungles. definitivo.tex; 30/11/1999; 16:52; p.7 8 G 1 1 2 g f 1 2 3 4 2 1 1 2 G 2 1 2 3 4 h 1 1 2 G... |

21 | Equational reasoning with 2-dimensional diagrams, Term rewriting (Font Romeux
- Lafont
- 1993
(Show Context)
Citation Context ...mations in enforced, thus providing an easy, equational characterisation of the notion of finite products. This is shown by the following result, which has become folklore in recent years (see, e.g., =-=[33, 41, 56]-=-). PROPOSITION 33. (Algebraic as gs-monoidal plus naturality). For a given signature \Sigma, the category C-Th(\Sigma) obtained by quotienting the associated gs-monoidal theory GS-Th(\Sigma) with the ... |

21 |
Natural Associativity and Commutativity
- MacLane
- 1963
(Show Context)
Citation Context .... Intuitively, symmetric monoidal categories formalise in categorical terms the basic notions of pairing and permutation, as shown by the many coherence results for these categories (originating from =-=[45]-=-). The following well-known fact provides a simple characterisation of symmetries that will be of use in the next section. LEMMA 12. (Symmetries are permutations). For any n 2 IlN, the set of symmetri... |

21 | Bisimilarity of open terms
- Rensink
(Show Context)
Citation Context ...; 30/11/1999; 16:52; p.32 33 on which most of the theory is based; the categorical , discussed to some extent in the previous subsection; and the logical one, for which we refer to [47, 48], but also =-=[29, 55]-=-. It is worth stressing that the logical and the categorical views are very strongly related, the former providing often a more syntactical presentation of the latter. Various papers in the literature... |

20 | From action calculi to linear logic
- Barber, Gardner, et al.
- 1998
(Show Context)
Citation Context ...mmunity of action calculi, since they provide a categorical characterisation of some sort of data flow structures (see Hasegawa [26] and Jeffrey [34]): we refer to Pavlovic [50] and to Barber et alii =-=[3]-=- for an overview. A combination of their approach and ours, with a specific application to term rewriting, is due to Miyoshi [49]. 10 A multi-algebra A over a one-sorted signature \Sigma has a carrier... |

20 |
Term graph reduction
- Barendregt, Eekelen, et al.
- 1987
(Show Context)
Citation Context ...nly in operational style. Furthermore, there is no common agreement on the way term graphs should be defined. In fact, they have been represented as directed graphs satisfying a number of constraints =-=[4]-=-, as suitable labeled hyper-graphs called jungles [32], or as sets of recursive equations [1], among others. This paper proposes a first contribution towards a categorical theory of term graph rewriti... |

20 |
Foundations of equational deduction: A categorical treatment of equational proofs and uni algorithms
- Rydeheard, Stell
- 1987
(Show Context)
Citation Context ...classical one (described in terms of redexes and substitutions [39]), the logical one (we think of Meseguer's rewriting logic [48]), and the categorical one, based on algebraic (Cartesian) 2-theories =-=[53, 57]-=-. While the operational presentation is the best suited for implementation purposes, both the logical and the categorical ones provide an inductive definition of the rewrite relation over terms, that ... |

18 | Relating two categorical models of term rewriting
- Corradini, Gadducci, et al.
- 1995
(Show Context)
Citation Context ...tion' of duplication and garbage collection) to be considered as a faithful description of the actual behaviour of a concurrent machine implementing the rewriting process over a distributed structure =-=[14, 21]-=-. 6.5. A logical view of term graphs In the introduction we mentioned three equivalent presentations of term rewriting, namely the operational , based on redexes and substitutions, definitivo.tex; 30/... |

18 | Implementing term rewriting by jungle evaluation
- Hoffmann, Plump
- 1991
(Show Context)
Citation Context ...mmon agreement on the way term graphs should be defined. In fact, they have been represented as directed graphs satisfying a number of constraints [4], as suitable labeled hyper-graphs called jungles =-=[32]-=-, or as sets of recursive equations [1], among others. This paper proposes a first contribution towards a categorical theory of term graph rewriting, which parallels the categorical description of ter... |

16 |
An abstract formulation for rewrite systems
- Power
- 1989
(Show Context)
Citation Context ...classical one (described in terms of redexes and substitutions [39]), the logical one (we think of Meseguer's rewriting logic [48]), and the categorical one, based on algebraic (Cartesian) 2-theories =-=[53, 57]-=-. While the operational presentation is the best suited for implementation purposes, both the logical and the categorical ones provide an inductive definition of the rewrite relation over terms, that ... |

