Logical distinction between sets and algebraic structures

[Werg22]

Let's say we have a set S, and a function f : S -> S. Now let S be endowed with a binary operation, forming a group G. Is it correct to write f : G - > G?

Up to now I have been operating on the assumption that yes, although G is not technically a set, there is little harm in being sloppy and use G to designate its underlying set, S.

However someone has recently told me that this is not correct. f : G - > G is different from f : S - > S. I was referred to category theory, of which I admittedly know nothing.

Is this true?

 

[Hurkyl]

There are a variety of syntactic conventions... but I expect you were told something more conceptual: a map of groups really ought to be a homomorphism. Only certain set functions S -> S correspond to group homomorphisms G -> G.

 

[mathman]

A set is is the generic term for a collection of things. Group members, vectors, probability events, etc. are all elements of sets.