Category theory : product and coproduct

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SUMMARY

The discussion focuses on the concepts of coproduct and product in category theory, specifically referencing the definitions provided on Wikiversity. It highlights that not all categories possess products for every pair of objects, illustrated by a category with three objects and two arrows. The participant questions the uniqueness of morphisms in relation to the product of objects A and C, noting that while the product AxC exists, A and B do not have a product. This indicates a deeper exploration of morphism relationships and product definitions in category theory.

PREREQUISITES
  • Understanding of basic category theory concepts
  • Familiarity with morphisms and objects in categories
  • Knowledge of product and coproduct definitions
  • Ability to interpret diagrams in category theory
NEXT STEPS
  • Study the formal definitions of products and coproducts in category theory
  • Learn about specific examples of categories that do not have products
  • Explore the role of morphisms in defining products and coproducts
  • Investigate the implications of uniqueness in morphism relationships
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Students and researchers in mathematics, particularly those focusing on category theory, as well as educators seeking to clarify the concepts of products and coproducts for their students.

Sumanta
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Hi,

I was trying to understand coproduct and product as defined in category theory from the website

http://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Products_and_Coproducts.

Before I could even think of sth difficult there are some simple things which I don't seem to understand. If anybody could kindly explain it would be helpful.

Not every category has products for all pairs of objects (i.e. 'has all products'). For example in the category with 3 objects and 2 arrows (+identity arrows) shown at right, the product of A and C is the object A together with morphisms.

http://upload.wikimedia.org/wikipedia/commons/f/f8/Simple_category.svg.

I see that t\pi_{1}g = \pi_{2}g where the t is the morphism between A and C. But does this somehow fail the uniqueness of g. Not sure how did he get the product as A with the two morphisms identity and t.

Excuse me as not being able to see the Greek letters even after I typed from the latex help
 
Last edited:
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The point of that diagram is that A and B does not have a product. AxC does exist and is equal to A.
 

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