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An isomorphism is more than a bijection, unless it is between sets and the category is the one of all functions from one set to another. Isomorphism is about p
by Bahamut 10y ago
An isomorphism is more than a bijection, unless it is between sets and the category is the one of all functions from one set to another.
Isomorphism is about preserving relations as well. For example, consider the ring of integers and the field of rational numbers. One can construct a bijection between one to the other (they both have the same cardinality as the natural numbers), but they are significantly different constructs. The bijection would not be able to preserve the operations - in fact, the rational numbers as a ring is significantly different, and unless one wants to use information about the cardinality, a bijection between the two is otherwise worthless.
- greydius 10y agoCorrect. A bijection implies isomorphism in the category of sets. Thus an isomorphism exists. I suppose we need to convince ourselves that algorithms and abstract syntax trees do indeed form sets. (Exclude such things as "the algorithm that computes the set of all algorithms", etc.)
- Bahamut 10y agoBut that is typically not how the word is used because to substitute bijection with isomorphism, it would only make sense when talking about cardinality - that is not how you used it.