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March 10, 2007

Snowglobe Models

Posted by John Baez

Last week in my course on Classical vs Quantum Computation, James Dolan said a bit about funny nonstandard models of typed λ-theories — like the typed λ-theory for high school calculus. He calls these ‘snowglobe models’, because they live ‘inside’ the usual world of sets — but in their own self-contained miniature universe, like a snowglobe:

I’d like to talk about these a bit before I forget!

Lately in my course I’ve been focusing on ‘classical’ computation, hence closed monoidal categories where the tensor product is the cartesian product. However, the idea of a snowglobe model is more general, so let me talk about it more generally.

Suppose TT is a closed symmetric monoidal category, as defined here. Very roughly, this a category where any two objects xx and yy have a tensor product xyTx \otimes y \in T and also an ‘object of maps’ from xx to yy, denoted hom(x,y)Thom(x,y) \in T. The latter should not be confused with the usual set of morphisms Hom(x,y)Hom(x,y)! It’s often called the ‘internal hom’, since it lives inside TT.

We can think of TT as a theory. A model of this theory in some other closed monoidal category CC is a functor

F:TCF: T \to C

preserving all the relevant structure. Namely:

  • The functor FF must be symmetric monoidal, as defined here. Very roughly, this means FF preserves tensor products.
  • The functor FF must be closed, meaning that for any pair of objects x,yTx,y \in T, the god-given morphism F(hom(x,y))hom(F(x),F(y))F(hom(x,y)) \to hom(F(x),F(y)) is an isomorphism.

Now, suppose we have a functor

F:TCF: T \to C

which is symmetric monoidal but not closed. We can think of this as a defective model — but it gives a perfectly fine ‘snowglobe model’ in some other category!

Here’s how.

First, let’s form a kind of ‘hybrid’ of TT and CC, having objects of TT as its objects, but the objects F(hom(x,y))CF(hom(x,y)) \in C as its ‘hom-objects’.

This hybrid — let’s call it C˜\tilde{C} — does not begin life as a category, because instead of a set of morphisms from xx to yy, it just has an object in CC, namely F(hom(x,y))F(hom(x,y)). It’s what people call a category enriched over CC, as defined here.

The terminology is a little confusing to beginners: it’s important to realize that a ‘category enriched over CC’ is not an actual category. But, just as Pinocchio, who began life as a puppet, eventually became a real boy, C˜\tilde{C} can with sufficient effort become a real category!

To do it, for each pair of objects x,yC˜x,y \in \tilde{C} we define a morphism f:xyf: x \to y to be a morphism in CC from II (the unit object of CC) to F(hom(x,y))F(hom(x,y)). This sometimes called a point of the hom-object F(hom(x,y))F(hom(x,y)). These points form an actual set, and with some work C˜\tilde{C} becomes an actual category.

In fact, now that C˜\tilde C is a category, it even becomes a closed symmetric monoidal category in an obvious way!

I know what some of you are thinking:

I felt like that too, the first time Jim showed me this sort of construction a few years ago. But really, once you grok the definitions, the construction is not hard! (Unless, of course, I’m making some mistake.) This is part of the fun of category theory — it gives your brain a real workout, but in the end it all seems perfectly obvious.

Anyway: now our original functor

F:TCF: T \to C

factors into a first stage

F 1:TC˜F_1 : T \to \tilde{C}

and a second stage

F 2:C˜C.F_2 : \tilde{C} \to C.

And, F 1F_1 is a closed symmetric monoidal functor! This is more or less by construction. F 1F_1 preserves the tensor product because FF did. But, F 1F_1 also preserves the internal hom, because the internal hom in C˜\tilde C is borrowed from CC.

So,

F 1:TC˜F_1 : T \to \tilde{C}

is a model of our theory TT: a snowglobe model. It’s called a snowglobe model because while it comes from a defective model that lives in CC, it really lives inside the miniature universe C˜\tilde{C}.

To really appreciate this construction, you need to look at some examples, like the theory of high-school calculus. But, I’m a bit too tired for that right now. For now, I just wanted to get this down in writing.

I should add for the experts that this factorization of FF into two stages is related to the theory of Postnikov towers, as explained in Section 3.1 here.


