4 ms·
I am a bit confused, I had thought there could not be multiple isomorphisms from object A to object B. I thought an Iso was an epi and mono, doesn't that mean
by runT1ME 12y ago
I am a bit confused, I had thought there could not be multiple isomorphisms from object A to object B. I thought an Iso was an epi and mono, doesn't that mean the arrows must all be the same? Thanks in advance if someone wants to go through all the ways I'm misunderstanding. :)
- nrinaudo 12y agoif f is an isomorphism between A and B, it simply means that there exists f-1 such that f(f-1) = idB and f-1(f) = idA. I don't think anything in this definition allows us to conclude that there can only be one such f between A and B. Indeed, if A and B are sets of identical cardinality, I'm pretty sure you can create n! isomorphims between the two (where n=|A|). Drawing a picture of of two sets with two elements each should let you see it.
- runT1ME 12y agoOh. I think I was confused, arrows can be equal (extensionally so), but not the 'same'.
- orbifold 12y agoA group is a category with a single object and whose morphisms are all isomorphisms. There are plenty of groups, so you can't be right :). A morphisms can however have at most one inverse.