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April 2, 2008

2-Structure Types

Posted by David Corfield

Structure types (aka species) are functors F:FinSet 0Set, F: FinSet_0 \to Set, where FinSet 0FinSet_0 is the groupoid of finite sets and isomorphisms. For example, we could look at the FF which sends the nn-element set to the set of its orderings, which has cardinality n!n!.

We talked before about the forgetful functor from PointedSet to Set, and how this is used to pullback functors to construct things like the action groupoid. We can, of course, also do the same thing in the case of structure types. Pulling back our FF above, we see sitting above the nn element set, the set of orderings of that set with morphisms between them corresponding to permutations down below.

Might we expect there to be a similar story one level up? Here we would be interested in 2-functors F:FinGpd 0Gpd, F: FinGpd_0 \to Gpd, where FinGpd 0FinGpd_0 is the 2-groupoid of finite groupoids, equivalences, and natural isomorphisms. Examples include the identity 2-functor and the terminal 2-functor.

Just as we can take a skeletal category for FinSet 0FinSet_0 by selecting one set of each finite cardinality, now instead of FinGpd 0FinGpd_0 we can take the 2-groupoid with objects finite multisets of finite groups. For a single copy of a group GG, we then have automorphisms of GG as 1-morphisms, and as 2-morphisms from 1 G1_G to itself, elements of the centre of GG (see 3 here).

Does anything like this appear in combinatorics? One thing we won’t have are the nice series expansions of structure types. But we do still have straightforward equivalents for the sum and product operations. As for the composite of structure types GG and HH, recall we define a structure GHG \circ H by saying a GHG \circ H-structure on a set SS consists of a way of partitioning SS into disjoint parts, putting a GG-structure on the set of parts, and putting an HH-structure on each part. Similarly with 2-functors GG and HH we could partition a finite groupoid into a set of subgroupoids and put a GG- structure on the set of parts, considered as a discrete groupoid, and an HH-structure on each part.

We also talked about pulling back 2-functors along the 2-category classifier, from PointCat +CatPointCat^+ \to Cat. In the case under consideration, we might consider pulling back these 2-structure types along PointGpd +GpdPointGpd^+ \to Gpd.

Posted at April 2, 2008 9:45 AM UTC

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52 Comments & 2 Trackbacks

Re: 2-Structure Types

So we might look for an analogue of the structure type which assigns the set of orderings to a set. This is the one represented

1+X+X 2+... 1 + X + X^2 + ...

I think we could make a case for the 2-functor which assigns to a groupoid GG the underlying groupoid of the 2-group AUT(G)AUT(G).

Posted by: David Corfield on April 3, 2008 8:41 AM | Permalink | Reply to this

Re: 2-Structure Types

the forgetful functor from PointedSet to Set, and how this is used to pullback functors to construct things like the action groupoid.

Now that I finally got this point you have all my attention.

But I am maybe not following this statement:

Pulling back our FF above, we see sitting above the nn element set, the set of orderings of that set with morphisms between them corresponding to permutations down below.

Why is that? If the representation ρ:BGSet \rho : \mathbf{B} G \to Set happens to be on a set with cardinality not a factorial, then there is no fiber of the ordering structure type sitting above it at all. So then the pullback FinSet 0 orderings BG ρ Set \array{ && FinSet_0 \\ && \;\;\;\;\;\;\;\downarrow^{orderings} \\ \mathbf{B}G &\stackrel{\rho}{\to}& Set } does not even exist. Am I misunderstanding something?

Posted by: Urs Schreiber on April 3, 2008 9:47 AM | Permalink | Reply to this

Re: 2-Structure Types

This is how I’m seeing things for structure types:

PointedSet forget Finset 0 F Set \array{ && PointedSet \\ && \;\;\;\;\;\;\;\downarrow^{forget} \\ Finset_0 &\stackrel{F}{\to}& Set }

Let’s look at the FF which sends a set to the set of orderings. Now just focus on the 4 element set sitting in FinSet 0FinSet_0. This gets sent to a 24 element set in Set.

In the fibre above this last set there are 24 pointed sets, each containing the 24 orderings with one ordering selected as the point. Now, pulling this back, we have above our original 4 element set, 24 objects corresponding to the orderings.

I haven’t mentioned the arrows, but there’s nothing surprising there.

Posted by: David Corfield on April 3, 2008 11:20 AM | Permalink | Reply to this

Re: 2-Structure Types

This is how I’m seeing things for structure types:

PointedSet forget FinSet 0 F Set \array{ && PointedSet \\ && \;\downarrow^{forget} \\ FinSet_0 &\stackrel{F}{\to}& Set }

Oh, I see. I had thought you wanted to pull back FF along a representation (hm, checking, I see that you did say: “pulling back our FF above”) instead of pulling back the universal Set-bundle along FF.

Okay, never mind, now I see what you mean.