I’ve corrected this post thanks to comments by Robin Houston, Todd Trimble and James Dolan. Some of their comments won’t make much sense without knowing what my mistakes had been, so here they are:

  1. I had written hom(F(x),F(y))hom(F(x),F(y)) in lots of places where I should have written F(hom(x,y))F(hom(x,y)).
  2. I had defined a symmetric monoidal functor F:TCF: T \to C to be ‘closed’ if F(hom(x,y))hom(F(x),F(y)).F(hom(x,y)) \cong hom(F(x),F(y)). In fact, there’s a god-given morphism F(hom(x,y))hom(F(x),F(y))F(hom(x,y)) \to hom(F(x),F(y)) and FF is closed if this is invertible.

    To get this god-given morphism, start with the evaluation ev:hom(x,y)xyev: hom(x,y) \otimes x \to y in TT. Then apply FF: F(ev):F(hom(x,y)x)F(y)F(ev): F(hom(x,y) \otimes x) \to F(y) Then use the fact that FF preserves tensor products up to a specified isomorphism to get the morphism F(hom(x,y))hom(F(x),F(y)).F(hom(x,y)) \to hom(F(x),F(y)).

Posted at March 10, 2007 12:05 AM UTC

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13 Comments & 2 Trackbacks

Re: Snowglobe Models

This looks very interesting, but I must be missing something because it isn’t quite making sense.

The way you’ve explained it, the intermediate stage where C˜\tilde{C} is viewed as a CC-enriched category seems to be redundant. A morphism xyx\to y in C˜\tilde{C} is a point of hom(F(x),F(y))\hom(F(x),F(y)), or in other words an arrow F(x)F(y)F(x)\to F(y) in CC!

So this factorisation is the usual factorisation of a functor into a bijection-on-objects followed by a full-and-faithful functor. But then I don’t see why C˜\tilde{C} is closed, unless the functor FF was closed in the first place, which rather defeats the object!

Any elucidation would be much appreciated.

Also, a little nit or query. You wrote

  • The functor FF must be closed, meaning precisely that for any pair of objects x,yTx,y\in T we have F(hom(x,y))hom(F(x),F(y))F(\hom(x,y))\cong \hom(F(x),F(y)).

I think the definition of closed functor is actually more subtle than that: the monoidality of the functor automatically induces a natural transformation F(hom(A,B))hom(FA,FB)F(\hom(A,B))\to\hom(F A,F B), and a (symmetric monoidal) closed functor is one for which this natural transformation is invertible. I suppose it’s possible that your condition implies mine: is that true?

(On a complete tangent, I wonder whether you could be persuaded that the linear logic-inspired notation xyx\multimap y is a much more pleasant way to denote the internal hom than hom(x,y)\hom(x,y), which is easily confused with Hom(x,y)(x,y). Another advantage of this notation is that it is readily adapted to the non-symmetric case, where there are two internal homs, by using a left-pointing variant – with the arguments in reverse order – for one of them. (This is unfortunately harder to do on the web, because the left-pointing version doesn’t have a Unicode codepoint. It’s easy in Latex, by rotating or reflecting the right-pointing one.))

Posted by: Robin on March 10, 2007 5:12 PM | Permalink | Reply to this

Re: Snowglobe Models

Robin, I followed what you said and I think you are right. I mentioned your remark to Jim Dolan, and he thinks the way it’s supposed to go is that the enrichment is given by C˜(x,y)=F(y x)\tilde{C}(x, y) = F(y^x) where xx and yy are objects of C˜\tilde{C} (i.e., objects of TT). So, a morphism f:xyf: x \to y in C˜\tilde{C} is now identified with a morphism [f]:IF(y x)[f]: I \to F(y^x) in CC.

We may as well sketch out some details (any mistakes should be attributed to me and not Jim). Composition in C˜\tilde{C} is given by

III[f][g]F(y x)F(z y)F(y xz y)F(comp)F(z x)I \cong I \otimes I \stackrel{[f] \otimes [g]}{\to} F(y^x) \otimes F(z^y) \cong F(y^x \otimes z^y) \stackrel{F(comp)}{\to} F(z^x)

where we have used the monoidal structure on FF. Identities are similarly easily constructed and use the unit datum for monoidal structure on FF. The axioms for monoidal structure ensure that associativity and unit axioms hold in C˜\tilde{C}.

