A weak coherence theorem and remarks to the Oka theory

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (only this case was considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the "abstract coherence theorem" found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture proves also the correctness of the modified description.

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The Yoneda Lemma and Coherence

2-Dimensional Categories ◽

10.1093/oso/9780198871378.003.0008 ◽

2021 ◽

pp. 305-330

Author(s):

Niles Johnson ◽

Donald Yau

Keyword(s):

Coherence Theorem ◽

Local Equivalence ◽

Yoneda Lemma

In this chapter, the Yoneda Lemma and the Coherence Theorem for bicategories are stated and proved. The chapter discusses the bicategorical Yoneda pseudofunctor, a bicategorical version of the Yoneda embedding for a bicategory, which is a local equivalence, and the Bicategorical Yoneda Lemma. A consequence of the Bicategorical Whitehead Theorem and the Bicategorical Yoneda Embedding is the Bicategorical Coherence Theorem, which states that every bicategory is biequivalent to a 2-category.

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A coherence theorem for pseudonatural transformations

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2016.09.005 ◽

2017 ◽

Vol 221 (5) ◽

pp. 1146-1217 ◽

Cited By ~ 1

Author(s):

Maxime Lucas

Keyword(s):

Coherence Theorem

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Experimental display of the extended polarization coherence theorem

Optics Letters ◽

10.1364/ol.44.001052 ◽

2019 ◽

Vol 44 (4) ◽

pp. 1052 ◽

Cited By ~ 2

Author(s):

P. Sánchez ◽

J. Gonzales ◽

V. Avalos ◽

F. Auccapuclla ◽

E. Suarez ◽

...

Keyword(s):

Coherence Theorem

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Experimental observation of the polarization coherence theorem

Optics Letters ◽

10.1364/ol.44.000159 ◽

2018 ◽

Vol 44 (1) ◽

pp. 159 ◽

Cited By ~ 7

Author(s):

Bhaskar Kanseri ◽

Sethuraj K. R.

Keyword(s):

Experimental Observation ◽

Coherence Theorem

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The number of proofs for a BCK-formula

Journal of Symbolic Logic ◽

10.2307/2275222 ◽

1993 ◽

Vol 58 (2) ◽

pp. 626-628 ◽

Cited By ~ 2

Author(s):

Yuichi Komori ◽

Sachio Hirokawa

Keyword(s):

Normal Form ◽

Modus Ponens ◽

Basic Notion ◽

Type Assignment ◽

Natural Deduction System ◽

Cartesian Closed Categories ◽

Image Position ◽

Coherence Theorem ◽

Cartesian Closed ◽

Necessary And Sufficient

In this note, we give a necessary and sufficient condition for a BCK-formula to have the unique normal form proof.We call implicational propositional formulas formulas for short. BCK-formulas are the formulas which are derivable from axioms B = (a → b) → (c → a) → c → b, C = (a→b→c)→b→a→c, and K = a→b→a by substitution and modus ponens. It is known that the property of being a BCK-formula is decidable (Jaskowski [11, Theorem 6.5], Ben-Yelles [3, Chapter 3, Theorem 3.22], Komori [12, Corollary 6]). The set of BCK-formulas is identical to the set of provable formulas in the natural deduction system with the following two inference rules.Here γ occurs at most once in (→I). By the formulae-as-types correspondence [10], this set is identical to the set of type-schemes of closed BCK-λ-terms. (See [5].) A BCK-λ-term is a λ-term in which no variable occurs twice. Basic notion concerning the type assignment system can be found [4]. Uniqueness of normal form proofs has been known for balanced formulas. (See [2,14].) It is related to the coherence theorem in cartesian closed categories. A formula is balanced when no variable occurs more than twice in it. It was shown in [8] that the proofs of balanced formulas are BCK-proofs. Relevantly balanced formulas were defined in [9], and it was proved that such formulas have unique normal form proofs. Balanced formulas are included in the set of relevantly balanced formulas.

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