Posted by: Urs Schreiber on April 3, 2008 11:33 AM | Permalink | Reply to this

Re: 2-Structure Types

Concerning moving up the categorical ladder here, I just want to point out again that there is one puzzle left which you might have more success thinking about than I had so far:

we have seen that the action groupoid V//GV//G of a representation ρ:BGSet\rho : \mathbf{B}G \to Set is the pullback of the universal SetSet-bundle:

V s 1pt V//G T ptSet BG ρ Set \array{ V &\to& s^{-1}pt \\ \downarrow && \downarrow \\ V//G &\to& T_{pt} Set \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Set }

But we also know that the left column here usefully continues on further down, where it starts inclreasing in categorical dimension

V s 1pt V//G T ptSet BG ρ Set BEG BBG. \array{ V &\to& s^{-1}pt \\ \downarrow && \downarrow \\ V//G &\to& T_{pt} Set \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Set \\ \downarrow \\ \mathbf{B E} G \\ \downarrow \\ \mathbf{B B }G \,. }

(The last item BBG\mathbf{B B}G has the obvious interpretation if GG is abelian. If not, there should still be something playing its role, as we are discussion with Tim Porter. But feel free to ignore BBG\mathbf{B B}G and let’s concentrate on BEG\mathbf{B E}G).

One idea was to use V s 1pt V//G T ptSet BG ρ Set codisc BEG Cat BBG, \array{ V &\to& s^{-1}pt \\ \downarrow && \downarrow \\ V//G &\to& T_{pt} Set \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Set \\ \downarrow && \;\;\downarrow^{codisc} \\ \mathbf{B E} G &\to& Cat \\ \downarrow \\ \mathbf{B B }G } \,,

where codisccodisc sends each set to the codiscrete groupoid over it (one unique morphism per ordered pair of objects).

That works in some of the applications I have, but seems to be unsatisfactory for other reasons. Or I don’t fully understand it yet.

Yet another idea is this: concentrate on representations with values in GAct=GSetG Act = G Set: sets equipped with GG-action.

Then we can form BG ρ GAct BEG SetGMod \array{ \mathbf{B}G &\stackrel{\rho}{\to}& G Act \\ \downarrow && \downarrow \\ \mathbf{B E}G &\to& Set\downarrow G-Mod }

Here SetGSet\downarrow G is the category of sets over GG which we equip with the monoidal structure the way John taught us: {a g}×{b h}={(a,b) gh}. \{a_g\} \times \{b_{h}\} = \{ (a,b)_{g\cdot h}\} \,.

Then SetGModSet\downarrow G-Mod has objects categories with a module structure over this, morphisms functors respecting that module structure and 2-morphisms not just what you might expect, but also transformations that may change the GG-labels.

Then GActSetGMod G Act \to Set\downarrow G - Mod

sends each GG-set TT to the category SetTSet\downarrow T of sets over TT, which is equipped with the obvious SetGSet\downarrow G-module structure induced from the cartesian product of sets and the action of GG on TT.

This alos looks like it is pointing in some right directions. But I am really not sure about this at the moment.

If you have any thoughts on this, please let me know.

Posted by: Urs Schreiber on April 3, 2008 11:29 AM | Permalink | Reply to this

Re: 2-Structure Types

Some more candidates for 2-structure types: ordering the components of a groupoid and partially ordering hom sets compatible with composition.

Posted by: David Corfield on April 3, 2008 3:55 PM | Permalink | Reply to this

Re: 2-Structure Types

Are there any restrictions on the classes of examples you are looking at?

Here is one, which happens to factor through finite groupoids:

since FinGrpdFinGrpd is closed, for any finite groupoid CC we get

Hom FinGrpd(C,):FinGrpdFinGrpd Hom_{FinGrpd}(C,-) : FinGrpd \to FinGrpd

The analog at the level of sets is for any finite set SS the functor Hom FinSet(S,):FinSetFinSet Hom_{FinSet}(S,-) : FinSet \to FinSet which is, in words: “form the set of SS-indexed subsets”.

Posted by: Urs Schreiber on April 3, 2008 4:05 PM | Permalink | Reply to this

Re: 2-Structure Types

By the way, what seems to be of crucial interest are the endomorphisms of the identity on SetSet, CatCat, etc. Not sure how that interprets in terms of structure types, though.

Well, I should admit the following: I do not really know (or don’t remember, certainly John must have explained it) what structures types are used for except the way they appear in John’s and Jeffrey’s work: as categorifications of the rig [z]\mathbb{N}[z] of formal power series with natural number coefficients.

In that interpretation, the identity functor corresponds to 1+z+z 2+. 1 + z + z^2 + \cdots \,.

Hm, what might auto-equivalences of that identity functor tell us, from this point of view?

Posted by: Urs Schreiber on April 3, 2008 4:13 PM | Permalink | Reply to this

Re: 2-Structure Types

Is that right about the identity functor?

If an nn element set is sent to an nn-element set, there will be nn pointed sets fibred above. Isn’t this the structure type pointed set, corresponding to z+z 2+z 3/2!+... z + z^2 + z^3/2! +...

Then the functor sending a set to all orderings corresponds to 1+z+z 2+... 1 + z + z^2 +...

For another example, the functor from any set to the singleton {*}\{*\} corresponds to e z=1+z+z 2/2!+... e^z = 1 + z + z^2/2! +...

Good to wonder about the point of all this. I guess structure types are very much about the actions of symmetric groups, so 2-structure types are about the actions of symmetric 2-groups.

But then why in the first case is it good to collect them together to give series, while this is not obvious in the 2- case?

