In algebraic K-theory, the K-theory of a category C (usually equipped with some kind of additional data) is a sequence of abelian groups Ki(C) associated to it. If C is an abelian category, there is no need for extra data, but in general it only makes sense to speak of K-theory after specifying on C a structure of an exact category, or of a Waldhausen category, or of a dg-category, or possibly some other variants. Thus, there are several constructions of those groups, corresponding to various kinds of structures put on C. Traditionally, the K-theory of C is defined to be the result of a suitable construction, but in some contexts there are more conceptual definitions. For instance, the K-theory is a 'universal additive invariant' of dg-categories and small stable ∞-categories. The motivation for this notion comes from algebraic K-theory of rings. For a ring R Daniel Quillen in introduced two equivalent ways to find the higher K-theory. The plus construction expresses Ki(R) in terms of R directly, but it's hard to prove properties of the result, including basic ones like functoriality. The other way is to consider the exact category of projective modules over R and to set Ki(R) to be the K-theory of that category, defined using the Q-construction. This approach proved to be more useful, and could be applied to other exact categories as well. Later Friedhelm Waldhausen in extended the notion of K-theory even further, to very different kinds of categories, including the category of topological spaces. (Wikipedia).
Category Theory 3.1: Examples of categories, orders, monoids
Examples of categories, orders, monoids.
From playlist Category Theory
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http://jscategory.wordpress.com/source-code/
From playlist Category theory for JavaScript programmers
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The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. The Farrell-Jones Conjecture predicts that the K-theory of group rings RG can be computed in terms of K-theory of group rings RV where V varies over the collection of virtually cycli
From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
Category Theory: The Beginner’s Introduction (Lesson 1 Video 2)
Lesson 1 is concerned with defining the category of Abstract Sets and Arbitrary Mappings. We also define our first Limit and Co-Limit: The Terminal Object, and the Initial Object. Other topics discussed include Duality and the Opposite (or Mirror) Category. Follow me on Twitter: @mjmcodr
From playlist Category Theory: The Beginner’s Introduction
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http://jscategory.wordpress.com/source-code/
From playlist Category theory for JavaScript programmers
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From playlist Global Noncommutative Geometry Seminar (Americas)
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From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
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From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
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From playlist HIM Lectures: Junior Trimester Program "Topology"
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For questions and discussions of the lecture please go to our discussion forum: https://www.uni-muenster.de/TopologyQA/index.php?qa=k%26l-conference This lecture is part of the event "New perspectives on K- and L-theory", 21-25 September 2020, hosted by Mathematics Münster: https://go.wwu
From playlist New perspectives on K- and L-theory
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From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory
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For questions and discussions of the lecture please go to our discussion forum: https://www.uni-muenster.de/TopologyQA/index.php?qa=k%26l-conference This lecture is part of the event "New perspectives on K- and L-theory", 21-25 September 2020, hosted by Mathematics Münster: https://go.wwu
From playlist New perspectives on K- and L-theory
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PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
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The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Arthur Bartels: K-theory of group rings The Farrell-Jones Conjecture predicts that the K-theory of group rings RG can be computed in terms of K-theory of group rings RV where V vari
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