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I'm playing a bit with category theory, and I've noticed that it is actually not rare when an object $C$ of a category $\mathcal{C}$ is isomorphic to its square $C \times C$.

An object $C$ of a category $\mathcal{C}$ is therefore called idempotent whenever it is isomorphic to its (categorical) product. Trivial examples are given by the terminal object $1$ of any category, or infinite sets in Sets. One can think of a bit less trivial examples in Grp (say, $C = \displaystyle \bigoplus_{n \in \mathbb{N}} \mathbb{Z}$) or in Rings (say, $\mathbb{R}^\omega$ or even $R^\omega$ for any ring). More interesting examples are given in Fields, where for any prime $p$, one can easily show that $\mathbb{Z}_p$ is idempotent.

Call coidempotent the dual of the previous notion. Again, one can easily think of trivial examples.

What I'm actually interested in is some non trivial examples of bi-idempotent objects (both idempotent and coidempotent), in some category where the notion of product and coproduct aren't coinciding. Of course, a strict initial object $0$ is always a bi-idempotent object, and it'll be the same for any idempotent object in a category where finite products and finite coproduct coincides (like any abelian category).

Anyone has any non trivial examples of such objects?

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  • $\begingroup$ Infinite sets, assuming AC. $\endgroup$ – Qiaochu Yuan Oct 24 '16 at 21:55
  • $\begingroup$ I gave it already, but its too trivial. Sets is just too "lax" $\endgroup$ – sure Oct 24 '16 at 21:55
  • $\begingroup$ The category of fields has no products. $\endgroup$ – Berci Oct 24 '16 at 22:53
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    $\begingroup$ Sure it does have some. Feel free to write the product of $\mathbb{Z}_2$ with itself and find that it's given by ... $\mathbb{Z}_2$. $\endgroup$ – sure Oct 24 '16 at 23:06
  • $\begingroup$ Cool example, re: fields. $\endgroup$ – Kevin Carlson Oct 24 '16 at 23:17

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