Posted by: David Corfield on April 3, 2008 5:10 PM | Permalink | Reply to this

Re: 2-Structure Types

Isn’t this the structure type pointed set, corresponding to z+z 2+z 3/2!+ z + z^2 + z^3/2! + \cdots

You are right, I forgot the factorial coefficients. Did go back to page p.5 of Jeffrey’s article to remind me about the rules of this game now.

But could you remind me what you mean by “the structure type pointed set”?

But then why in the first case is it good to collect them together to give series, while this is not obvious in the 2- case?

In the first case we could usefully collect them in a series mainly because we have that equivalence classes of sets are in bijection with natural numbers; and series are really maps from the natural numbers to the coefficent set.

So when we pass to groupoids we get “series” which are not indexed by natural numbers, but by isomorphism classes of groupoids. Or (I can’t tell right now without much further thinking) maybe by their groupoid cardinalities.

Hm, maybe we get nets this way?

Posted by: Urs Schreiber on April 3, 2008 5:25 PM | Permalink | Reply to this

Re: 2-Structure Types

The pointed set structure type just adds the extra structure of equipping a set with a pointed element.

The “series” for groupoids would be indexed by equivalence classes of groupoids, multisets of groups.

Posted by: David Corfield on April 3, 2008 5:35 PM | Permalink | Reply to this

Re: 2-Structure Types

Are there any restrictions on the classes of examples you are looking at?

None that I know of. Yours seems interesting. Can we also go the other way around Hom FinGrpd(,C)Hom_{FinGrpd} (-, C)?

Posted by: David Corfield on April 3, 2008 5:24 PM | Permalink | Reply to this

Re: 2-Structure Types

Urs wrote:

I do not really know (or don’t remember, certainly John must have explained it) what structures types are used for except the way they appear in John’s and Jeffrey’s work: as categorifications of the rig [z]\mathbb{N}[z] of formal power series with natural number coefficients.

A structure type is any type of structure we can put on finite sets. If we can put this structure on an nn-element set in a na_n ways, its generating function is defined to be

na nn!z n \sum_n \frac{a_n}{n!} z^n

As David pointed out, this power series doesn’t usually lie in [z]\mathbb{N}[z]. It lies in a somewhat larger rig consisting of formal power series where the nnth coefficient can have an n!n! in the denominator.

More importantly, one of the main jobs of combinatorics is to count structures on finite sets. In other words: compute generating functions of structure types!

For example, here’s a sort of question that people in combinatorics enjoy:

How many ‘derangements’ does an nn-element set have: that is, permutations with no fixed points?

This question, and millions like it, are easily answered using the technology of structure types and their generating functions.

In fact the generating function in the example above happens to be

e z1z\frac{e^{-z}}{1 - z}

and the answer to the question is:

The integer closest to n!/en!/e.

For more details see my homework assignment entitled Let’s get deranged!, and the answers by Jeffrey Morton, Derek Wise and other folks attending the Winter 2004 Quantum Gravity Seminar — where you can also see many other examples worked out in the course notes and homeworks. This stuff is lots of fun!

Combinatorists originally used generating functions to count structures on finite sets by seeing which operations on structure types correspond to adding their generating functions, multiplying them, differentiating them, and so on. Later Joyal came along and showed that most of these operations can be categorified, becoming operations on the structure types themselves.

It later became clear that structure types are just a special case of stuff types, and stuff types work better for categorifying the quantum harmonic oscillator. That’s what you’re more likely to care about — but the connection to combinatorics is what makes the subject so charming.

Posted by: John Baez on April 4, 2008 3:02 AM | Permalink | Reply to this

Re: 2-Structure Types

David wrote:

Might we expect there to be a similar story one level up? Here we would be interested in 2-functors

F:FinGpd 0Gpd F:FinGpd_0 \to Gpd

Of course David knows there is such a story one level up. We get a 2-functor of the above sort whenever we have any type of ‘structure’ we can put on finite groupoids. Given a finite groupoid XX, we define F(X)F(X) to be the groupoid of all structures of this type on XX.

Indeed, we can can run around taking our favorite structure types and trying to boost them up a level and get structure 2-types!

I’ll do one right now. Everyone reading this should do their own, and then David can collect them and write a paper on the subject.

My favorite structure type is being an NN-colored finite set. This is the structure type

F:FinSet 0Set F : FinSet_0 \to Set

given by

F(X)=N X F(X) = N^X

for some set NNm which we call the set of ‘colors’. An ‘NN-coloring’ of the finite set XX is just a function from XX to the set of NN.

Can we boost this example up a level? Sure! Take a groupoid NN and define the 2-structure type being an NN-colored finite groupoid:

F:FinGpd 0Gpd F: FinGpd_0 \to Gpd

by the same formula:

F(X)=N X F(X) = N^X

We can march on… but before we do, we should think a bit.

One problem, as David notes, is that the set of isomorphism (or equivalence) classes of finite groupoids is much less tractable than the set of isomorphism classes of finite sets. So, the ‘generating functions’ of structure 2-types are much harder to explicitly describe than for structure types.

We should also carefully ponder the yoga of properties, structure and stuff. We can equip finite sets with extra properties, structure, or stuff. Structure types are nice, but stuff types are better for many purposes. These are (weak) 2-functors

F:FinSet 0GpdF : FinSet_0 \to Gpd

Similarly, we can equip finite groupoids with properties, structure, stuff, or 2-stuff. Which of these corresponds to David’s ‘2-structure types’?

