Proof Of Coherence Research Articles

Coherence for bicategorical cartesian closed structure

AbstractWe prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence for bicategories with finite bilimits to show that every bicategory with bicategorical cartesian closed structure is biequivalent to a 2-category with 2-categorical cartesian closed structure. Then we show how to extend this result to a Mac Lane-style “all pasting diagrams commute” coherence theorem: precisely, we show that in the free cartesian closed bicategory on a graph, there is at most one 2-cell between any parallel pair of 1-cells. The argument we employ is reminiscent of that used by Čubrić, Dybjer, and Scott to show normalisation for the simply-typed lambda calculus (Čubrić et al., 1998). The main results first appeared in a conference paper (Fiore and Saville, 2020) but for reasons of space many details are omitted there; here we provide the full development.

Mechanization of coherence and adequacy: Being extrinsic extended to subtyping

• Mechanization of Reynolds' bracketing theorem. • Mechanized proof of coherence for an intrinsic semantics. • Adequacy of an operational semantics with respect to the extrinsic denotational semantics by means of biorthogonality. • The adequacy of the operational semantics with respect to the intrinsic semantics follows from the adequacy for the extrinsic semantics and bracketing. Operational and denotational are two common approaches to specify the semantics of programming languages: the former is more suitable for expressing a particular evaluation strategy and the latter is more abstract. If there are both an operational and a denotational semantics for a programming language, one expects that the operational semantics be adequate with respect to the denotational one (this ensures that a program whose denotational semantics is some mathematical function computes that function). Moreover, there are two styles of denotational semantics for typed languages: the extrinsic one gives a denotation for any expression (even if ill-typed), while in the intrinsic only well-typed expressions are meaningful. In this paper we mechanize in Coq a typed, call-by-value language by specifying its operational semantics and giving intrinsic and extrinsic denotational semantics, both using domain theory. We also prove that the denotational semantics are equivalent; this is interesting because it leads to a direct proof of coherence for the intrinsic semantics. Finally, we prove the adequacy of the operational semantics with respect to the denotational semantics. As far as we know, this is the first mechanization of Reynolds' bracketing theorem and also the use of biorthogonality with extrinsic semantics instead of intrinsic semantics. We also show that the methodology is modular by extending those results for a larger language with more types (Booleans and products) and subtyping.

Categorified Cyclic Operads

In this paper, we introduce a notion of categorified cyclic operad for set-based cyclic operads with symmetries. Our categorification is obtained by relaxing defining axioms of cyclic operads to isomorphisms and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form “all diagrams of canonical isomorphisms commute”. Our coherence results come in two flavours, corresponding to the “entries-only” and “exchangeable-output” definitions of cyclic operads. Our proof of coherence in the entries-only style is of syntactic nature and relies on the coherence of categorified non-symmetric operads established by Dosen and Petric. We obtain the coherence in the exchangeable-output style by “lifting” the equivalence between entries-only and exchangeable-output cyclic operads, set up by the second author. Finally, we show that a generalization of the structure of profunctors of Benabou provides an example of categorified cyclic operad, and we exploit the coherence of categorified cyclic operads in proving that the Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd version.

Weak Cat-Operads

An operad (this paper deals with non-symmetric operads)may be conceived as a partial algebra with a family of insertion operations, Gerstenhaber's circle-i products, which satisfy two kinds of associativity, one of them involving commutativity. A Cat-operad is an operad enriched over the category Cat of small categories, as a 2-category with small hom-categories is a category enriched over Cat. The notion of weak Cat-operad is to the notion of Cat-operad what the notion of bicategory is to the notion of 2-category. The equations of operads like associativity of insertions are replaced by isomorphisms in a category. The goal of this paper is to formulate conditions concerning these isomorphisms that ensure coherence, in the sense that all diagrams of canonical arrows commute. This is the sense in which the notions of monoidal category and bicategory are coherent. The coherence proof in the paper is much simplified by indexing the insertion operations in a context-independent way, and not in the usual manner. This proof, which is in the style of term rewriting, involves an argument with normal forms that generalizes what is established with the completeness proof for the standard presentation of symmetric groups. This generalization may be of an independent interest, and related to matters other than those studied in this paper. Some of the coherence conditions for weak Cat-operads lead to the hemiassociahedron, which is a polyhedron related to, but different from, the three-dimensional associahedron and permutohedron.

