Reconciling Awodey, Aluffi & Leinster's Products as Categories

[Math Amateur]

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

 

[steenis]

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
Adamek, definition 10.19. page 168
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.

 

[Math Amateur]

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
Adamek, definition 10.19. page 168
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