There is a functor F˜:TC˜\tilde{F}: T \to \tilde{C} which is the identity on objects, and sends an arrow f:xyf: x \to y in TT to

IF(I)F(f )F(y x)I \cong F(I) \stackrel{F(f^{\prime})}{\to} F(y^x)

where f :Iy xf^{\prime}: I \to y^x is mated with ff. Functoriality again uses monoidal structure on FF.

Define tensors and exponentials of objects in C˜\tilde{C} as the corresponding tensors and exponentials in TT. This does indeed give a closed monoidal structure according to the hom-set isomorphisms

C˜(xy,z)C(I,F(z xy))C(I,F((z y) x))C˜(x,z y)\tilde{C}(x \otimes y, z) \cong C(I, F(z^{x \otimes y})) \cong C(I, F((z^y)^x)) \cong \tilde{C}(x, z^y)

and at this point I’m feeling pretty convinced that the darn thing works.

Also, a little nit or query.

Your nit is right of course; I’m guessing John was just writing in somewhat abbreviated style.

I wonder whether you could be persuaded that the linear logic-inspired notation x⊸y is a much more pleasant way to denote the internal hom than hom(x,y)

I agree with you that the hom-notation is overloaded here, and something else should be used. I’ve used the lollipop on occasion myself, also \Rightarrow, but most often (as here) I use exponential notation (esp. when dealing with closed symmetric monoidal categories). In cases of biclosed structure I’ve also used a division symbol (forward slash and backward slash).

For what it’s worth, I’m not crazy about lollipop, but I guess I sometimes find it handy in cases where I want to distinguish linear implication from cartesian implication (which sometimes happens).

Posted by: Todd Trimble on March 10, 2007 11:54 PM | Permalink | Reply to this

Re: Snowglobe Models

After posting the above, it occurred to me that this snowglobe construction is an instance of a change-of-base construction which I think would be considered “well-known” in Australia, among other places.

For example, if VV and WW are monoidal categories and F:VWF: V \to W is a lax monoidal functor, then there is a change-of-base

F˜:VCatWCat\tilde{F}: V–Cat \to W–Cat

from VV-enriched categories to WW-enriched categories. In particular, if we take V=TV = T to be a closed monoidal category, then TT is enriched in itself by the internal hom, and a lax monoidal functor F:TCF: T \to C gives an enrichment of TT in CC: this is precisely the snowglobe enrichment.

A standard example of change-of-base is the lax monoidal functor hom(I,):VSethom(I, -): V \to Set. This gives the underlying category of a VV-enriched category; this example was implicit in my earlier post.

All this can be souped up: if TT is symmetric monoidal closed, then it can be regarded as symmetric monoidal closed as a TT-category (again via the internal hom enrichment). Then a lax symmetric monoidal functor F:TCF: T \to C transfers symmetric monoidal closed TT-categories to symmetric monoidal closed CC-categories, and this is also part of what is going on.

Even though the snowglobe could in this way be regarded as “well-known”, this doesn’t mean it has invariably been well-mined in particular cases!

Posted by: Todd Trimble on March 11, 2007 2:21 AM | Permalink | Reply to this

Re: Snowglobe Models

Thanks a lot, Todd. It all makes sense now. I still find the construction remarkable, even though it’s ‘just’ a special case of something I already knew.

(I think the implications of a lot of “well-known in Australia” theory are going to take a long time to be properly appreciated. To take one example, just think how much time elapsed between Street proving the bicategorical Yoneda lemma (1980), and Gordon and Power realising that it implies every bicategory is biequivalent to a 2-category (1991).)

Have you considered writing a book called Well-known in Australia: 23rd-century mathematics today! or something along those lines? I’d love to read it. :-)

Posted by: Robin on March 11, 2007 12:21 PM | Permalink | Reply to this

Re: Snowglobe Models

Thanks for catching those two mistakes, Robin. I’ve fixed them on the blog entry.

There are certainly advantages to using some funny notation to more emphatically distinguish between internal and external homs; the main reason I keep using hom(x,y)hom(x,y) and Hom(x,y)Hom(x,y) is that I started off using this notation in my course, so for consistency I want to keep using it throughout the course — including little tendrils of the course, like this blog entry.

I’ve sometimes seen [X,Y][X,Y] used for internal hom, and of course we can use Y XY^X in the cartesian case. I might use xyx \multimap y if I were trying to pretend I understood linear logic, but somehow I can’t carry it off with a straight face yet.