(The very question suggests that ‘2-structure types’ might turn out to be an awkward name.)

I know what I think the really good things are: the ‘2-stuff types’, that is, ways of equipping finite groupoids with extra 2-stuff. These are (weak) 3-functors

F:FinGpd 02Gpd F: FinGpd_0 \to 2Gpd

What’s an example? ‘Being a weak 2-group’ is a good warmup example. Making a finite groupoid into a 2-group is not just a matter of adding extra structure; we’re really adding extra stuff!

Why? Because the morphisms between weak 2-groups are functors between their underlying groupoids equipped with extra structure… and the 2-morphisms between these morphisms are natural transformations equipped with extra properties.

(This pattern is part of what I mean by ‘the yoga of properties, structure and stuff’.)

For a more full-fledged example, try ‘being the first of two finite groupoids’. Making a finite groupoid into the first of a pair of finite groupoids is not just a matter of adding extra structure or stuff: here we’re really adding extra 2-stuff!

Anyway, there’s a lot more to say, but this is doubtless already more than most people want to hear.

Posted by: John Baez on April 4, 2008 3:36 AM | Permalink | Reply to this

Re: 2-Structure Types

When trying to calculate cardinalities of these 2-groupoids of structured/stuffed groupoids presumably we need to wheel out your and Jim’s cardinality from p. 15 of From Finite Sets to Feynman Diagrams.

Could there be any sneaky way of calculating the cardinality of the 2-groupoid of finite groupoids?

If 2-stuff/2-structure types (or stuff/structure 2-types) were to be a good thing, should someone in combinatorics or representation theory have glimpsed them?

Posted by: David Corfield on April 5, 2008 1:07 PM | Permalink | Reply to this

Re: 2-Structure Types

Could there be any sneaky way of calculating the cardinality of the 2-groupoid of finite groupoids?

One big justification for the notion of groupoid cardinality is that it (or rather the “Leinster measure” it comes from) magically provides the right path integral measure for finite group gauge theories, thus making the entire path integral a categorical coend/colimit of the categorified action functional over the “groupoid of field configurations” #.

There is also a known path integral measure for finite 2-group gauge theory. In this entry I demonstrate that this is just the ordinary Leinster measure of the 1-groupoid obtained by taking isomorphism classes of 2-morphisms in the configuration 2-groupoid.

This suggests (to me) that the right canonical measure on 2-groupoids is the Leinster measure on the 1-groupoids obtained after one-step decategorification.

In general, it seems to me that we should develop a theory of \infty-category cardinality as a theory that relates the colimits of faithful functors over the \infty-category CC to the colimit of faithful functors over Disc(Obj(C))\mathrm{Disc}(\mathrm{Obj}(C)). Because that’s how the Leinster measure arises and (I think) how this entire cardinality business fits into quantum theory.

I am in the process of preparing notes related to that. For a sneak preview see this, where the statement I just made appears on p. 7 (but that page number will change, this is in the early stages of its develoment).

Posted by: Urs Schreiber on April 5, 2008 1:47 PM | Permalink | Reply to this

Re: 2-Structure Types

This suggests (to me) that the right canonical measure on 2-groupoids is the Leinster measure on the 1-groupoids obtained after one-step decategorification.

But this doesn’t fit well with the Baez-Dolan story of cardinality as an alternating product of homotopy groups. Consider the 2-groupoid with one object, one 1-morphism and two 2-morphisms. You would have its cardinality as 1. They would have it as 2.

Posted by: David Corfield on April 5, 2008 3:10 PM | Permalink | Reply to this

Re: 2-Structure Types

the Baez-Dolan story of cardinality as an alternating product of homotopy groups

Hm, thanks for pointing that out. Could you remind me of the precise definition of their cardinality for 2-groupoids?

I need figure out what happens when that definition is used to compute the path integral measure for the Yetter model.

Does it maybe amount, in that case, to the same answer I got by quotientng out isomorphism of 1-morphisms (though I doubt it, unless we have some restriction on the underlying 2-group, but I need to recall the precise definition of the baez-dolan cardinality first)?

(That quotienting out has more substance to it than it might seem on first sight: I am arguing that, in the context of QFTs like DW or Yetter models, it implements “integration without integration” #. But anyway.)

Maybe we are just dealing with two different concepts that happen to coincide for 1-groupoids? Hm, or it’s related to the nature of the transgression operation that I am thinking of:

maybe the 1-groupoid cardinality-weighted sum on the one-step decategorification of the transgressed functor [Π 1(X),BG 2][Π 1(X),tra][Π 1(X),B 3U(1)] [\Pi_1(X),\mathbf{B} G_2] \stackrel{[\Pi_1(X),tra]}{\to} [\Pi_1(X),\mathbf{B}^3 U(1)] (I am still referring to this consideration) corresponds to the Baez-Dolan 2-groupoid cardinality-weighted sum of non-transgressed [Π 1(X),BG 2]×Π 1(X)evBG 2traB 3U(1) [\Pi_1(X),\mathbf{B} G_2] \times\Pi_1(X) \stackrel{ev}{\to} \mathbf{B} G_2 \stackrel{tra}{\to} \mathbf{B}^3 U(1) ?