Assessing Proofs Via Oral Interviews

In this qualitative study, we explored how oral interviews can inform instructors about students’ understanding of abstract algebra and their ability to construct a proof in this setting. Our findings indicate that some students had a good understanding of the ideas needed for a subgroup proof, but could not write a coherent proof. One the other hand, these same students provided fairly sound verbal proofs. We discuss the implications of these results for teaching and future research.

RECURRENCE RELATIONS FOR ELLIPTIC SEQUENCES: EVERY SOMOS 4 IS A SOMOS LOWERCASE $k$

In his celebrated memoir, Morgan Ward's definition of elliptic divisibility sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial examples of such sequences. Even then, Ward's proof of the coherence of his definition relies on displaying a sequence of values of quotients of Weierstraß σ-functions. We give a direct proof of coherence and show, rather more generally, that a sequence defined by a so-called Somos relation of width 4 is always also given by three-term Somos relations of all larger widths 5, 6, 7, …. 2000 Mathematics Subject Classification 11B83, 11G05 (primary), 11A55, 14H05, 14H52 (secondary).

SPIN-STATISTICS THEOREM IN PATH INTEGRAL FORMULATION

We present a coherent proof of the spin-statistics theorem in path integral formulation. The local path integral measure and Lorentz-invariant local Lagrangian, when combined with the Green functions defined in terms of time ordered products, ensure causality regardless of statistics. The Feynman m-iε prescription ensures the positive energy condition regardless of statistics, and the abnormal spin-statistics relation for both the spin-0 scalar particles and spin-1/2 Dirac particles is excluded if one imposes the positive norm condition in conjunction with Schwinger's action principle. The minus commutation relation between one Bose and one Fermi field arises naturally in the path integral. The Feynman m-iε prescription also ensures a smooth continuation to Euclidean theory, for which the use of the Weyl anomaly is illustrated to exclude the abnormal statistics for the scalar and Dirac particles not only in four-dimensional theory but also in two-dimensional theory.

HYPERBOLICITY OF GROUPS WITH SUBQUADRATIC ISOPERIMETRIC INEQUALITY

The author gives a simple and coherent proof of Gromov's statement on hyperbolicity of groups with subquadratic isoperimetric inequality.

Teaching and Testing Mathematics Reading

When we discuss teaching undergraduate mathematics with our colleagues, we frequently distinguish two sorts of goals: first, we want students to be able to do computations related to the course and, second, we want students to know the theory from the course. In a calculus course, we want students to be able to integrate polynomials, so in class we tell them how, and on the exam; we give them polynomials to integrate. In linear algebra class, we want the students to know the theorems concerning determinants, so we prove the theorems from the book, and on the exam, we ask them to prove that if A is an invertible matrix, then det(A-1) = 1/det(A). Our hallway conversations elicit general agreement that most of our students do learn to do the calculations we put on the exam and most don't master the theory to the point that they can write a coherent proof of any but the most trivial theorems. I want to argue in this note that there is another goal, between these in accessibility, that should be a conscious part of our teaching effort:

The Coherence of Theism

Richard Swinburne’s new book “is concerned solely with the central core of theistic belief, that God exists, that there is a God. It is not concerned primarily with whether this belief is true or with whether \ye can know it to be true, but with the prior question of what it means and whether it is coherent.” (p. 1) The discussion begins with a general analysis (Chapters 4-6) of the way in which talk of God is to be understood, whether it is analogical, propositional and so forth. There then follows (Chapters 7-15) a consideration of what Swinburne takes to be involved in the claim that there is a God. Swinburne concludes that theism may be coherent, but that a certain understanding of divine attributes is necessary to sustain this conclusion. In particular, God’s eternity cannot be equated with timelessness, and both ‘omnipotent’ and ‘omniscient’ need to be understood in more narrow a fashion than is sometimes the case. Swinburne also maintains that the coherence of theism may only be properly affirmed where there is good, inductive evidence for the truth of theism. There is no direct proof of coherence. Consequently, “those theists who claim to believe that there is a God ‘by faith’, in a sense of the latter expression which entails that they do not have good inductive grounds for this belief, will, if I am right, have to face the consequence that they do not have good grounds for believing that the claim which they make is a coherent or logicallly possible one.” (p. 296)So much for the overall argument of the book. What about the cogency of its details? It seems to me that Swinburne’s whole discussion is spoiled by a fundamental error of approach, a pervasive wrong move. On this I shall comment presently.