Another problem with xyx \multimap y is it seems to terrify people who don’t know any linear logic. I think folks are a bit more likely to have some idea of what’s meant by ‘hom’, even if it’s not quite the right idea. I try hard to make things seem easy.

Posted by: John Baez on March 12, 2007 6:33 PM | Permalink | Reply to this

Re: Snowglobe Models

This is a transcription of an email from James Dolan which gives some intuition and motivation for snowglobe models. (I’m having technical trouble pasting it in in a smooth way, so I’m copying it out – it won’t look much like one of his emails because there will be some capital letters involved.)

“It’s true that the “snowglobe” idea is related to base-change in the way that you described, but the point was supposed to be to answer questions like:

• What is the underlying products theory T T^{\prime} of a products-exponential theory TT?

• What is the underlying operad T T^{\prime} of a tensor-products-and-duals theory TT?

And in both cases I think that a nice answer (interpreted in an appropriate sense) is given by “a model of T T^{\prime} is essentially a snowglobe model of TT”. Also as you indicated, I think that there are special aspects of this particular case of base-change that offhand I don’t see showing up in the general case, such as the idea of getting a factorization system from it.

“Anyway, yes, that’s one of the ways that I described the snow-globe idea when I first thought about it: use the composite of F:XYF: X \to Y and hom(1,):YSethom(1, -): Y \to Set as a base-change to convert XX from being self-enriched to being unenriched; this alternate de-enrichment of XX then serves as the “image” in a certain “image-factorization” of FF.

“I guess that the idea is supposed to be something like this: start with a “defective” interpretation of your theory; defective in the sense that it doesn’t preserve internal homs. Then it’s interesting that the defectiveness can be completely isolated in the right half of a functorial “image-factorization”, with the map from the domain to the image being a perfectly non-defective model:

TheorysnowglobemodelRealUniverseTheory \stackrel{snowglobe model}{\to} Real Universe

=

TheorygenuinemodelSnowglobeUniverseRealUniverseTheory \stackrel{genuine model}{\to} Snowglobe Universe \to Real Universe

“A snowglobe model is actually a genuine model, but encapsulated inside a snowglobe universe”.

“I guess another way of thinking about what’s going on here is that we’re exploiting the fact that even though preservation of internal homs isn’t required of a base-change, you can transport the structure of having internal homs across a base-change.

“Also as I mentioned earlier, I made only a very brief attempt to find out what the standard name is for “the snowglobe universe factorization of a snowglobe model”. I’d be interested to find out about other work that incorporates this idea. I always had a vague feeling that certain standard ways of factorizing geometric morphisms between toposes were related to snowglobe factorization, but I never actually succeeded in making a connection. I’m still vaguely hopeful that such a connection might exist and might for example help in understanding ideas about “proving independence results in topos-based set theory by construction of non-standard (=?= snowglobe??) models”, or something like that.” [End of email]

(Just a few notes: I think “products-exponential theory” refers to cartesian closed categories, and “tensor-products-and duals” refers to compact closed categories. The word “operad” is being used in a multi-typed sense.)

It all sounds very interesting. I’m wondering if John or James would be willing to amplify on this with some examples?

Posted by: Todd Trimble on March 12, 2007 4:29 PM | Permalink | Reply to this

Re: Snowglobe Models

Todd wrote:

After posting the above, it occurred to me that this snowglobe construction is an instance of a change-of-base construction which I think would be considered “well-known” in Australia, among other places.

Right. If I’d had more energy and hadn’t bungled it up, I would have explained that this ‘snowglobe model’ construction uses the composite of two base changes. You’ve already explained how, but let me take another crack at it, just for people too slow to follow what you said!

Let’s use the term ‘VV-category’ as short for ‘category enriched over VV’. This concept makes sense whenever VV is a symmetric monoidal category — we could get away with less, but let’s not. Very roughly, a VV-category is a gizmo like a category, but where for any pair of objects we have an object in VV of morphisms, hom(x,y)Vhom(x,y) \in V, instead of the usual set of morphisms.

If we have a symmetric monoidal functor

F:VWF: V \to W

we can use it to turn VV-categories into WW-categories: just hit hom(x,y)Vhom(x,y) \in V with FF and get something in WW. This is called ‘base change’.

We can also talk about symmetric monoidal closed VV-categories, and base change works for these too.