15 |
Feedback and generalized logic
- Bainbridge
- 1976
(Show Context)
Citation Context ...30. (The algebraic theory for a first-order signature). Let us consider the signature \Sigma e = \Sigma 0 [ \Sigma 1 , where \Sigma 0 = fag and \Sigma 1 = ffg. By definition, the hom-set Th(\Sigma e )=-=[2; 1]-=- contains all the elements of the term algebra T \Sigma e (x 1 ; x 2 ), e.g., the variables x 1 and x 2 (regarded as the projectionss2 1 ;s2 2 ), the terms a, f(x 1 ; x 2 ), f(a; x 1 ), and so on. Com... |

14 |
Tiles for Concurrent and Located Calculi
- Ferrari, Montanari
- 1997
(Show Context)
Citation Context ...d accordingly in a straightforward manner, and a characterisation result for term graphs over a polyadic signature would be obtained. The extension to polyadic signatures has been used for example in =-=[20]-=-. 6.3. Cyclic term graphs In the present paper we only considered acyclic term graphs. By allowing term graphs with cycles, one can represent in a finitary way certain structures that arise dealing wi... |

14 |
Models of Sharing Graphs
- Hasegawa
- 1997
(Show Context)
Citation Context ...iding often a more syntactical presentation of the latter. Various papers in the literature discuss term graph rewriting using a logical presentation in terms of inference rules and axioms, including =-=[26, 27, 49]-=-. As far as the topic of the present paper is concerned, we just want to recall the inference rule that defines term graphs as an extension of standard first order terms. Roughly, the usual syntax is ... |

12 |
On flowchart theories: Part II. The nondeterministic case
- Stefanescu
- 1987
(Show Context)
Citation Context ... graphs, a topic that has surfaced many times in the literature on algebraic specifications. A comprehensive contribution to this topic is given by the works on flownomial algebras by Stefanescu (see =-=[19, 17]-=-, and [18] for a survey). As flow graphs, he considers structures that are richer than ours, namely, using our terminology,scyclic hyper-graphs (see Sections 6.2 and 6.3); on the categorical side, he ... |

12 |
Axiomatizing permutation equivalence in the - calculus
- Laneve, Montanari
- 1992
(Show Context)
Citation Context ...s sequential composition. Furthermore, the generated rewriting sequences are subject to an equivalence that coincides with the so-called permutation equivalence [7], due to the axioms of 2-categories =-=[21, 42]-=-. A similar construction can be followed for term graph rewriting as well, as it is shown in [10]. The idea is to add to the gs-monoidal category generated by \Sigma cells representing the rules of a ... |

11 |
Towards a proof theory of rewriting: the simply typed 2λcalculus
- Hilken
- 1996
(Show Context)
Citation Context ...; 30/11/1999; 16:52; p.32 33 on which most of the theory is based; the categorical , discussed to some extent in the previous subsection; and the logical one, for which we refer to [47, 48], but also =-=[29, 55]-=-. It is worth stressing that the logical and the categorical views are very strongly related, the former providing often a more syntactical presentation of the latter. Various papers in the literature... |

8 |
Algebra of flownomials
- Stefanescu
- 1994
(Show Context)
Citation Context ...achieved by enriching gs-monoidal categories with a feedback structure (already studied in the algebraic specification community: See [2] for earlier references, and especially the work of Stefanescu =-=[19, 18]-=-) using the recently introduced notion of trace [35], thus yielding traced gsmonoidalscategories. In this case, the correspondence with terms has to be reconsidered, because in general a cyclic term g... |

7 | A view on implementing processes: Categories of circuits
- Hensel, Spooner
- 1995
(Show Context)
Citation Context ...l-behaved' category of partial maps. To this aim, they introduce p-categories, which were very close to ours if we required r to be natural. Also related to ours are the copy-categories introduced in =-=[28]-=-. Walters et alii investigate the algebraic structure of relations, regarded as arrows of suitable categories of spans [36, 37]. They have on the operational side an interpretation in terms of process... |

7 |
On partial algebras
- Hoenke
- 1977
(Show Context)
Citation Context ...ons of) standard terms are obtained. The analysis of the categorical structure of classes of algebras of partial maps is the main topic of the work by Hoenke and other former East Germany algebraists =-=[8, 30, 31]-=-. They introduce dht-symmetric monoidal categories, which can be regarded as gs-monoidal ones with a zero-object, and where transformation r is natural while ! in general is not. A functorial presenta... |