Wednesday, November 21, 2012

The category of sets and its opposite

In the book Categories and sheaves by Kashiwara and Shapira, I found a nice argument for the fact that the category of sets is not equivalent to its opposite: they write « every morphism to the initial object is an isomorphism ». Of course!

In the category Sets, the initial object is the empty set, which means that for every set $A$, there is a unique map from $\emptyset$ to $A$. Now, if we reverse the process, namely, if we consider a set $A$ and a map $f\colon A\to \emptyset$, we see that $A$ must be empty and $f$ is a bijection, hence an isomorphism in the category of sets.

In the opposite category, all arrows are reversed, the initial object becomes the terminal object, etc. Isomorphisms are (reversed) maps which have an inverse, so isomorphisms are still given by bijections. A terminal object of Sets is one-element set, $\{x\}$ (you could take the set $1=\{\emptyset\}$ if, like von Neumann, you believe that numbers are sets). Indeed, there is a unique map from any set to a one-element set. Reverse the process again and consider a set $A$ and a map $f\colon \{x\} \to A$. This amounts to choosing an element of $A$, but such maps are not bijections in general, unless $A$ has itself only one element.


  1. The first sentence of the 2nd paragraph has the meaning of $\emptyset$ being initial reversed.