# Reconciling Awodey, Aluffi & Leinster's Products as Categories

• MHB
• Math Amateur
In summary, Paolo Aluffi, Steve Awodey, and Tom Leinster have different approaches to the construction of a product category. Aluffi seems to differ from the approach of Awodey and Leinster, and the product of two categories is not defined using the universal mapping property.
Math Amateur
Gold Member
MHB
The books by Awodey, Aluffi and Leinster have different approaches to the construction of a product category ... at least Awodey seems to differ from the approach of Aluffi and Leinster ...

Can someone explain how to reconcile the approach of Awodey with the approach of Aluffi/Leinster ... presumably the two approaches are actually the same ...

The books I am referring to are as follows:

Algebra: Chapter 0 by Paolo Aluffi

Category Theory by Steve Awodey

Basic Category Theory by Tom LeinsterThe approaches to the construction of a product category are as follows:

Awodey

https://www.physicsforums.com/attachments/8361
https://www.physicsforums.com/attachments/8362Aluffi

View attachment 8363Leinster

View attachment 8364
View attachment 8365
As I mentioned above ... my question is as follows:

How do we reconcile the approach of Awodey to the construction of a product category with the approach of Aluffi/Leinster ... ?
Hope someone can help ...

Peter

Basically, you are mixing up theory and examples. The “official” definition of the product of two (or more) objects in a category is for all authors the same, as is should be. Compare:

Leinster, definition 5.1.1. page 108
Awodey, definition 2.15. page 39
Simmons, definition 2.5.2. page 50
The most important feature of a product is the universal mapping property.
Allufi: you mentioned section 5.4. on page 35 of Aluffi. Here he defines the universal mapping propery in an example using sets. Replacing “set” with “object”, you get an “official” definition of the universal property and the product. He does not state that definition explicitly (I could not find it) which is strange, because he mentions the definition of a coproduct on page 36.

Then there is the product of two categories $C$ and $D$, Awodey 1.6. page 14. This product is defined as a Cartesian product. Why aren’t the categories $C$ and $D$ considered as objects in $CAT$, the category of categories, and the product of $C$ and $D$ defined using the universal property ?. I do not know. Maybe someone who knows something of categories can answer this.

steenis said:
Basically, you are mixing up theory and examples. The “official” definition of the product of two (or more) objects in a category is for all authors the same, as is should be. Compare:

Leinster, definition 5.1.1. page 108
Awodey, definition 2.15. page 39
Simmons, definition 2.5.2. page 50
The most important feature of a product is the universal mapping property.
Allufi: you mentioned section 5.4. on page 35 of Aluffi. Here he defines the universal mapping propery in an example using sets. Replacing “set” with “object”, you get an “official” definition of the universal property and the product. He does not state that definition explicitly (I could not find it) which is strange, because he mentions the definition of a coproduct on page 36.

Then there is the product of two categories $C$ and $D$, Awodey 1.6. page 14. This product is defined as a Cartesian product. Why aren’t the categories $C$ and $D$ considered as objects in $CAT$, the category of categories, and the product of $C$ and $D$ defined using the universal property ?. I do not know. Maybe someone who knows something of categories can answer this.

Thanks for the above post, steenis ...

I am still reflecting on it ... and the category of Product and Coproduct ...

Peter

## 1. What is the concept of "products" in category theory?

In category theory, a product is a universal construction that allows for the combination of two or more objects or morphisms in a category. It is a generalization of the Cartesian product in set theory.

## 2. How do Awodey, Aluffi & Leinster reconcile the concept of products in category theory?

In their works, Awodey, Aluffi & Leinster reconcile the concept of products by providing a categorical description of products as a universal construction. They also highlight the importance of the terminal object and the notion of commutativity in the definition of products.

## 3. What is the significance of reconciling Awodey, Aluffi & Leinster's products as categories?

Reconciling Awodey, Aluffi & Leinster's products as categories is significant because it provides a unified and abstract perspective on the concept of products. It also allows for the application of category theory in various fields, including mathematics, computer science, and physics.

## 4. Are there any limitations to the reconciliation of products as categories?

Although the reconciliation of products as categories is a powerful tool in category theory, it does have some limitations. For example, it may not be applicable to certain categories, such as infinite categories or categories with a weakly initial object.

## 5. How can the concept of products in category theory be applied in practical settings?

The concept of products in category theory can be applied in various practical settings, such as in database management, functional programming, and topology. It provides a powerful tool for analyzing and understanding the relationships between objects and morphisms in a given category.

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