Categories and Surreals: Disordered Thoughts

I’ve been going through André Joyal’s category-theoretic construction of the surreal numbers, futzing around to see how restricting the construction to yield only the real numbers affects things. (See my earlier remarks on this and/or Mark Chu-Carroll’s discussion for background information.) If I were an actual mathematician, I’d probably be done by now. Instead, I have a headful of half-baked notions, which I’d like to spill out into the Intertubes (mixing my metaphors as necessary).

Why would a physics person even care about the surreal numbers? Well, ultimately, my friends and I were going through Baez and Dolan’s “From Finite Sets to Feynman Diagrams” (2000), which touches upon the issue that subtraction can be a real pain to interpret categorically. If you interpret the natural numbers as the decategorification of FinSet, then addition is easy: you’re just talking about a coproduct. But subtraction and negation — oh, la!

If [tex]0[/tex] denotes the initial object which serves as our “additive identity,”

[tex]A + 0 \cong A,[/tex]

then if two objects [tex]A[/tex] and [tex]B[/tex] satisfy the relation

[tex]A + B \cong 0,[/tex]

we can prove that [tex]A[/tex] and [tex]B[/tex] are in fact both initial objects themselves. This is easy to see in FinSet, where the coproduct of [tex]A[/tex] and [tex]B[/tex] is just the elements from [tex]A[/tex] pooled together with the elements of [tex]B[/tex]. The only way to have that “pooling together,” or disjoint union, be the empty set is for both [tex]A[/tex] and [tex]B[/tex] to be empty sets. We can, though, deduce the same thing in general, purely arrow-oriented terms.

To wit: for any object [tex]X[/tex], there is one and only one morphism from [tex]A + B[/tex] to [tex]X[/tex], by the definition of “initial object.” However, this morphism is the same as a morphism from [tex]B[/tex] to [tex]X[/tex] together with an arrow from [tex]A[/tex] to [tex]X[/tex], so both [tex]A[/tex] and [tex]B[/tex] are initial.

[tex]A + B \cong 0 \Longleftrightarrow A \cong 0 \hbox{ and } B \cong 0.[/tex]

So, while natural numbers (Armstrong, Chu-Carroll) have a natural origin in the decategorification of FinSet, integers appear to be outside our grasp. Baez and Dolan write,

This does not mean that categorifying the integers is doomed to fail; it just means that we must take a more generous attitude towards what counts as success. Schanuel [26] has proposed one approach; Joyal [17] and the authors [6] have advocated another, based on ideas from homotopy theory.

The upshot of this approach appears to be that “to properly categorify subtraction, we need to categorify not just once but infinitely many times.” I had a vague notion that the difficulty of categorifying the integers means that the interpretation of numbers as “counting things in a box” is not the natural way to approach negative quantities. It’s easier to get at negative numbers by thinking about a number line — just move in the opposite direction. Now, the “sign expansion” form of a surreal number is just a sequence of up and down arrows indicating the steps necessary to reach a position on a number line.

With sign expansions, if you can build fractions you can build negative numbers, because fractions are obtained in dyadic style: you “overshoot” and then reverse your steps.

To go from this view of the surreals to the combinatorial game version, replace each up-arrow with a black stick and each down-arrow with a red stick, and then play the take-away game called Hackenbush. Each Hackenbush configuration must have the property that you can get from any node to the “ground” by following a chain of edges. The game has two players, Left and Right; Left can delete any black edge while Right can remove any red edge. The players alternate, removing one edge each move, and each time any edges not connected to the ground must be removed. The goal is to get the other guy into a position where they cannot move. A surreal sign-expansion is just a long, skinny Hackenbush chain.

Joyal defines a category which we can call Game. The objects of Game are two-player games, or more precisely, game positions defined by their permissible moves. A game [tex]G[/tex] is an ordered pair of sets [tex]G_L[/tex] and [tex]G_R[/tex],

[tex]G = (G_L, G_R),[/tex]

where [tex]G_L[/tex] are the positions which the Left player could take if they opened the game, and likewise [tex]G_R[/tex] are the positions which Right could achieve if they had the move. The “empty game”

[tex]0 = (\{\}, \{\})[/tex]

has no opening positions. Whoever is first to move, loses! We form the opposite or negation of a game by swapping the players’ roles:

[tex]-G = (-G_R, -G_L),[/tex]

an operation we must understand recursively. The sum of two games [tex]H[/tex] and [tex]G[/tex] is defined by placing both [tex]H[/tex] and [tex]G[/tex] on the table. The first player to move chooses in which of the two sub-games to play and then makes a legal move; the second player then does the same. (There’s a formal, set-theoretic description of this process which is described in more detail here.)

The objects of the category Game are two-player games, and the morphisms are strategies. Joyal says, “A winning strategy is a rule that dictates (to the player who uses it during a play) a choice of position with which to respond to the position chosen by the opponent. A player who applies a winning strategy will be the winner, because his opponent could not be the last to play.” A morphism [tex]f: G \rightarrow H[/tex] is a winning strategy for Left in the game [tex]H – G[/tex].

Joyal goes into some details on identity strategies and the composition of strategies which I’ll elide for now. The upshot is that games [tex]H[/tex] and [tex]G[/tex] can be called equivalent if there exist [tex]f: H \rightarrow G[/tex] and [tex]g: G \rightarrow H[/tex]. Furthermore, the existence of a winning strategy from [tex]H[/tex] to [tex]G[/tex] can be taken to define an ordering upon their equivalence classes. If [tex]f: H \rightarrow G[/tex] exists, then [tex][H] \leq [G][/tex]. Equivalence classes of games then give an abelian group which obeys the following rules:

[tex][H] + [G] = [H + G][/tex]
[tex][H] + [0] = [H][/tex]
[tex][H] + [-H] = [0][/tex]

This abelian group is the group of surreal numbers.

Here, we see that negative integers have a perfectly reasonable interpretation: they are equivalence classes which contain games in which Left has no options to play.

Now, as Conway tells us, you can construct the reals by imposing an extra condition on the surreals. Basically, you want the numbers you construct to be “bounded” or “confined” by the integers, and to exclude the infinitesimals which appear along with the infinities on [tex]\aleph[/tex] day. A game x is defined to be real if [tex]-n < x < n[/tex] for some integer [tex]n[/tex], and if [tex]x[/tex] falls in the same equivalence class as the game [tex](\{x - 1, x - \frac{1}{2}, x - \frac{1}{4}, \ldots\}, \{x + 1, x + \frac{1}{2}, x + \frac{1}{4}, \ldots\}).[/tex] At least the first of these requirements has, I believe, a ready interpretation in terms of a commuting diagram. You'll note that we've dodged the isomorphism problem, because the object [tex]0[/tex] is no longer an initial object: arrows only exist from [tex]0[/tex] to non-negative numbers. Oops. Looks like it's time to get back to work — more will follow on this, particularly if any real math people have anything to say. At some point, I might even be able to explain either of the two disjoint reasons we found ourselves interested in the Baez and Dolan paper in the first place!