Dualizability in Low-Dimensional Higher Category Theory

These lecture notes form an expanded account of a course given at the Summer School on Topology and Field Theories held at the Center for Mathematics at the University of Notre Dame, Indiana during the Summer of 2012. A similar lecture series was given in Hamburg in January 2013. The lecture notes are divided into two parts. The first part, consisting of the bulk of these notes, provides an expository account of the author's joint work with Christopher Douglas and Noah Snyder on dualizability in low-dimensional higher categories and the connection to low-dimensional topology. The cobordism hypothesis provides bridge between topology and algebra, establishing important connections between these two fields. One example of this is the prediction that the $n$-groupoid of so-called `fully-dualizable' objects in any symmetric monoidal $n$-category inherits an O(n)-action. However the proof of the cobordism hypothesis outlined by Lurie is elaborate and inductive. Many consequences of the cobordism hypothesis, such as the precise form of this O(n)-action, remain mysterious. The aim of these lectures is to explain how this O(n)-action emerges in a range of low category numbers ($n \leq 3$). The second part of these lecture notes focuses on the author's joint work with Clark Barwick on the Unicity Theorem, as presented in arXiv:1112.0040. This theorem and the accompanying machinery provide an axiomatization of the theory of $(\infty,n)$-categories and several tools for verifying these axioms. The aim of this portion of the lectures is to provide an introduction to this material.Comment: 65 pages, 8 figures. Lecture Note

A Generalized Contou-Carr\`ere Symbol and its Reciprocity Laws in Higher Dimensions

We generalize the theory of Contou-Carr\`ere symbols to higher dimensions. To an $(n+1)$-tuple $f_0,\dots,f_n \in A((t_1))\cdots((t_n))^{\times}$, where $A$ denotes a commutative algebra over a field $k$, we associate an element $(f_0,\dots,f_n) \in A^{\times}$, compatible with the higher tame symbol for $k = A$, and earlier constructions for $n = 1$, by Contou-Carr\`ere, and $n = 2$ by Osipov--Zhu. Our definition is based on the notion of \emph{higher commutators} for central extensions of groups by spectra, thereby extending the approach of Arbarello--de Concini--Kac and Anderson--Pablos Romo. Following Beilinson--Bloch--Esnault for the case $n=1$, we allow $A$ to be arbitrary, and do not restrict to artinian $A$. Previous work of the authors on Tate objects in exact categories, and the index map in algebraic $K$-theory is essential in anchoring our approach to its predecessors. We also revisit categorical formal completions, in the context of stable $\infty$-categories. Using these tools, we describe the higher Contou-Carr\`ere symbol as a composition of boundary maps in algebraic $K$-theory, and conclude the article by proving a version of Parshin--Kato reciprocity for higher Contou-Carr\`ere symbols.Comment: 55 pages, introduction completely rewritte

Reporting categories in urine test strip analysis: Croatian survey and call for action

Introduction: In line with the national recommendations, Croatian medical laboratories report urine test strip qualitative analysis results using a categorized scale with defined number of categories. Since concentration ranges for measured analytes have not been provided by national professional authority, it is up to the laboratories to define their own categories. The aim of study was to assess the comparability of concentrations assigned to different categories used in reporting the results of dipstick urinalysis in Croatian laboratories. Material and methods: A questionnaire was e-mailed to all Croatian medical laboratories (N = 195). They were asked to provide the number of categories and respective concentrations for each parameter. Data were described as numbers and percentages. Values above the upper reference range limit, which were assigned as normal and/or trace category, were considered as false negative. Results: Response rate was 71% (139/195). Seventy percent (98/139) of laboratories report their results with either higher (77/98; 79%) or lower (2/98; 2%) number of categories, relative to the national recommendation, whereas 19/98 (19%) report their results as concentrations. Great heterogeneity of reporting categories was observed. Multiple categories were assigned to same concentrations and there was a large overlap of concentrations for most categories. Considerable proportion of laboratories reported false negative results for ketones (42%), leukocytes (30%) and glucose (21%). Conclusions: The concentrations assigned to categories used to report the results of dipstick urinalysis are not comparable among Croatian medical laboratories. There is an urgent need for harmonization and standardization of reporting the results of urine dipstick analysis in Croatia

Capturing Goodwillie's Derivative

Recent work of Biedermann and R\"ondigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we focus on understanding the derivative as a right Quillen functor to a new model category. This is directly analogous to the behaviour of Weiss's derivative in orthogonal calculus. The immediate advantage of this new category is that we obtain a streamlined and more informative proof that the n-homogeneous functors are classified by spectra with an action of the symmetric group on n objects. In a later paper we will use this new model category to give a formal comparison between the orthogonal calculus and Goodwillie's calculus of functors.Comment: Final version, to appear. Substantially shortened from earlier version, with a significantly expanded introduction, new results and examples. 27 page