John Dillon, "Dexippus: On Aristotle Categories"
English | 2014 | ISBN: 1780933711, 0715622420 | PDF | pages: 156 | 8.0 mb
Dexippus, a pupil or follower of lamblichus, preserves a crucial moment in the Neoplatonist interpretation of Aristotle. Aristotle's Categories has been attacked by Descriptioninus, but Porphyry's defence proved decisive, so that the Categories was acceptable as compatible with Platonism and an essential introduction to the Neoplatonist curriculum.

Christopher L. Douglas, Christopher Schommer-Pries, "Dualizable Tensor Categories"
English | ISBN: 1470443619 | 2021 | 88 pages | PDF | 1 MB
"We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3- dimensional 3-framed local field theories. We also show that all finite tensor categories are 2-dualizable, and yield categorified 2-dimensional 3-framed local field theories. On the other hand, topological properties of 3-framed manifolds determine algebraic equations among functors of tensor categories. We show that the 1-dimensional loop bordism, which exhibits a single full rotation, acts as the double dual autofunctor of a tensor category. We prove that the 2-dimensional belt-trick bordism, which unravels a double rotation, operates on any finite tensor category, and therefore supplies a trivialization of the quadruple dual. This approach produces a quadruple-dual theorem for suitably dualizable objects in any symmetric monoidal 3-category. There is furthermore a correspondence between algebraic structures on tensor categories and homotopy fixed point structures, which in turn provide structured field theories; we describe the expectedconnection between pivotal tensor categories and combed fixed point structures, and between spherical tensor categories and oriented fixed point structures"-

Published 7/2022
MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz
Language: English | Size: 486.85 MB | Duration: 0h 47m
5 Fluid Categories

Sets, Logic and Categories by Peter J. Cameron
English | PDF | 1998 | 191 Pages | ISBN : 1852330562 | 14.8 MB
Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Gödel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.

English | 2022 | ISBN: 9811250928 | 486 pages | True PDF EPUB | 15.84 MB
This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad.In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.

Categories for the Working Mathematician By Saunders Mac Lane
1998 | 317 Pages | ISBN: 0387984038 | PDF | 6 MB
Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions.

Aristotle The Categories
pdf | 57.15 KB | English | Isbn:‎ 979-8673295076 | Author: Aristotle | Year: 2020