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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
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    Presumably the three "$n$ things" can be three different (but fixed) objects? Also, you should perhaps specify which of the (equivalent) definitions of "permutation" you are using.2012-08-17
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    Isomorphic objects in the category of groups. (Or equal objects, depending on your definition of permutation.)2012-08-17
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    What do you mean by permutations of $n$ things?2012-08-17

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