I am just thinking loudly, will have to figure this out myself. Thanks again for the hint.

Posted by: Urs Schreiber on April 5, 2008 3:52 PM | Permalink | Reply to this

Re: 2-Structure Types

In From Finite Sets to Feynman Diagrams they say:

The cardinality of a category CC equals that of its underlying groupoid C 0C_0. This suggests that this notion really deserves the name groupoid cardinality. It also suggests that we should generalize this notion to nn-groupoids, or even ω\omega-groupoids. Luckily, we don’t need to understand ω\omega-groupoids very well to try our hand at this! Whatever ω\omega-groupoids are, they are supposed to be an algebraic way of thinking about topological spaces up to homotopy. Thus we just need to invent a concept of the ‘cardinality’ of a topological space which has nice formal properties and which agrees with the groupoid cardinality in the case of homotopy 1-types. In fact, this is not hard to do. The key is to use the homotopy groups π k(X)\pi_k (X) of the space XX.

The homotopy cardinality of a topological space XX is defined as the alternating product

|X|=|π 1(X)| 1|π 2(X)||π 3(X)| 1... |X| = |\pi_1(X)|^{-1} |\pi_2(X)| |\pi_3(X)|^{-1} ...

when XX is connected and the product converges.

Posted by: David Corfield on April 5, 2008 4:19 PM | Permalink | Reply to this

Re: 2-Structure Types

Thanks, David.

So let’s see: if we restrict to sufficently strict higher groupoids – which we can do, I think, while staying with that definition of higher homotopy cardinality, since strict higher groupoids are sufficient to model nn-types – then we are talking about (n1)(n-1)-groupoid enriched categories.

Tom Leinster says that it is easy to generalize his definition of category cardinality to the case of enriched categories.

So the obvious question is:

i) what is the Leinster cardinality of nn-groupoids regarded as (n1)(n-1)-enriched nn-groupoids?

ii) how does it relate to homotopy cardinality of the realizations of these groupoids?

Anything known about this?

Posted by: Urs Schreiber on April 5, 2008 4:55 PM | Permalink | Reply to this

Re: 2-Structure Types

I can give some kind of answer to question (i), though I think you already know it. You may have been asking something deeper. Anyway:

Suppose, inductively, that you already have a definition of the cardinality of (n1)(n - 1)-groupoids. Take a (finite) nn-groupoid, regarded as a groupoid enriched in (n1)(n - 1)-groupoids (that’s what you meant, right?). Call its objects a 1,,a na_1, \ldots, a_n, then play the usual game: write down the matrix ZZ whose (i,j)(i, j)-entry is the cardinality of the (n1)(n - 1)-groupoid Hom(a i,a j)Hom(a_i, a_j), and define the cardinality of your nn-groupoid as the sum of the entries of Z 1Z^{-1}.

But perhaps I’ve misunderstood the question.

Posted by: Tom Leinster on April 6, 2008 3:45 AM | Permalink | Reply to this

Re: 2-Structure Types

Thanks Tom,

the formula involving the inverse of the matrix ZZ seems to be problematic for groupoids, because the inverse tends not to exist, right?

But I take it that the same reasoning applies also to the formula involving the weighting?

So if I have this finite strict 2-groupoid with

- n 0n_0 objects

- n 1n_1 1-morphisms emanating from each object

and

- n 2n_2 2-morphisms emanating from each 1-morphism.

What is its Euler characteristic?

For any two objects aa, bb, the groupoid Hom(a,b)Hom(a,b) has n 1/n 0n_1/n_0 objects on each of which n 2n_2 morphisms start. So |Hom(a,b)|=n 1n 01n 2. |Hom(a,b)| = \frac{n_1}{n_0}\frac{1}{n_2} \,. That means the weighting on the objects should be k:an 2n 1 k : a \mapsto \frac{n_2}{n_1}

and hence the Euler characteristic is n 0(n 1) 1n 2 n_0 (n_1)^{-1} n_2

Is that right?

Posted by: Urs Schreiber on April 6, 2008 8:29 AM | Permalink | Reply to this

Re: 2-Structure Types

One of the reasons for using weightings is precisely because of the non-invertability of the zeta matrix for non-skeletal groupoids. To calculate the groupoid cardinality, you can either

a) reduce the groupoid to a skeleton, and use the cardinality of the set (or, more generally, of the n-1-category) of automorphisms of the single object in each isomorphism class—à la Baez-Dolan; or

b) use Tom’s weightings, which distribute the Baez-Dolan cardinality of each isomorphism class across the various objects in that class, so they all get to share in a little bit of their common cardinality…

Posted by: Tim Silverman on April 6, 2008 2:35 PM | Permalink | Reply to this

Re: 2-Structure Types

Tim,

yes, thanks, I know. My question was about how Tom conceives the arbitrary enriched case. He told me “proceed as usual by inverting the Z-matrix replacing the cardinality of morphism sets by cardinality of morphism objects.”

And I just wanted to make sure if I can take this also as “proceed as usual using the weighting formula, replacing the cardinality of sets by … etc.”

Apparently we can, because doing so one can actually derive the Baez-Dolan homotopy cardinality of higher groupoids from the Leinster weighting formula for the enriched case, it seems. That’s the point of the little computation above, which is also in section 2.1.1 here.

