## Extracting a Proof of Coherence for Monoidal Categories from a Proof of Normalization for Monoids (1995)

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Venue: | In TYPES |

Citations: | 16 - 3 self |

### BibTeX

@INPROCEEDINGS{Beylin95extractinga,

author = {Ilya Beylin and Peter Dybjer},

title = {Extracting a Proof of Coherence for Monoidal Categories from a Proof of Normalization for Monoids},

booktitle = {In TYPES},

year = {1995},

pages = {47--61},

publisher = {Springer-Verlag}

}

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### Abstract

. This paper studies the problem of coherence in category theory from a type-theoretic viewpoint. We first show how a Curry-Howard interpretation of a formal proof of normalization for monoids almost directly yields a coherence proof for monoidal categories. Then we formalize this coherence proof in intensional intuitionistic type theory and show how it relies on explicit reasoning about proof objects for intensional equality. This formalization has been checked in the proof assistant ALF. 1 Introduction Mac Lane [18, pp.161--165] proved a coherence theorem for monoidal categories. A basic ingredient in his proof is the normalization of object expressions. But it is only one ingredient and several others are needed too. Here we show that almost a whole proof of this coherence theorem is hidden in a Curry-Howard interpretation of a proof of normalization for monoids. The second point of the paper is to contribute to the development of constructive category theory in the sense of Huet a...

### Citations

938 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...egorical notions, such as strict and relaxed monoidal categories in category theory, where a strict monoidal category is one where the monoidal laws hold up to equality and not only up to isomorphism =-=[18]-=-. 4.3 The Normalization Proof for Monoids There is no difficulty in principle in formalizing the normalization proof for monoids. See also Hedberg [13]. A minor point is that we define N as the set of... |

419 | Constructive Analysis - Bishop, Bridges - 1985 |

109 |
tricategories
- Power, Why
- 1995
(Show Context)
Citation Context ...hat the extracted proof is essentially a logical version of the proof of coherence for bicategories (in the special case of monoidal categories) given in the recent paper by Gordon, Power, and Street =-=[12]-=-. Their proof relies on Street's bicategorical Yoneda lemma. In our case a proof with similar structure was instead discovered by using the Curry-Howard interpretation which makes explicit the connect... |

77 | Inductive sets and families in Martin-Löf’s Type Theory and their set-theoretic semantics: An inversion principle for Martin-Löf’s type theory
- Dybjer
- 1991
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Citation Context ...ories is adequate with respect to the standard set-theoretic definition in the following sense. The essential idea is to use the naive interpretation in classical set theory of MartinL of type theory =-=[8]-=-. But we need to interpret the Hom-setoids as Hom-sets of equivalence classes rather than as set-theoretic setoids, and similarly for the interpretation of maps. 4.5 The Coherence Proof Since there ar... |

59 | Generalised algebraic theories and contextual categories - Cartmell - 1986 |

55 | Computational category theory - Rydeheard, Burstall - 1988 |

50 |
An inverse of the evaluation functional for typed -calculus
- Berger, Schwichtenberg
- 1991
(Show Context)
Citation Context ...l proof of normalization and the proof of coherence. The present work can be seen as an application of a certain approach to normalization in logical calculi: so-called "reduction-free" norm=-=alization [5, 7, 6, 4]-=-. The idea is to construct an appropriate model of the calculus and a function which inverts the interpretation function. Here the appropriate model is the category N N and inversion is application to... |

45 | Intuitionistic model constructions and normalization proofs
- Coquand, Dybjer
- 1997
(Show Context)
Citation Context ...l proof of normalization and the proof of coherence. The present work can be seen as an application of a certain approach to normalization in logical calculi: so-called "reduction-free" norm=-=alization [5, 7, 6, 4]-=-. The idea is to construct an appropriate model of the calculus and a function which inverts the interpretation function. Here the appropriate model is the category N N and inversion is application to... |

38 | Internal type theory
- Dybjer
(Show Context)
Citation Context ...uet and Saibi [16], who implemented part of elementary category theory in the proof assistant Coq. Here we extend the scope of constructive category theory to the area of coherence theorems (cf. also =-=[9]-=-). We have formalized our proof in Martin-Lof type theory and implemented it in the proof assistant ALF. An interesting aspect of this formalization is that the problem of reasoning about explicit pro... |

29 | From Semantics to Rules: A Machine Assisted Analysis
- Coquand
- 1993
(Show Context)
Citation Context ...l proof of normalization and the proof of coherence. The present work can be seen as an application of a certain approach to normalization in logical calculi: so-called "reduction-free" norm=-=alization [5, 7, 6, 4]-=-. The idea is to construct an appropriate model of the calculus and a function which inverts the interpretation function. Here the appropriate model is the category N N and inversion is application to... |

26 | Constructive category theory
- Huet, SaÄbi
- 1995
(Show Context)
Citation Context ...a Curry-Howard interpretation of a proof of normalization for monoids. The second point of the paper is to contribute to the development of constructive category theory in the sense of Huet and Saibi =-=[16]-=-, who implemented part of elementary category theory in the proof assistant Coq. Here we extend the scope of constructive category theory to the area of coherence theorems (cf. also [9]). We have form... |

