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We introduce Manifold tensor categories, which make precise the notion of a tensor category with a manifold of simple objects. A basic example is the category of vector spaces graded by a Lie group. Unlike classic tensor category theory, our setup keeps track of the smooth (and topological) structure of the manifold of simple objects. We set down the necessary definitions for Manifold tensor categories and a generalisation we term Orbifold tensor categories. We also construct a number of examples, most notably two families of examples we call Interpolated Tambara-Yamagami categories and Interpolated quantum group categories. Finally, we show conditions under which pointwise duality data in an Orbifold tensor category automatically assembles into smooth duality data. Our proof uses the classic Implicit function theorem from differential geometry., Comment: This document is an edited excerpt of the author's thesis

The computational thinking becomes more and more an essential part of modern teaching. In this context, the resources, including the collection of online lessons plan, videos and other resources were created to provide a better understanding of CT. But the most important are software resources, because of their flexibility and wide application tools. The aim of the present paper is to have an introduction trip focusing on the advantages of using powerful software like MAPLE in computational teaching. Some examples are focused on representing the objects and basic algebraic manipulation commands, very successfully used in teaching sciences.

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory, the global homotopy theory of Schwede (2018). Specifically, for a global ring spectrum R, we consider which classes of ring homomorphisms ηe W πe eR ! Se can be realised by a map ηW R ! S in the category of global R– modules, and what multiplicative structures can be placed on S. If ηe witnesses Se as a projective πe eR–module, then such an η exists as a map between homotopy commutative global R–algebras. If ηe is in addition étale or S0 is a Q–algebra, then η can be upgraded to a map of E1–global R–algebras or a map of G1–R–algebras, respectively. Various global spectra and E1–global ring spectra are then obtained from classical homotopy-theoretic and algebraic constructions, with a controllable global homotopy type.

We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be recovered in this way as categories of modules over a commutative semiring category (or $\infty$-category in the last case). This language provides a simultaneous generalization of the formalism of algebraic theories (operads, PROPs, Lawvere theories) and stable homotopy theory, with essentially a variant of algebraic K-theory bridging between the two., 51 pages; some proofs and notation clarified; this is the final version to appear in Algebraic and Geometric Topology

La Didáctica de la divisibilidad en Álgebra Superior “mediada por entornos informáticos” es un campo poco explorado pero interesante por razones ecológicas y por el alto grado de generali-zación requerido en los estudiantes. El objetivo general es describir y analizar la influencia del uso de recursos informáticos en la enseñanza y el aprendizaje de la divisibilidad, para Grado en Ingeniería Informática, identificando fenómenos didácticos y conflictos de significado, utilizando para ello las herramientas que propone el Enfoque Ontosemiótico. Se establecen dos tipos de hipótesis: unas encaminadas a caracterizar el significado institucional pretendido del máximo común divisor, mostrado en manuales de la asignatura de Matemática Discreta, poniendo de manifiesto la influencia del uso de los recursos informáticos y el tratamiento de las aplicaciones, en la enseñanza; y otras, dirigidas al análisis del aprendizaje de los estudiantes, sus dificultades y conflictos semióticos. The teaching of divisibility in Higher Algebra "mediated by computer technology" is a little-explored field, but interesting for ecological reasons in addition to the high degree of generalisation required in students. The general aim is to describe and analyse how the use of computer resources influences the teaching and learning of divisibility, for a degree in Computer Engineering, identifying didactic phenomena and conflicts of meaning, using the tools proposed for this by the onto-semiotic approach. Two types of hypothesis are established: those aimed at characterising the intended institutional meaning of the greatest common divisor, shown in manuals on Discrete Mathematics, highlighting how the use of computer resources and the treatment of applications influences the teaching; and those directed towards the analysis of the students' learning, their difficulties, and semiotic conflicts. Tesis Univ. Jaén. Departamento de Didáctica de las Ciencias. Leída el 16 de julio de 2018.

Letter concerning a position in the mathematics department at Utah Agricultural College, as well as a booklet describing O. T. Geckeler's qualifications.

Letter concerning a position in horticulture and botany at Utah Agricultural College, copies given to J. C. Blair, J. L. Snyder, and L. H. Bailey.

We show that an oo-category M with a closed left action of a monoidal oo-category V is completely determined by the V -valued graph of morphism objects resulting from closedness of the action equipped with the structure of a V -enrichment in the sense of Gepner-Haugseng. We prove a similar result when M is a V -enriched oo-category in the sense of Lurie, an operadic generalization of the notion of oo-category with closed action. Precisely, we prove that sending a V -enriched oo-category in the sense of Lurie to the V -valued graph of morphism objects refines to an equivalence chi between the oo-category of V -enriched oo-categories in the sense of Lurie and of Gepner-Haugseng.Moreover if V is a presentably k + 1-monoidal oo-category for 1 < k < oo, we prove that chi restricts to a lax k-monoidal functor between the oo-category of left V -modules in PrL, the symmetric monoidal oo-category of presentable oo-categories, endowed with the relative tensor product, and the tensor product of V -enriched oo-categories of Gepner-Haugseng. As an application of our theory we construct a lax symmetric monoidal embedding of the oo-category of small stable oo-categories into the oo-category of small spectral oo-categories. As a second application we produce an enriched Yoneda- embedding in the framework of Lurie's notion of enriched oo-categories.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).