Category theory | Monoidal categories
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint (i.e., making the category symmetric monoidal or even symmetric closed monoidal, respectively). Enriched category theory thus encompasses within the same framework a wide variety of structures including * ordinary categories where the hom-set carries additional structure beyond being a set. That is, there are operations on, or properties of morphisms that need to be respected by composition (e.g., the existence of 2-cells between morphisms and horizontal composition thereof in a 2-category, or the addition operation on morphisms in an abelian category) * category-like entities that don't themselves have any notion of individual morphism but whose hom-objects have similar compositional aspects (e.g., preorders where the composition rule ensures transitivity, or Lawvere's metric spaces, where the hom-objects are numerical distances and the composition rule provides the triangle inequality). In the case where the hom-object category happens to be the category of sets with the usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory. An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or an enriched category in M, or simply an M-category. Due to Mac Lane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as V-categories. (Wikipedia).
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Evaluating Limits Definition of Derivative
I work through 2 example evaluating the limit definition of a derivative. Check out http://www.ProfRobBob.com, there you will find my lessons organized by chapters within each subject. If you'd like to make a donation to support my efforts look for the "Tip the Teacher" button on my chann
From playlist Calculus (New)
If you are interested in learning more about this topic, please visit http://www.gcflearnfree.org/ to view the entire tutorial on our website. It includes instructional text, informational graphics, examples, and even interactives for you to practice and apply what you've learned.
From playlist Microsoft Excel
Open Source vs. Closed Source Software
In this video, you’ll learn more about the differences between open-source software and closed-source software. Visit https://edu.gcfglobal.org/en/basic-computer-skills/ for more technology, software, and computer tips. We hope you enjoy!
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Evaluate limits by expanding the power
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From playlist Sect2.3, Evaluating Limits Algebraically
Stable Homotopy Seminar, 14: The stable infinity-category of spectra
I give a brief introduction to infinity-categories, including their models as simplicially enriched categories and as quasi-categories, and some categorical constructions that also make sense for infinity-categories. I then describe what it means for an infinity-category to be stable and h
From playlist Stable Homotopy Seminar
Emily Riehl: On the ∞-topos semantics of homotopy type theory: All ∞-toposes have... - Lecture 3
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From playlist Topology
Higher Algebra 1: ∞-Categories
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From playlist Higher Algebra
Stable Homotopy Seminar, 8: The Stable Model Category of Spectra
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From playlist Stable Homotopy Seminar
Homotopy Category As a Localization by Rekha Santhanam
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
From playlist Dualities in Topology and Algebra (Online)
Duality in Higher Categories-I by Pranav Pandit
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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A topos-theoretic view of difference algebra
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From playlist Open Q&A
6B. RNA 2: Clustering by Gene or Condition and Other Regulon Data Sources Nucleic Acid ...
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The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories - Emily Riehl
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Higher algebra 4: Derived categories as ∞-categories
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From playlist Higher Algebra