Is the Tensor Product Associative?

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SUMMARY

The tensor product is associative up to isomorphism, as established in the discussion. To demonstrate this, one must utilize the Universal Property of tensor products, specifically through the use of commutative diagrams. The discussion references a specific resource, particularly page 31 of the document found at http://www.math.uga.edu/%7Eroy/845-3.pdf, which outlines how both (U⊗V)⊗W and U⊗(V⊗W) satisfy the Universal Property, leading to their isomorphism.

PREREQUISITES
  • Understanding of tensor products in linear algebra
  • Familiarity with Universal Properties in category theory
  • Knowledge of commutative diagrams
  • Basic concepts of isomorphism in mathematics
NEXT STEPS
  • Study the Universal Property of tensor products in detail
  • Explore commutative diagrams and their applications in category theory
  • Review examples of isomorphisms in linear algebra
  • Examine the specific resource mentioned for deeper insights on tensor products
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Mathematicians, students of linear algebra, and anyone interested in the properties of tensor products and category theory.

huyichen
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It is a well fact that tensor product is associative up to isomorphism, but how should I use Universal property(you know, diagrams that commute) to show that it is true?
 
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http://www.math.uga.edu/%7Eroy/845-3.pdf

especially page 31
 
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Show that both (U\otimes V)\otimes W and U\otimes (V\otimes W) both satisfy the universal property. By general nonsense they are then isomorphisc.
 

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