Categorical foundation for numbers
I'm exploring a specific foundation for number systems, using category theory, in order to achieve a particular property. But I'm a neophyte in matters categorical, and so I'm concerned doing so properly.
The old Russell-Whitehead way of defining the cardinals was as equivalence classes of sets having the same cardinality. One could describe a particular natural number, then, as the property of being in bijective correspondance with such-and-such a finite set. From these, one could construct negative numbers, rationals, etc. by set-theoretic constructions; but the cardinals would not then be a subset of these constructions. My approach is meant to be similar, but to define as large a class of numbers as possible in terms of properties similar to the Russell-Whitehead definitions.
I'm trying to do this using coproducts, using the concept of value attribution. Let A and B be objects in a category with a coproduct functor +. We do this in terms of "value attribution", that is, attributing values to objects or pairs of objects in terms of other objects. We would say e.g. that {1,2} is worth 2/3 of the set {4.5.6} in Sets; or that the difference between R^2 and R^3 is minus one copy of R in Vec_R; and so forth.
This allows us to define various numbers up to the real numbers, and also to define addition on them. This works out nicely because all of these can be described just using natural numbers. I could similarly define multiplication by category products (and then remark that e.g. for the category of finite sets, multiplication is equivalent to repeated addition); this might give me the usual arithmetic on numbers up to the reals. (Exponentiation may not be quite as nice, but I'm not worrying about that qwuite yet.)
My problem: I'm stuck on how to extend my approach to the complex numbers. Anyone have an idea?
In my next post, I'll describe how I'm going how about defining the various number systems.
Niel Apr 9 Natural numbers.
-- A is worth zero B if A is isomorphic to an initial object. (We say A ~ 0B.)
-- A is worth one B if A is isomorphic to B. (We say A ~ B.)
-- A is worth two B if A + A is isomorphic to B. (We say 2A ~ B.)
etc.
Integers.
-- The difference between B and C is worth zero A if B ~ C.
-- The difference between B and C is worth two A if B ~ C + 2A.
-- The difference between B and C is worth minus two A if B + 2A ~ C.
etc. Note that the difference between B and an initial object in a category in terms of an object A is the same as the value of B in terms of A: thus, we have an equivalence of relationships for the fixed value of C being an initial object.
Rationals.
-- We say that A is worth n/d of B, for n,d > 0, if dA ~ nB.
-- We say that the difference between A and C is worth n/d of B:
. . . for n positive, if dA ~ dB + nC;
. . . for n negative, if dA + nC ~ dB;
. . . for n=0, if dA ~ dB.
etc. Note that for d=1, these definitions reduce to the definitions for natural numbers and integers given above; and that similar fractions yield equivalent propositions.
Ordering of the above number systems.
All of the above have meaningful instantiations in, e.g. the category Sets. We can also extend the above notions in order to establish a total ordering on the numbers, via monomorphisms. This will prove useful in order to clearly define the reals via coproducts.
-- Generally, we will say of objects that A <= B if there is a monomorphism from A into B.
-- For natural numbers n and m, we have n <= m if nA <= mA. This extends directly to integers, where n <= m for n,m positive if B + nA <= B + mA. For negative integers, we only fiddle with which side of the mapping the coproduct is, depending on the sign of the integer.
-- For positive rational numbers, a/b <= c/d if (da)A <= (bc)A; differences are treated similarly to the case for integers.
Niel Apr 9 Real numbers.
-- Let r be a positive real given by r = r0 + r1/d + r2/d^2 + etc; where d is a positive integer, r0 a natural number, and each subsequent r_k is in {0, . . . , d-1}. Then B is worth r of A if:
. . . r0 A <= B <= r0 A + A
. . . (d)(r0) A + r1 A <= 2 B <= (d)(r0) A + r1 A + A
. . . (d^2)(r0) A + (d)(r1) A + r2 A <= (d^2) B <= (d^2)(r0) A + (d)(r1) A + r2 A + A
etc.
-- For reals in general, we define a difference between B and C in terms of A using a similar device as above for possibly-negative integers and possibly-negative rationals, only here taking a sequence of statements about monomorphisms rather than single statements about isomorphisms.
Note that for real numbers which happen to be rational, this again can possibly be made equivalent to the previous definitions. Certainly, some expressions of real numbers will be equivalent to some rational number: in particular, terminating d-adic expansions for any positive d. But unlike the previous definitions of numbers, showing that a proposiotion about real values for some repeating expansion is equivalent to a proposition about a rational value may be non-trivial. Hopefully, the usual analysis could be carried off just as well with the present definitions, but I haven't ventured that far.
I'm quite stuck on how I could express complex numbers in this way. So far, I've gotten away only with propositions about coproducts, and products of natural numbers. To be able to express an idea such as i^2 + 1 = 0, I need to introduce something else to cope with "adding a multiple of i" and "multiplying i by itself". (Describing how to multiply two negative numbers using these definitions will be difficult enough.)
Any ideas?
Guilherme Apr 30 Didn't read through all you wrote, but shouldn't you be using topoi for this?
Sam Apr 30 I'm not sure I understand what exactly you want to do, so what I am going to say may be completely off-topic, but if you start with a topos T, you can define a natural number object in T as an object N with a global section 0: 1 -> N and a morphism s:N -> N such that for any other object A with a global section
x:1 -> A and morphism q:A -> A, there exist a unique morphism u:N -> A such that u o 0 = x and
u o s = q o u. (that defines 0, successor and recursion).
If such an object exist, you can then build Z and R in whatever usual way you like, using the internal (Mitchell-Benabou) language of the topos.
Niel May 1 Thanks for the responses.
My impression was that topoi are primarily important for describing logics, although it can hardly be surprising that there follows a formulation of the natural numbers for each one. Perhaps that would be an important starting point.
I am completely ignorant of category theoretic constructions of number systems, except for pedestrian concepts (such as the existence of a functor mapping e.g. integral domains to their quotient fields), as well as a vague idea that it has probably been done by someone before. I will read up a bit on topoi.
* * *
Just to repeat what I'm interested in (a bit more concisely, this time):
My objective is to express things such as rationals, reals, complex numbers, quaternions, etc. by use of propositions, each of which generalize the one that comes before. The idea is that not only should there be an image of the natural numbers in the rationals (for example), but the definition of a rational number should properly subsume the definition of a natural number as a mathematical object. The typical set-theoretic construction fails to do this.
For example, a given natural number N would be described as a relationship R that may hold between some pairs of objects in some category: R would probably be expressed in terms of monomorphisms and coproducts. The same number N can be described as an integer (also as a relationship, R', between two objects, in terms of monomorphisms and coproducts), a rational number (again as a relationship, R''), a real number (as a relationship R'''), and so on. The objective is for the relational propositions R, R', R'', R''', etc. to be equivalent relational propositions.
We can define the numbers as referring to these relations; in which case, e.g. the "natural number 5" will be identical to "the rational number 5", as the relations for these two objects will be equivalent