Category theory : product and coproduct

In summary, Category theory defines coproduct and product as objects that represent the combination of two objects in a category. However, not all categories have products for every pair of objects. In a category with 3 objects and 2 arrows, the product of A and C is the object A with two morphisms. This may not follow the uniqueness property of morphisms, but it is necessary in order to have a product in this category.
  • #1
Sumanta
26
0
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[itex]\pi[/itex][itex]_{1}[/itex]g = [itex]\pi[/itex][itex]_{2}[/itex]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
 
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  • #2
The point of that diagram is that A and B does not have a product. AxC does exist and is equal to A.
 

1. What is a product in category theory?

A product in category theory is a construction that represents the combination of two or more objects in a category. It is denoted as A × B, where A and B are objects in the category. The product of two objects is defined as the object that satisfies certain universal properties, making it unique up to isomorphism.

2. What is a coproduct in category theory?

A coproduct in category theory is a construction that represents the choice between two or more objects in a category. It is denoted as A + B, where A and B are objects in the category. The coproduct of two objects is defined as the object that satisfies certain universal properties, making it unique up to isomorphism.

3. What is the difference between a product and a coproduct in category theory?

The main difference between a product and a coproduct in category theory is the direction of information flow. In a product, information flows from the product to its components, while in a coproduct, information flows from the components to the coproduct. Additionally, the universal properties satisfied by each construction are slightly different, reflecting their different purposes.

4. How are products and coproducts used in category theory?

Products and coproducts are used to understand and analyze relationships between objects in a category. They provide a way to break down complex objects into simpler components and understand how those components interact. Additionally, they allow for the generalization of mathematical concepts and ideas, making it easier to study and compare different mathematical structures.

5. What are some examples of products and coproducts in category theory?

In the category of sets, the product of two sets A and B is the cartesian product A × B, and the coproduct is the disjoint union A + B. In the category of groups, the product of two groups G and H is the direct product G × H, and the coproduct is the free product G * H. Products and coproducts can also be found in other categories, such as topological spaces, vector spaces, and many more.

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