Have a look and let me know if you follow my derivation (which is possibly discussed elsewhere already, but I haven’t seen it).

Posted by: Urs Schreiber on April 6, 2008 6:33 PM | Permalink | Reply to this

Re: 2-Structure Types

Oops! I thought, Great, a question easy enough for me to answer, and leapt in.

I don’t entirely follow your reasoning above, but that may be because, due to rather intermittent decent internet access recently, I’m reading and posting from my phone, which curiously has no mathml capability—making it difficult to distinguish between multiplication and division. (And yes, my predictive text now predicts html tags…)

But I’d reason like this: starting at the top, each 1-morphism has a set of 2-morphisms. Being an ordinary set, this has a cardinality of n 2n_2. Moving one level down, this means that each 1-morphism has a cardinality (enriched or groupoid, take your pick) of n 2 1n_2^{-1}. Hence the groupoid of 1-morphisms from an object to itself has cardinality n 1n 2 1n_1 n_2^{-1}. Hence, finally, each object has cardinality n 1 1n 2n_1^{-1} n_2, and thus the entire 2-groupoid has cardinality n 0n 1 1n 2n_0 n_1^{-1} n_2.

Which is the same answer you got, only I don’t understand how you got there, since you introduce n 0n_0 right at the start.

Apologies if this is all painfully obvious to you once again.

Posted by: Tim Silverman on April 6, 2008 11:46 PM | Permalink | Reply to this

Re: 2-Structure Types

Hi Tim,

one difference is that I don’t (want to) assume from the start the special situation that every kk-morphism is an automorphism.

The general situation should always be reducible to this one, but it isn’t clear to me if for nn-groupoid cardinality I can assume that before the fact.

So I considered the situation where all I knew was that the number of kk-morphisms emanating anywhere did not depend on the source (k1)(k-1)-morphisms, because this is what the nn-groupoids of the form ωCat(Π n(X),BG (n)) \omega\mathrm{Cat}(\Pi_n(X),\mathbf{B}G_{(n)}) look like.

Since we have a strict nn-groupoid, if there are n 1n_1 1-morphisms emanating from each object, this implies that there are n 1(n 0) 1n_1 (n_0)^{-1} automorphisms of each object.

I suppose now you recognize the line of reasoning I had.

Posted by: Urs Schreiber on April 7, 2008 2:44 AM | Permalink | Reply to this

Re: 2-Structure Types

OK, misread what you wrote, and our formulas don’t agree. In fact I think, though with some trepidation, that yours is wrong: your n 0n_0 should be the number of components of the groupoid, not the number of objects. Similarly, correcting (or expanding) what I said earlier, if there are several 2-isomorphic 1-morphisms, n 1n_1 only counts 1-components, not 1-morphisms. The whole weighting business should sort this out automatically (at all levels).

As for having multiple objects in an enriched environment, every instinct that I have about linear algebra is screaming at me that the enrichment can’t possibly make any difference to the basic calculation, but I haven’t checked, so I could be wrong. Or I could have misunderstood you again. :-(

Posted by: Tim Silverman on April 7, 2008 9:27 PM | Permalink | Reply to this

Re: 2-Structure Types

Tim, wait, I think there is just an subtlety with our notation conventions:

my n in_i is not your n in_i, in general. In particular, it is not π i(A)\pi_i(A), in general (for AA the nn-groupoid).

I was talking about cardinalities of 2-groupoids which have fixed constant number n kn_k of kk-morphisms emanating at every source (k1)(k-1)-morphism.

It’s a simple computation then: we can assume 1-connectedness (and sum connected components afterwards) and find

a) that each Hom-groupoid A(a,b)A(a,b) has n 1(n 0) 1n_1 (n_0)^{-1} objects on each of which n 2n_2 morphisms start, hence has cardinality |A(a,b)|=(n 0) 1n 1(n 2) 1. |A(a,b)| = (n_0)^{-1} n_1 (n_2)^{-1} \,. A weighting for this matrix is a(n 1) 1n 2 a \mapsto (n_1)^{-1} n_2 leading to a cardinality |A|=n 0(n 1) 1n 2. |A| = n_0 (n_1)^{-1} n_2 \,.

We get from this to the formula you have in mind by determining the π i\pi_i:

for instance for a connected 1-groupoid with n 0n_0 objects and n 1n_1 1-morphisms emanating from each we have π 0=1 \pi_0 = 1 and π 1=n 1(n 0) 1. \pi_1 = n_1 (n_0)^{-1} \,. So the formulas do agree.

Posted by: Urs Schreiber on April 7, 2008 10:36 PM | Permalink | Reply to this

Re: 2-Structure Types

So n 0n_0 is the number of objects in each component? I’m afraid I still don’t see what n 0n_0 has to do with the cardinality of the hom-groupoid, which I still expect to be n 1n 2 1n_1 n_2^{-1}. I don’t see why n 0n_0 should come into it anywhere, since the number of objects in a component isn’t homotopy-invariant. Or have I misunderstood yet again? (This feels like the sort of conversation that would be a lot easier if we could sit down together with a pencil and paper.)

Oh, and, just to check, when you say 1-connected, do you mean connected or simply connected? If the former, I think I’d say 2-connected!

