### Can 1+1 Have More Than Two Points?

#### Posted by John Baez

I feel I’ve asked this before… but now I *really* want to know. Christian Williams and I are working on cartesian closed categories, and this is a gaping hole in my knowledge.

**Question 1.** Is there a cartesian closed category with finite coproducts such that there exist more than two morphisms from $1$ to $1 + 1$?

Cartesian closed categories with finite coproducts are a nice context for ‘categorified arithmetic’, since they have $0$, $1$, addition, multiplication and exponentiation. The example we all know is the category of finite sets. But *every* cartesian closed category with finite coproducts obeys what Tarski called the ‘high school algebra’ axioms:

$x + y \cong y + x$

$(x + y) + z \cong x + (y + z)$

$x \times 1 \cong x$

$x \times y \cong y \times x$

$(x \times y) \times z \cong x \times (y \times z)$

$x \times (y + z) \cong x \times y + x \times z$

$1^x \cong 1$

$x^1 \cong x$

$x^{(y + z)} \cong x y \times x z$

$(x \times y)^z \cong x^z \times y^z$

$(x^y)^z \cong x^{(y \times z)}$

together with some axioms involving $0$ which for some reason Tarski omitted: perhaps he was scared to admit that in this game we want $0^0 = 1$.

So, one way to think about my question is: *how weird can such a category be?*

For all I know, the answer to Question 1 could be “no” but the answer to this one might still be “yes”:

**Question 2.** Let $C$ be a cartesian closed category with finite coproducts. Let $N$ be the full subcategory on the objects that are finite coproducts of the terminal object. Can $N$ be inequivalent to the category of finite sets?

In fact I’m so clueless that for all I know the answer to Question 1 could be “no” but the answer to *this* one might still be “yes”:

**Question 3.** Is there a cartesian closed category with finite coproducts such that there exist more than three morphisms from $1$ to $1 + 1 + 1$?

Or similarly for other numbers.

Or how about this?

**Question 4.** Is there a category with finite coproducts and a terminal object such that there exist more than two morphisms from $1$ to $1 + 1$?

Just to stick my neck out, I’ll bet that the answer to this last one, at least, is “yes”.

## Re: Can 1+1 Have More Than Two Points?

In the category Set^2 of pairs of sets, the terminal object is the pair (1,1), and “1+1” is the object (2,2). But hom((1,1),(2,2)) = 4.