26 | The groupoid model refutes uniqueness of identity proofs - Hofmann, Streicher - 1994 |

22 | Categorical reconstruction of a reduction free normalization proof
- Altenkirch, Hofmann, et al.
- 1995
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Citation Context |

8 |
Galois - a theory development project. Report for the 1993 Turin meeting on the Representation of Mathematics in Logical Frameworks. Available at URL: www.cs.man.ac.uk/˜petera/papers.html
- Aczel
(Show Context)
Citation Context ...e an explicit injection J : N ! M . (There are no "true" subsets in type theory.) Hence the normalization function is defined as follows: Nf (a) = J([[a]](e)): 4.4 Monoidal Categories We fol=-=low Aczel [1]-=-, Huet and Saibi [16], and Dybjer and Gaspes [10] and formalize a notion of category in intuitionistic type which does not have equality of objects as part of the structure. A category thus consists o... |

8 |
A user's guide to
- Altenkirch, Gaspes, et al.
- 1994
(Show Context)
Citation Context ...nistic Type Theory We have formalized the coherence proof in Martin-Lof intuitionistic type theory using the proof assistant ALF developed in Goteborg by Coquand, Magnusson, Nordlander, and Nordstrom =-=[3]-=-. When we formalize the free monoid in type theory we introduce explicit proofs that two elements a and b of a monoid are equal. These proofs correspond to arrow expressions in the free monoidal categ... |

8 |
Elimination of extensionality and quotient types in martin-löf’s type theory
- Hofmann
- 1994
(Show Context)
Citation Context ...nce relation which will play the role of equality on a certain set (a book equality in AUTOMATH terminology) . Extensional equality of functions in a set A ! B is one example. We shall follow Hofmann =-=[14]-=- and call such pairs of sets and equivalence relationsssetoids. It is necessary to work with setoids, since one cannot form a new set by taking the quotient of a set with respect to the equivalence re... |

7 |
Normalizing the associative law: An experiment with Martin-Lof's type theory
- Hedberg
- 1991
(Show Context)
Citation Context ...e subset N of normal binary words is the least set such that e 2 N and if n 2 N and x 2 X then n\Omega x 2 N . We shall analyze the proof of the following "obvious" normalization theorem (se=-=e Hedberg [13]-=-): Theorem 1. There is a function (algorithm) Nf : M ! N , such that asb iff Nf (a) = Nf (b). A simple way to construct such a function is by using that N N together with function composition and the ... |

4 | Implementing a category of sets in ALF
- Dybjer, Gaspes
- 1994
(Show Context)
Citation Context ...o "true" subsets in type theory.) Hence the normalization function is defined as follows: Nf (a) = J([[a]](e)): 4.4 Monoidal Categories We follow Aczel [1], Huet and Saibi [16], and Dybjer a=-=nd Gaspes [10]-=- and formalize a notion of category in intuitionistic type which does not have equality of objects as part of the structure. A category thus consists of a set of objects, but setoids of arrows indexed... |

2 | A comparison of HOL and ALF formalizations of a categorical coherence theorema
- Agerholm, Beylin, et al.
- 1996
(Show Context)
Citation Context ... about related work. The ALF-implementation can be found on the web [11]. More discussion and a comparison with an implementation of the same proof in HOL can be found in Agerholm, Beylin, and Dybjer =-=[2]-=-. 2 Normalization for Monoids Let M be the set of binary words with variables in the set X, that is, the least set such that e 2 M x 2 M for any x 2 X a\Omega b 2 M for any a; b 2 M Write asb if a and... |

2 |
Initiation `a la Th'eorie des Cat'egories. Notes de cours du DEA Fonctionnalit'e, Structures de Calcul et Programmation donn'e `a l'Universit'e Paris VII en 1983-84 et
- Huet
- 1984
(Show Context)
Citation Context ...gory N N and inversion is application to unit. Another proof of coherence in this style is Lafont's for cccs [17]. We would also like to mention the proof of coherence for monoidal categories by Huet =-=[15]-=-. In contrast to ours his approach is reduction-based and uses the method of Knuth-Bendix completion from rewriting theory. Acknowledgement. The first author was supported by a grant from the Swedish ... |

2 |
Categories & Machines. Implantation de Langages de Programmation guid'ee par la Logique Cat'egorique
- Logique
- 1988
(Show Context)
Citation Context ... function which inverts the interpretation function. Here the appropriate model is the category N N and inversion is application to unit. Another proof of coherence in this style is Lafont's for cccs =-=[17]-=-. We would also like to mention the proof of coherence for monoidal categories by Huet [15]. In contrast to ours his approach is reduction-based and uses the method of Knuth-Bendix completion from rew... |

2 | Implementing a category of sets - Dybjer, Gaspes - 1993 |

2 | Uniqueness and internal decidability in type theory - Hedberg - 1995 |