Comments (3)
An isomorphism is a "structure-preserving" function. It can mean different things for different structures.
In the case of sets, bijection is the same thing as isomorphism simply because sets have no structure. (https://math.stackexchange.com/questions/90923/isomorphism-of-sets)
So, in this context, both words are equivalent, but since my aim is not to talk about sets specifically, but to generalize the notion of isomorphism for other objects, I use "isomorphism". If you want to know where I am going with this, skip to the last chapter:
https://abuseofnotation.github.io/category-theory-illustrated/06_functors/#categorical-isomorphisms
from category-theory-illustrated.
Yes, it is my impression as well, that two terms are roughly equivalent.
Why do you think "bijection" is more appropriate than "isomorphism"?
from category-theory-illustrated.
Isomorphism is (to my knowledge) used in context of monoids and groups which have structure. It implies two things:
- the function is bijective (injective and surjective i.e. invertible),
- the function is homomorphism (that is:
f(a) * f(b) = f(a * b)
, where*
is the group's operator).
So all isomorphism are bijective, but not all bijections are isomorphic.
In the case of first chapter, sets don't even have the operation, which means it does not really make sense to talk about them being homomorphic, and by extension, isomorphic.
from category-theory-illustrated.
Related Issues (19)
- OR and AND diagram labels are reversed HOT 2
- poset used without being defined HOT 1
- Chapter 02 - `isomorphism.svg` image is broken HOT 3
- Monoid associativity doesn't imply that inverses exist HOT 6
- Illustrations for logical operators are hard to read HOT 3
- Please may you make a libre license official? HOT 4
- Singlton Example HOT 4
- Grammar: it's > its HOT 3
- For or: p → p ∨ q and p → p ∨ q HOT 1
- Is the arrow direction wrong here? HOT 1
- Image of is letter function - incorrect mappings HOT 1
- A few typos HOT 1
- minor typos HOT 2
- About ePub format output HOT 5
- small error HOT 1
- Initial object diagram is backwards HOT 1
- What is the software you used for creating the figure? HOT 2
- Content's license HOT 4
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from category-theory-illustrated.