Now, suppose we have a symmetric monoidal closed category TT, a symmetric monoidal category CC, and a symmetric monoidal functor F:TCF: T \to C.

We can do base change twice, starting with TT, to get a snowglobe model of TT:

  1. Since TT is closed it becomes enriched over itself. So, think of TT as symmetric monoidal closed TT-category! Use the symmetric monoidal functor F:TCF: T \to C to turn this into a symmetric monoidal closed CC-category, say C˜\tilde C.
  2. Next, use the symmetric monoidal functor points: C Set x Hom(I,x) \begin{matrix} points: &C& \to &Set \\ & x &\mapsto& Hom(I,x) \end{matrix} to turn C˜\tilde C into a symmmetric monoidal closed SetSet-category. This is just a symmetric monoidal closed category! In my blog article I again called this C˜\tilde C.

The really fun part should be seeing what this does in examples. Suppose we take TT to be the lambda-theory of high school calculus (viewed as a cartesian closed category). Then James and I hope there’s a nice symmetric monoidal functor

F:TSetF : T \to Set

sending RR (the commutative ring object in our theory) to \mathbb{R} (the real numbers). This FF won’t be closed.

Note that in this case, F(hom(R,R))F(hom(R,R)) is just the polynomial functions from \mathbb{R} to \mathbb{R}. So, in our ‘snowglobe model’, all real-valued functions on \mathbb{R} are polynomials!

This is what gives the snowglobe model the feel of a ‘miniature toy universe’. It’s like a simplified universe suitable for high-school students.

Posted by: John Baez on March 12, 2007 7:25 PM | Permalink | Reply to this

Re: Snowglobe Models

Note that in this case, F(hom(R,R))F(hom(R,R)) is just the polynomial functions from ℝ to ℝ. So, in our ‘snowglobe model’, all real-valued functions on ℝ are polynomials!

Hm, that’s rather a nice hint! And of course there’s nothing special about the reals here; there should be a similar game for any commutative ring.

For example, the ring \mathbb{Z} should correspond to the product-preserving functor hom(1,):TSethom(1, -): T \to Set (the set of constants hom(1,R)hom(1, R) in the theory TT is just the set of integers). On similar grounds, for the product-preserving functor F=hom(R n,)F = hom(R^n, -), where hom(R n,R)hom(R^n, R) is the set of nn-ary operations in the theory, one would expect F(R)F(R) to be the polynomial ring [x 1,...,x n]\mathbb{Z}[x_1, ..., x_n].

So at this point we could try some Lawvere algebraic theory trickery to get at any commutative ring AA. Forgive my getting technical here (and I’m not as up on this as I’d like), but I hope it could go something like this: any finitely generated AA is finitely presented as a reflexive coequalizer

[x 1,...,x m][x 1,...,x n]A\mathbb{Z}[x_1, ..., x_m] \stackrel{\to}{\to} \mathbb{Z}[x_1, ..., x_n] \to A

and AA would therefore be the value F(R)F(R) of a reflexive coequalizer FF of set-valued product-preserving functors

hom(R m,)hom(R n,)Fhom(R^m, -) \stackrel{\to}{\to} hom(R^n, -) \to F

whereby FF is also product-preserving. Then, any commutative ring AA is a filtered colimit of finitely generated rings, and the corresponding filtered colimit of product-preserving functors is product-preserving.

So if I haven’t goofed up, for any commutative ring AA there is a product-preserving functor F:TSetF: T \to Set such that F(R)=AF(R) = A, and here F(hom(R,R))F(hom(R, R)) should be the polynomial ring A[x]A[x] just as you suggest. Something like this should be true, I think.

Posted by: Todd Trimble on March 13, 2007 5:15 AM | Permalink | Reply to this

Re: Snowglobe Models

While the argument sketched in the post above can be made tight, it’s a little on the technical and syntactic side, and I’d bet John and Jim had something much more conceptual and model-oriented in mind.

Let’s see: suppose we consider the topos [\mathbb{R}-Alg, Set] of presheaves on spectra of finitely presented commutative algebras over \mathbb{R}. This topos is of course cartesian closed. The spectrum hom([x],)hom(\mathbb{R}[x], -) is the geometric line L=L = \mathbb{R}, which of course is a commutative ring object. The set of points 1L L1 \to L^L of the internal hom is the external hom, hom(L,L)hom(L, L), where

hom(L,L)Alg([x],[x])[x],hom(L, L) \cong \mathbb{R}–Alg(\mathbb{R}[x], \mathbb{R}[x]) \cong \mathbb{R}[x],

just as John suggested.

