# Math Help - Category of Sets and its opposite

1. ## Category of Sets and its opposite

Hello, I apologize if this is the wrong section to post this in.

There is an exercise Category Theory by Awodey that asks whether some categories are isomorphic. One is the category of sets and its opposite. I figured it wouldn't be since a function need not have an inverse. Although, I kept thinking such a functor preserved objects, but then, there are probably functors that don't do that and so maybe something else could work. But a quick google showed that these are not isomorphic, but I don't know where to begin to show that....

I thought maybe there's a result but the book hasn't gotten very far with anything so I'm guessing this should be straight forward?

I'm sorry, I'm totally new to category theory :S

2. ## Re: Category of Sets and its opposite

the arrows of Set^op are still functions, we just write f:A<--B instead of f:A-->B.

the easiest thing i can think of to show they are not isomorphic, is that Set^op has only one terminal object, whereas Set has many.

3. ## Re: Category of Sets and its opposite

Aaah ok, then I should show that a functor that is an isomorphism preserves terminal objects (and initial objects), which I can (thankfully!) do. Thank you!

I have an aside, though. I know that there are categories that are isomorphic to their dual. Also, if an object is terminal its dual is initial. But a functor isomorphism preserves these things. Does that mean that such categories have that terminal objects are initial and viceversa? Or am I wrongly assuming that a functor doesn't necessarily send to the respective dual in the opposite category?