Shadows and traces in bicategories

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs "noncommutative" traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a "shadow." In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate "cylindrical" type of string diagram, which we justify formally in an appendix.Comment: 46 pages; v2: reorganized and shortened, added proof for cylindrical string diagrams; v3: final version, to appear in JHR

Abstract Tensor Systems as Monoidal Categories

The primary contribution of this paper is to give a formal, categorical treatment to Penrose's abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for traced symmetric monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda

On the word problem for SP-categories, and the properties of two-way communication

International audienceThe word problem for categories with free products and coproducts (sums), SP-categories, is directly related to the problem of determining the equivalence of certain processes. Indeed, the maps in these categories may be directly interpreted as processes which communicate by two-way channels. The maps of an SP-category may also be viewed as a proof theory for a simple logic with a game theoretic intepretation. The cut-elimination procedure for this logic determines equality only up to certain permuting conversions. As the equality classes under these permuting conversions are finite, it is easy to see that equality between cut-free terms (even in the presence of the additive units) is decidable. Unfortunately, this does not yield a tractable decision algorithm as these equivalence classes can contain exponentially many terms. However, the rather special properties of these free categories -- and, thus, of two-way communication -- allow one to devise a tractable algorithm for equality. We show that, restricted to cut-free terms s,t : X --> A, the decision procedure runs in time polynomial on |X||A|, the product of the sizes of the domain and codomain type

Calculating Colimits Compositionally

We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages

Comparison of high-specific-activity ultratrace 123/131I-MIBG and carrier-added 123/131I-MIBG on efficacy, pharmacokinetics, and tissue distribution

Metaiodobenzylguanidine (MIBG) is an enzymatically stable synthetic analog of norepinephrine that when radiolabled with diagnostic ((123)I) or therapeutic ((131)I) isotopes has been shown to concentrate highly in sympathetically innervated tissues such as the heart and neuroendocrine tumors that possesses high levels of norepinephrine transporter (NET). As the transport of MIBG by NET is a saturable event, the specific activity of the preparation may have dramatic effects on both the efficacy and safety of the radiodiagnostic/radiotherapeutic. Using a solid labeling approach (Ultratrace), noncarrier-added radiolabeled MIBG can be efficiently produced. In this study, specific activities of >1200 mCi/micromol for (123)I and >1600 mCi/micromol for (131)I have been achieved. A series of studies were performed to assess the impact of cold carrier MIBG on the tissue distribution of (123/131)I-MIBG in the conscious rat and on cardiovascular parameters in the conscious instrumented dog. The present series of studies demonstrated that the carrier-free Ultratrace MIBG radiolabeled with either (123)I or (131)I exhibited similar tissue distribution to the carrier-added radiolabeled MIBG in all nontarget tissues. In tissues that express NETs, the higher the specific activity of the preparation the greater will be the radiopharmaceutical uptake. This was reflected by greater efficacy in the mouse neuroblastoma SK-N-BE(2c) xenograft model and less appreciable cardiovascular side-effects in dogs when the high-specific-activity radiopharmaceutical was used. The increased uptake and retention of Ultratrace (123/131)I-MIBG may translate into a superior diagnostic and therapeutic potential. Lastly, care must be taken when administering therapeutic doses of the current carrier-added (131)I-MIBG because of its potential to cause adverse cardiovascular side-effects, nausea, and vomiting

The Platonic 'Theages' : an introduction, commentary and critical edition

The Theages poses a number of problems for the interpreter of Plato and the Platonic dialogue. Traditionally, the most controversial one concerns the authenticity of the work: is Plato its author, and what criteria may be considered valid and important for settling the debate over authorship. But there are numerous other questions of at least equal significance. What is the purpose for which this dialogue was written, and what is its meaning. Is it merely a patchwork, as is commonly assumed, or does it display a structural unity. How does the Socrates of this work compare with the same character in other Socratic compositions, and what literary qualities can be attributed to the author's portrayal of the dialogue's other personae. How are we to evaluate the lengthy section in the Theages on Socrates' "divine sign". When was this dialogue written. What is its relation to the other works in the Platonic Corpus, to Socratic literature generally, and to philosophical interests at the time of its composition. The introduction and some of the appendices to this thesis attempt to offer answers to these questions, both through a comprehensive review and assessment of the critical literature on the Theages, and through the use of new evidence, argumentation, and interpretation. At the same time, a basis for the analyses offered here (and for future examinations of the Theages) is provided in this study by a detailed line-by-line commentary on the text. The text on which this commentary depends has been established from a fresh collation of all known manuscripts, early printed editions, and ancient testimonia, containing all or part of the Theages. This thesis represents the first attempt, in any language, to undertake the above programme of work on a definitive scale

Nets, relations and linking diagrams

In recent work, the author and others have studied compositional algebras of Petri nets. Here we consider mathematical aspects of the pure linking algebras that underly them. We characterise composition of nets without places as the composition of spans over appropriate categories of relations, and study the underlying algebraic structures.Comment: 15 pages, Proceedings of 5th Conference on Algebra and Coalgebra in Computer Science (CALCO), Warsaw, Poland, 3-6 September 201

A Physicist's Proof of the Lagrange-Good Multivariable Inversion Formula

We provide yet another proof of the classical Lagrange-Good multivariable inversion formula using techniques of quantum field theory.Comment: 9 pages, 3 diagram

The fusion algebra of bimodule categories

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F. As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.Comment: 16 page

Higher Algebraic Structures and Quantization

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in a d+1 dimensional topological theory to manifolds of dimension less than d+1. We then ``construct'' a generalized path integral which in d+1 dimensions reduces to the standard one and in d dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.Comment: 62 pages + 16 figures (revised version). In this revision we make some small corrections and clarification