I was unable to understand John Baez' golden object problem, nor his description of the solutions. He refuses to tell us what 'nice' means, but let me at least propose that to be 'tolerable' a solution must be an object in a category, and John doesn't tell us what category is involved in either of the solutions; at least I couldn't find a specification of the objects, nor the maps, so I found the descriptions 'intolerable', in the technical sense defined above. He is very generous, allowing one to use a category with both plus and times as extra monoidal structures. (Does anyone know an example of interest in which the plus is not coproduct?) This freedom is unnecessary; a little algebra plus Robbie Gates' theorem provides a solution G to G^2additional equations, in an extensive category (with coproduct as plus, cartesian product as times). Briefly, here it is. A primitive fifth root of unity z is a root of the polynomial 1+X+X^2+X^3+X^4, hence satisfies 1+z+z^2+z^3+z^4+zwhich is of the 'fixed point' form p(z)0. Gates' theorem then says that the free distributive category containing an object Z and an isomorphism from p(Z) to Z is extensive, and its Burnside rig B (of isomorphism classes of objects) is, as one would hope, N[X]/(p(X)equations. Since the degree of p is greater than 1, an easy general theorem tells us (from the joint injectivity of the Euler and dimension homomorphisms) that two polynomials agree at the object Z if and only if either they are the same polynomial or both are non-constant and they agree at the number z.Now the 'algebra': the golden number is 1+z+z^4. So G satisfies G^2unexpected equations, because the relation X^2polynomial in N[X] to a linear polynomial, and these reduced forms have distinct Euler characteristics, i.e. differ at z. Thus the homomorphism from N[X]/(X^2I wanted. Since in the category of sets, any nasty old infinite set satisfies the golden equation and many others, I have taken the liberty of interpreting 'nice' to mean at least 'satisfying no unexpected equations'. One could ask for more; the construction above has produced a distributive, but not extensive, category whose Burnside rig is N[X]/(X^2(If it were extensive, it would be closed under taking summands, but every object in the larger category is a summand of G.) I don't know whether there is an extensive category with N[X]/(X^2Burnside rig; perhaps one or both of the examples John mentioned would do, if I knew what they were. While I'm airing my confusions, can anyone tell me what 'categorification' means? I don't know any such process; the simplest exanple, 'categorifying' natural numbers to get finite sets, seems to me rather 'remembering the finite sets and maps which gave rise to natural numbers by the abstraction of passing to isomorphism classes'. Finally, a note to John: While you're trying to give your audience some feeling for the virtues of n-categories, couldn't you give them a little help with ny being a little more precise about objects and maps? Greetings to all, and thanks for your patience while I got this stuff off my chest, Steve Schanuel
Categorification is a bit like quantization: it isn't a construction so much as a desideratum for a relationship between one thing and another (in the case of categorification an (n+1)-categorical structure and an n-categorical structure; in the case of quantization a quantum mechanical system and a classical mechanical system). Categorification wants to find a higher-dimensional categorical structure corresponding to a lower-dimensional one by weakening equations to natural isomorphisms and imposing new, sensible, coherence conditions. In general, for the original purpose for which it was proposed--constructions of TQFT's and models of quantum gravity--one wants the highest categorical level to have a linear structure (hence Baez wanting tensor product and a sum it distributes over, rather than cartesian product and coproduct). Specific lower-dimensional categories with structure are 'categorified' by finding a higher-dimensional category with the new structure which 'lies over' the lower dimensional one in the way an additive monoidal category lies over its Grothendieck rig. For instance any (k-linear) monoidal category with monoid of isomorphism classes M is a categorification of M, and more generally (k-linear) monoidal categories are a categorification of monoids. A simple example shows why it is not a construction: commutative monoids (as rather special categories with one object) admit two different categorifications: symmetric monoidal categories and braided monoidal categories (each regarded as a kind of bicategory with one object). There is a good argument for regarding braided monoidal categories as the 'correct' categorification: the Eckmann-Hilton theorem ('a group in GROUPS is an abelian group' or, really as the proof shows, 'a monoid in MONOIDS is a commutative monoid') 'categorifies' to: A monoidal category in MONCAT is a braided monoidal category.
Steve Schanuel wrote:
a category with both plus and times as extra monoidal structures. (Does anyone know an example of interest in which the plus is not coproduct?)