There’s a handy little algebra [d]/(d 2)\mathbb{R}[d]/(d^2) whose spectrum DD is a walking tangent vector, and which we can use to help us define a derivative operator L LL LL^L \to L^L. I’ll just sketch how this works: one may calculate L DL×LL^D \cong L \times L in the topos. The action of the internal functor () D(-)^D on the morphism object L LL^L gives a map

L L(L D) (L D)(L×L) L×LL^L \to (L^D)^{(L^D)} \cong (L \times L)^{L \times L}

which by transposition gives

L L×L×LL×LL^L \times L \times L \to L \times L

and this is a realization of the following map in the theory of high-school calculus TT:

(ϕ,p,v)ϕ(p),D(ϕ)(p)v.(\phi, p, v) \mapsto \langle \phi(p), D(\phi)(p) \cdot v \rangle.

In other words, if we set v=1v = 1 in the second coordinate, we get a map L L×LLL^L \times L \to L which in turn gives, by transposing back, an operator D:L LL LD: L^L \to L^L which satisfies all the axioms John gave for the theory TT of high-school calculus.

Summarizing then: we have constructed a genuine (i.e., cartesian closed) model

T[Alg,Set]T \to [\mathbb{R}–Alg, Set]

and the composite

T[Alg,Set]hom(1,)SetT \to [\mathbb{R}–Alg, Set] \stackrel{hom(1, -)}{\to} Set

is a snowglobe model which however is not closed. But, the snowglobe factorization presumably gives the thing John and Jim were hoping for.

By changing the topos (e.g., to a topos of smooth spaces, as in SDG, as John suggested), we can construct other snowglobe models F:TSetF: T \to Set where again F(R)=F(R) = \mathbb{R}, but where F(R R)F(R^R) is the set of C C^{\infty} functions \mathbb{R} \to \mathbb{R}. There are lots of interesting possibilities.

Posted by: Todd Trimble on March 14, 2007 2:18 PM | Permalink | Reply to this

Re: Snowglobe Models

Excellent!

Your second construction is a lot easier for me to understand than your first one. And, it does indeed seem to give the kind of thing Jim and I were hoping for: a product-preserving functor

F:C λTh(Calc)SetF: C_{\lambda Th(Calc)} \to Set

(where λTh(Calc)\lambda Th(Calc) is the λ\lambda-theory for high school calculus and C λTh(Calc)C_{\lambda Th(Calc)} is the corresponding cartesian closed category) with the property that

F(R)=F(R) = \mathbb{R}

and

F(hom(R,R))={polynomialfunctionsf:}.F(hom(R,R)) = \{polynomial functions f: \mathbb{R} \to \mathbb{R} \} .

So, we can imagine our idealized high-school calculus students as living in the corresponding snowglobe model: a miniature universe nestled inside the category of sets, where all functions from \mathbb{R} to \mathbb{R} are polynomials!

Now it should be doable, and somewhat entertaining, to work out things like

F(hom(hom(R,R),R)),F(hom(hom(R,R),R)),

that is, ‘all functions on the space of functions’ in this miniature universe.

Posted by: John Baez on March 16, 2007 2:37 AM | Permalink | Reply to this

Re: Snowglobe Models

Speaking of various snowglobe models in which to do calculus, John wrote:

Now it should be doable, and somewhat entertaining, to work out things like F(hom(hom(R,R),R)), that is, ‘all functions on the space of functions’ in this miniature universe.

I want to take up the ‘challenge’ here. (It’s not hard; just a matter of technique. But it is therefore technical.)

In the particular snowglobe model John was referring to, we had calculated that all functions RRR \to R are polynomial. John is suggesting we kick it up a notch, and understand ‘functionals’ R RRR^R \to R in the model.

