Category theory | Mathematical terminology
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by mean of a universal morphism (see , below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see , below). Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that would need boring verifications otherwise. For example, given a commutative ring R, the field of fractions of the quotient ring of R by a prime ideal p can be identified with the residue field of the localization of R at p; that is (all these constructions can be defined by universal properties). Other objects that can be defined by universals properties include: all free objects, direct products and direct sums, free groups, free lattices, Grothendieck group, completion of a metric space, completion of a ring, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification, tensor products, inverse limit and direct limit, kernels and cokernels, quotient groups, quotient vector spaces, and other quotient spaces. (Wikipedia).
Complete Derivation: Universal Property of the Tensor Product
Previous tensor product video: https://youtu.be/KnSZBjnd_74 The universal property of the tensor product is one of the most important tools for handling tensor products. It gives us a way to define functions on the tensor product using bilinear maps. However, the statement of the universa
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Universal Law of Gravitation - Part 2 | Physics | Don't Memorise
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Tensor Product Basis With the Universal Property
Tensor product universal property explanation: https://youtu.be/vZzZhdLC_YQ Generating set proof: https://youtu.be/KnSZBjnd_74?t=1437 timestamp 23:57 If we have a basis for each of two vector spaces (or modules over a commutative ring) V and W, then we can use that to form a basis for the
From playlist Tensor Products
What is the complement of a set? Sets in mathematics are very cool, and one of my favorite thins in set theory is the complement and the universal set. In this video we will define complement in set theory, and in order to do so you will also need to know the meaning of universal set. I go
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Emily Riehl: On the ∞-topos semantics of homotopy type theory: The simplicial model of...- Lecture 2
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Proof: Uniqueness of the Tensor Product
Universal property introduction: https://youtu.be/vZzZhdLC_YQ This video proves the uniqueness of the tensor product of vector spaces (or modules over a commutative ring). This uses the universal property of the tensor product to prove the existence of an isomorphism (linear bijection) be
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Google Analytics has been a game twister for many businesses. In this Google Analytics 4 video, we are going to understand, what is google analytics 4, importance of google analytics, how to identify google analytics property for your website. The most important part we are going to learn
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Galois Representations 4 by Shaunak Deo
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Yonatan harpaz : The universal property of topological Hochschild homology
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Mod-03 Lec-07 The Samkhya Philosophy - III
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