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Jacob Zelko (Jul 25 2022 at 03:32):[Beginner Question]:
Background: In Category Theory for the Sciences, Spivak defines an aspect as, "the way of viewing a thing, x; a particular way in which x can be regarded or measured." He gives the example that a molecule has a molecular mass so a molecular mass is an aspect of a molecule. Later, he goes on to say that every function has an image. I follow this all and this makes sense to me.
Problem: What I get confused on is an exercise he poses where he says:
For each of the following types, write a function for which it is the image, or write “not clearly useful as an image type.”
There are many types he gives but the one I am going to use is a used book. In this case, I took a used book to be an aspect that has the relationship:
a book -- was owned by --> a person
Which I would read as "A book which was owned by a person" which I think describes the aspect of a used book as an image type of books.
Question: Somehow, this doesn't sit right as when I draw out a small olog to describe this process, the paths do not commute leading me think I am wrong in my reasoning somewhere. Could anyone explain to me why a used book is not a valid image type? Thanks! image.png
Morgan Rogers (he/him) (Jul 25 2022 at 06:30):The "is" arrow should go the other way and the upper function doesn't make sense given that not every book is owned by someone?
Spencer Breiner (Jul 25 2022 at 12:10):Agreed. Ownership of a book should be a bidirectional relationship ([[span]])
Book <- Ownership -> Person.
Jacob Zelko (Jul 25 2022 at 15:19):This makes sense - thanks @Morgan Rogers (he/him) and @Spencer Breiner . Out of curiosity, my intuition when the olog didn't make sense - is this a good way of "checking" myself when working through questions such as these? Because, as y'all indicate, I was right in my thinking that I was wrong. Just couldn't pinpoint why with the language I currently knew/understood. Thanks!
John Baez (Jul 25 2022 at 16:37):I don't understand your question, but it's definitely good to keep in mind that sometimes when you're confused, it's because the book is confused.