Posted by: Tim Silverman on April 8, 2008 10:11 PM | Permalink | Reply to this

Re: 2-Structure Types

n 0n_0 is the total number of objects in the 2-groupoid. n 1n_1 the total number of 1-morphisms starting at each object. n 2n_2 the total number of 2-morphisms starting at each 1-morphism.

Assume first that there is just one connected component (later we can trivially sum over connected components).

If there are n 1n_1 morphisms starting at every object in a groupoid, then there are n 1/n 0n_1/n_0 morphisms in every Hom-groupoid.

Do you see this? Somehow it seems this is the point where for some reason a communication problem arises.

Take a simple example:

A connected groupoid with 2 objects aa,bb, one trivial and one nontrivial automorphism of every object object. Then there must be precisely two morphism from aa to bb: by the assumption of connectedness there is at least one, precomposing with the nontrivial automorphism gives the other one and any two differ either by the trivial or the nontrivial automorphism.

So then there are precisely 4 morphisms starting at every object. 2=n 1/n 0=4/22 = n_1/n_0 = 4/2 are in every Hom-set Hom(a,a)Hom(a,a), Hom(a,b)Hom(a,b), Hom(b,a)Hom(b,a), Hom(b,b)Hom(b,b).

That’s the reason why the weighting on a groupoid is at each object one over the number of morphisms starting there.

You know all this. I am just saying this since some misunderstanding is bugging us here…

Posted by: Urs Schreiber on April 9, 2008 12:39 AM | Permalink | Reply to this

Re: 2-Structure Types

I don’t see why n 0n_0 should come into it anywhere, since the number of objects in a component isn’t homotopy-invariant.

Yes. But n 1n_1 is neither! Only their quotient is (for a 1-groupoid). Their quotient is the number of automorphisms of each object.

Maybe you keep thinking that my n 1n_1 is the number of automorphisms of any object. But it’s not. It’s the total number of morphisms whose source is that given object. Automorphisms and non-automorphisms.

Posted by: Urs Schreiber on April 9, 2008 1:07 AM | Permalink | Reply to this

Re: 2-Structure Types

OK, now I understand. Sorry to have been so persistent in reading what I thought you ought to have written rather than what you actually wrote. And thank you for your patience.

What’s the reason for expressing an invariant expression in terms of two invariant ones? I looked at your diffcoh paper to see if this would make it clearer (so you may count your patience not completely wasted :-)) but it didn’t make it seem clearer to me. Or maybe I have just got used to thinking in a certain way and don’t readily change.

Posted by: Tim Silverman on April 9, 2008 7:32 PM | Permalink | Reply to this

Re: 2-Structure Types

Tim, thanks for bearing with me! I appreciate it.

What’s the reason for expressing an invariant expression in terms of two invariant ones?

As I tried to indicate, I am interested in the cardinality of (finite, for the moment) nn-groupoids CC which are of the special form of Hom-groupoids

C=hom(S,BG) C = hom(S, \mathbf{B}G)

where SS is any (finite, for the moment) nn-groupoid and BG\mathbf{B} G is any (finite, for the moment) nn-groupoid with a single object.

I want to know the Leinster measure, i.e. the weighting on these beasts.

At least in the special cases that I was looking at, these nn-groupoids have the special property that the number of kk-morphisms emanating at each (k1)(k-1)-morphism is a fixed constant just depending on BG\mathbf{B} G. And that number is easy to determine.

So I began writing down formulas for the cardinality of these nn-groupoids in terms of the numbers of emanating kk-morphisms.

This is of interest for the following reason:

For n=1n=1, CC is the configuration space of Dijkgraaf-Witten theory, and the Leinster measure on CC happens to reproduce the DW path integral measure.

For n=2n=2, CC is the configuration space of the Yetter model, and the Leinster measure on CC happens to reproduce the path integral measure for that, as shown by Martins and Porter.

To show that last statement, I needed that formula dμ:an 1 1n 2d\mu : a \mapsto n_1^{-1} n_2 for the weighting on CC.

This relation between the Leinster measure and path integral measures is interesting. It suggests that the path integral for non-finite gauge theories might have a good abstract-nonsense definition in terms of colimits of functors.

I looked at your diffcoh paper to see if this would make it clearer (so you may count your patience not completely wasted :-))

Great, thanks. :-)

but it didn’t make it seem clearer to me.

This morning (meaning \sim 12 hours ago) I made an addition to the relevant section, describing in more detail how the numbers n 1n_1 and n 2n_2 are computed for the Yetter case.

But thanks for the info that the main point didn’t come across clearly. I’ll add another remark to that effect.

Posted by: Urs Schreiber on April 9, 2008 8:51 PM | Permalink | Reply to this

Re: 2-Structure Types

Yeah, sorry Urs, of course you’re right: the inverse tends not to exist when you’re dealing with (enriched) groupoids. So you should use weightings. In a bit of a hurry now - may come back to this in a while.

Posted by: Tom Leinster on April 6, 2008 7:59 PM | Permalink | Reply to this

Re: 2-Structure Types

Tom,

no problem, thanks for your message.

I understand that it is the obvious definition, but I just wanted to check with you, to be sure.

I am also in a hurry, have a few hours of sleep left before I need to catch the plane to start my tour d’april (Aarhus/DK \to Notre Dame \to UPenn ).