Since it’s been a while since we discussed this, let me recall the way this model worked. We had a ‘lambda theory’ TT of high-school calculus; actually TT is construed as the free cartesian closed category generated from some terms which allow for a commutative ring RR together with a differentiation operator

D:R RR RD: R^R \to R^R

satisfying some reasonable axioms. The snowglobe model we were considering decomposed into two parts:

  • An honest model (i.e. a functor preserving cartesian closed structure) TET \to E. Here we took EE to be the topos of functors AlgSet\mathbb{R}-Alg \to Set, where Alg\mathbb{R}-Alg is the category of finitely generated commutative algebras over \mathbb{R}; this is a topos near and dear to algebraic geometry. There are of course other possibilities (e.g., taking EE to be a smooth topos, or the quasitopos of Chen smooth spaces).
  • A product-preserving functor ESetE \to Set. Here we took hom E(1,):ESethom_E(1, -): E \to Set, where 1 is terminal in EE.

The FF John is referring to is the composite of these functors, F:TSetF: T \to Set. The objects of the snowglobe model are objects of TT. If xx and yy are objects, the morphisms xyx \to y in the snowglobe model are by definition elements of F(y x)F(y^x).

As we said, we are trying to calculate morphisms R RRR^R \to R inside the snowglobe. Letting LL (for ‘line’) be the value of RR in the true model EE, we are thus trying to compute the set

F(R (R R))=hom E(1,L (L L))hom E(L L,L).F(R^{(R^R)}) = hom_E(1, L^{(L^L)}) \cong hom_E(L^L, L).

Now, in the algebraic geometry context EE, the geometric line object LL in EE is the underlying-set functor U:AlgSetU: \mathbb{R}-Alg \to Set, which is representable:

L=hom([x],)L = hom(\mathbb{R}[x], -)

The first thing we need to understand is L LL^L. It’s not hard; in fact it’s fun to calculate, but I’ll just give the answer: L LL^L is the functor which takes an algebra AA to the set of polynomials UA[x]U A[x]. (A few cryptic hints: one can compute this by using the Yoneda lemma, the adjunction between product and exponentiation in cartesian closed categories, and one’s knowledge of how to calculate coproducts in \mathbb{R}-Alg.)

So the problem of calculating the maps L LLL^L \to L is the problem of computing the set of set-valued transformations (natural in algebras AA) of the form

UA[x]UA.U A[x] \to U A.

Again this is not hard; the idea is just to express the functor L LL^L as a colimit of representables in a tractable way. The set UA[x]U A[x] is a colimit of set inclusions

UAUA 2UA 3...UA \to U A^2 \to U A^3 \to ...

where UAU A is considered the set of constant polynomials with coefficients in AA, UA 2U A^2 the set of linear polynomials with coefficients in AA, and so on. In terms of polynomial coefficients, the inclusion map

UA nUA n+1U A^n \to U A^{n+1}

takes (a 0,...,a n1)(a_0, ..., a_{n-1}) to (a 0,...,a n1,0)(a_0, ..., a_{n-1}, 0).

Now, each functor AUA nA \mapsto U A^n is a representable

hom([x 0,...,x n1],),hom(\mathbb{R}[x_0, ..., x_{n-1}], -),

and the inclusion maps UA nUA n+1U A^n \to U A^{n+1} are thereby induced by surjections

ϕ n:[x 0,...,x n][x 0,...,x n1]\phi_n: \mathbb{R}[x_0, ..., x_n] \to \mathbb{R}[x_0, ..., x_{n-1}]

where ϕ n(x i)=x i\phi_n(x_i) = x_i for ii less than nn, and ϕ n(x n)=0\phi_n(x_n) = 0.

It follows (with the help of Yoneda) that the set of transformations we are after,

colim nhom([x 0,...,x n],)U,colim_n hom(\mathbb{R}[x_0, ..., x_n], -) \to U,

is thus the projective limit of the system of surjections Uϕ nU\phi_n. What is that limit?

I think of it like this: it’s the set of formal infinite sums

p 0(x 0)+x 1p 1(x 0,x 1)+x 2p 2(x 0,x 1,x 2)+...p_0(x_0) + x_1 p_1(x_0, x_1) + x_2 p_2(x_0, x_1, x_2) + ...

where the p ip_i are arbitrary polynomials in the indicated variables. Thus, I am claiming that all functionals in our snowglobe model can be expressed in this form. We may as well give some examples.

