At pag 15.
Theorem 1.6 Every category C with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions.
The following Remark 1.7 is:
"This shows us what is wrong with the naive notion of a 'concrete' category of sets and functions: while not every category has special sets and functions as its objects and arrows, every category is isomorphic to such a one. Thus the only special properties such categories can posses are the ones that are categorically irrelevant, such as features of the objects that do not affect the arrows in any way"
Theorem 1.6 is the generalization to categories of Cayley theorem for groups (every group is isomorphic to a group of permutations), which is useful because the isomorphism allows to study the group of permutations and obtain information on every other group.
- My Problem is the sentence in bold. One one side isomorphic in a way means that two things are the same, so how can one have no categorical interest while on the other side of the iso there's something that is of categorical interest (in the sense of the properties they possess)?
- Also it seems to me like Awodey is saying that studying concrete categories is categorically useless, because of the generalized Cayley, while for groups the group of permutations I'd say acquires importance thanks to Cayley's theorem. I am not understanding why for categories happens the opposite than what happens for groups.