Here are two examples that I've come across previously of rig categories in which the plus is not coproduct: (i) the category of finite sets and bijections, with + and x inherited from the category of sets; (ii) discrete rig categories, which are of course the same thing as rigs.
This freedom is unnecessary; a little algebra plus Robbie Gates' theorem provides a solution G to G^2=G+1 which satisfies no additional equations, in an extensive category (with coproduct as plus, cartesian product as times).
If you *do* allow yourself the freedom to use any rig category then an even simpler solution exists, also satisfying no additional equations: just take the rig freely generated by an element G satisfying G^2 = G + 1 and regard it as a discrete rig category.
Since in the category of sets, any nasty old infinite set satisfies the golden equation and many others, I have taken the liberty of interpreting 'nice' to mean at least 'satisfying no unexpected equations'.
I'd interpret "nice" differently. (Apart from anything else, the trivial example in my previous paragraph would otherwise make the golden object problem uninteresting.) "Nice" as I understand it is not a precise term - at least, I don't know how to make it precise. Maybe I can explain my interpretation by analogy with the equation T = 1 + T^2. A nice solution T would be the set of finite, binary, planar trees together with the usual isomorphism T -~-> 1 + T^2; a nasty solution would be a random infinite set T with a random isomorphism to 1 + T^2. (Both these solutions are in the rig category Set with its standard + and x.) I regard the first solution as nice because I can see some combinatorial content to it (and maybe, at the back of my mind, because it has a constructive feel), and the second as nasty because I can't. I'm not certain what I think of the solution given by the set of not-necessarily-finite binary planar trees (nice?), or by the set of binary planar trees of cardinality at most aleph_5 (probably nasty). Maybe the finding of a "nice" solution is similar in spirit to the finding of a "concrete interpretation" of a combinatorial identity. As an extremely simple example, consider the identity saying that each entry in Pascal's triangle is the sum of the two above it, (n+1 choose r) = (n choose r-1) + (n choose r). This is a doddle to prove, but all the same you'd be missing something if you didn't know the standard "concrete interpretation": choosing r objects out of n+1 objects amounts to EITHER choosing the first one and then choosing r-1 of the remaining n OR ... . Even if the challenge of finding a "nice solution" or "concrete interpretation" isn't made precise, I think there is a shared sense of what would count as an answer, and finding an answer is in general not straightforward. Best wishes, Tom
On 7 Mar 2004, at 19:43, Tom Leinster wrote:
I'd interpret "nice" differently. (Apart from anything else, the trivial example in my previous paragraph would otherwise make the golden object problem uninteresting.) "Nice" as I understand it is not a precise term - at least, I don't know how to make it precise. Maybe I can explain my interpretation by analogy with the equation T = 1 + T^2. A nice solution T would be the set of finite, binary, planar trees together with the usual isomorphism T -~-> 1 + T^2; a nasty solution would be a random infinite set T with a random isomorphism to 1 + T^2. (Both these solutions are in the rig category Set with its standard + and x.) I regard the first solution as nice because I can see some combinatorial content to it (and maybe, at the back of my mind, because it has a constructive feel), and the second as nasty because I can't. I'm not certain what I think of the solution given by the set of not-necessarily-finite binary planar trees (nice?), or by the set of binary planar trees of cardinality at most aleph_5 (probably nasty).
From a computer science point of view, both the first "nice" solution (finite binary trees) and the second "nice?" solution (possibly non-finite binary trees) are canonical, in the sense that the first is the carrier of the initial algebra for the endofunctor 1+X^2 on Set, while the second is the carrier of its final coalgebra. All the best, Pawel.
I would like some references to model structures on the category of toposes/locales (thinking of them as generalized spaces), perhaps even the category of internal toposes of a given (boolean?) topos. What I know of is about model structures on the category of simplicial objects in a topos. Along the same lines, does it make sense to ask about ``internally simplicial objects'' in a topos with an NNO (i.e., do such toposes have an internal category that looks like the category of internally ``finite sets with linear order and order preserving maps'')? Nath Rao
participants (5)
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David Yetter -
Pawel Sobocinski -
Stephen Schanuel -
Tom Leinster -
Vidhyanath Rao