  • The functional R RRR^R \to R which takes ff to f(2)f(2) takes the polynomial a 0+a 1x+a 2x 2+...+a nx na_0 + a_1 x + a_2 x^2 + ... + a_n x^n to a 0+2a 1+4a 2+...+2 na na_0 + 2a_1 + 4a_2 + ... + 2^n a_n. It is represented by the infinite formal sum x 0+2x 1+4x 2+...x_0 + 2x_1 + 4x_2 + ...
  • The functional R RRR^R \to R which takes ff to (Df)(1)(Df)(1) takes a 0+a 1x+a 2x 2+...+a nx na_0 + a_1 x + a_2 x^2 + ... + a_n x^n to a 1+2a 2+...+na na_1 + 2a_2 + ... + n a_n. It is represented by the infinite formal sum x 1+2x 2+3x 3+...x_1 + 2x_2 + 3x_3 + ...
  • The functional R RRR^R \to R which takes ff to f(1) 2f(1)^2 takes a 0+a 1x+a 2x 2+...+a nx n)a_0 + a_1 x + a_2 x^2 + ... + a_n x^n) to a 0 2+2a 0a 1+(2a 0a 2+a 1 2)+...+a n 2a_0^2 + 2a_0 a_1 + (2a_0 a_2 + a_1^2) + ... + a_n^2. It is represented by the formal infinite sum x 0 2+x 1(2x 0+x 1)+x 2(2x 0+2x 1+x 2)+...x_0^2 + x_1(2x_0 + x_1) + x_2(2x_0 + 2x_1 + x_2) + ...
Posted by: Todd Trimble on June 2, 2007 4:23 AM | Permalink | Reply to this
Read the post Classical vs Quantum Computation (Week 16)
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Tracked: March 12, 2007 8:12 PM

Re: Snowglobe Models

This is part of the fun of category theory — it gives your brain a real workout, but in the end it all seems perfectly obvious.

This points to a way out of the philosophical difficulty of foundations. This difficulty arises from expecting on the one hand, that first principles must be sufficiently contentful so that they have many consequences we know to be true, while, on the other hand, that they should be obvious.

If you expect this being obvious to apply to any rational being, then it’s unlikely that the principles should have sufficient content. E.g., “Things which equal the same thing also equal one another.”

But as John suggests, clarity can arise in those who work hard on a certain domain. Principles arise in traditions of enquiry. You’re not going to find category theoretic ideas obvious unless you knuckle down to the literature.

Of course, people can also err in their assessment of these principles as being able to withstand any challenge.

Posted by: David Corfield on March 13, 2007 1:29 PM | Permalink | Reply to this

Re: Snowglobe Models

David wrote:

This difficulty arises from expecting on the one hand, that first principles must be sufficiently contentful so that they have many consequences we know to be true, while, on the other hand, that they should be obvious.

[…]

But as John suggests, clarity can arise in those who work hard on a certain domain. Principles arise in traditions of enquiry.

In other words, the ‘foundations’ of a subject are not the principles that come first to mind; they’re laid bare only after hard work.

Aristotle noted something similar at the beginning of his Physics:

When the objects of an inquiry, in any department, have principles, conditions, or elements, it is through acquaintance with these that knowledge, that is to say scientific knowledge, is attained. For we do not think that we know a thing until we are acquainted with its primary conditions or first principles, and have carried our analysis as far as its simplest elements. Plainly therefore in the science of Nature, as in other branches of study, our first task will be to try to determine what relates to its principles.

The natural way of doing this is to start from the things which are more knowable and obvious to us and proceed towards those which are clearer and more knowable by nature; for the same things are not ‘knowable relatively to us’ and ‘knowable’ without qualification. So in the present inquiry we must follow this method and advance from what is more obscure by nature, but clearer to us, towards what is more clear and more knowable by nature.

Now what is to us plain and obvious at first is rather confused masses, the elements and principles of which become known to us later by analysis.

We start with what’s ‘more knowable and obvious to us’, but our goal is to find what’s ‘clearer and more knowable by nature’. These are not at all the same!

In mathematical logic, for example, the concept of ‘set’ was far from immediately accessible: it had to be laid bare through lots of work. Only in retrospect does it seem obvious and natural. If we forget this, and claim that any idea we don’t instantly understand can’t be ‘foundational’, we’re locking ourselves in the prison of what just happens to seem obvious to us now.

Posted by: John Baez on March 16, 2007 2:19 AM | Permalink | Reply to this
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Excerpt: On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
Tracked: March 19, 2007 9:10 PM

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