I have a nit-picky question about how the word "object" (as in "mathematical object") is generally used/understood. I'll ask by way of a simple, specific example.
Consider 1) the set of permutations of $n$ things, and 2) the set of bijective functions mapping $n$ things to $n$ things. If you separately explain 1) and 2) to someone with no prior knowledge, it's not immediately obvious that they are basically the same thing (otherwise I've chosen a bad example).
Would you say,
- 1) and 2) are two equivalent objects
- 1) and 2) are two different ways of looking at the same object
or,
- both the above usages are ok, depending on what you want to emphasize