If I weren’t, I think I could start to write up a proof which derives the homotopy cardinality formula from your formula for the enriched setting by working recursively with nn-groupoid-enrichment in direct generalization of the simple computation above.

But I feel this must have been considered before. Has it?

Another thing, certainly just as obvious, but I want to check anyway: is it right that your theorem 3.1 survives the passage to the arbitrary VV-enriched context?

I.e. is it true for VV a suitable monoidal category with cardinality operation ||:(V,)(,+) |\cdot| : (V,\oplus) \to (\mathbb{Q},+) and AA any small VV-category with VV-weighting dμ:Obj(A) d\mu : Obj(A) \to \mathbb{Q} and F:AV F : A \to V a sum of VV-representable functors that | aAF(a)|=|colimF|= Obj(A)|F()|dμ:= aObj(A)|F(a)|dμ(a) |\int^{a \in A} F(a)| = |colim F| = \int_{Obj(A)} |F(-)|\; d\mu := \sum_{a \in Obj(A)} |F(a)| \; d\mu(a) ? Or else, under which conditions is this true?

Should be immediate from the obvious direct generalization of the simple proof you give on p. 16.

Posted by: Urs Schreiber on April 6, 2008 9:43 PM | Permalink | Reply to this

Re: 2-Structure Types

Let’s see what happens with the one object 2-groupoid, i.e., 2-group, AUT(G)AUT(G) for some finite group GG.

This has one object, and aut(G)aut(G) worth of 1-morphisms.

A 2-morphism between ϕ\phi and ψ\psi is a group element gg making the following diagram commute:

G ϕ G g g G ψ G \array{ G &\stackrel{\phi}{\to}& G \\ g\downarrow && g\downarrow \\ G &\stackrel{\psi}{\to}& G }

So, for all hGh \in G, ϕ(h)=g 1ψ(h)g\phi(h) = g^{-1} \cdot \psi(h) \cdot g.

There are |G||G| 2-morphisms emerging from each 1-morphism. Out of the identity automorphism |Z(G)||Z(G)| go to each member of Inn(G)Inn(G) (isomorphic to G/Z(G)G/Z(G)).

So according to your reckoning,

|AUT(G)|=1|aut(G)| 1|G|=|Z(G)|/|Out(G)|. |AUT(G)| = 1 \cdot |aut(G)|^{-1} \cdot |G| = |Z(G)|/|Out(G)|.

This seems right, as if we consider AUT(G)AUT(G) as a 1-groupoid, it has |Out(G)||Out(G)| connected components, and there are |Z(G)||Z(G)| arrows from an object to itself, hence groupoid cardinality |Out(G)|/|Z(G)||Out(G)|/|Z(G)|, the reciprocal of our previous answer.

Posted by: David Corfield on April 7, 2008 2:57 PM | Permalink | Reply to this

Re: 2-Structure Types

I guess we could ask Klein 2-geometry kinds of question already in this simple case, looking for sub-2-groups of AUT(G)AUT(G) fixing various ‘shapes’ in GG.

But we have also wondered before about preserving shapes in topological groupoids. For example, we might take a space, XX, of objects, each endowed with a vertex group GG, and no other arrows. Then symmetries will combine an automorphism of XX with a continous choice of automorphism of GG for each point.

Natural transformations will involve a choice of element of GG for conjugate action, which presumably could vary if GG were continuous.

I suppose the question then is whether there are any interesting shapes to preserve beyond simple blending of shapes of XX and shapes of GG.

Posted by: David Corfield on April 11, 2008 11:55 AM | Permalink | Reply to this

Re: 2-Structure Types

There must surely be a lot of existing work on AUT(G)AUT(G)’s action on GG. If we were interested as part of Klein 2-geometry, I guess we’d be more concerned with transitive actions. Clearly the action on GG is not transitive, and we’d need to look at things like conjugacy classes.

Perhaps more systematic would be to think of the lattice of subgroups of GG. Here’s a 1996 review by Ralph Freese of Subgroup lattices of groups by Roland Schmidt. Freese mentions results which

… led to the hope that lattice theory might prove to be a powerful tool in group theory. In the introduction to his book Suzuki concluded from his theorem that if GG is a simple group, then GG is determined by the lattice of subgroups of G×GG \times G that “we might have a possibility to apply lattice theoretical considerations to solve the classiffication problem of finite simple groups.” However this hope was not realized; much more powerful techniques, primarily character theory and “local analysis”, were used. Similarly in Abelian group theory Baer’s lattice theory techniques are no longer used… So group theory and lattice theory went their separate ways. (For that matter, group theory nowadays has little in common with Abelian group theory.) Group theory had other techniques and lattice theory had its own deep problems to work on, and most of the applications of lattice theory to algebra were in the field of universal algebra.

In the last several years some connections between lattice theory and group theory have resurfaced.

Hmm, powers of GG, just like we needed powers of XX here.

Posted by: David Corfield on April 15, 2008 9:10 AM | Permalink | Reply to this

Re: 2-Structure Types

So according to your reckoning,

|AUT(G)|=1|aut(G)| 1|G|=|Z(G)|/|Out(G)| |AUT(G)| = 1 \cdot |aut(G)|^{-1} \cdot |G| = |Z(G)| / |Out(G)|

This seems right,

Just to let you know: yes, I agree that’s the formula one gets from Tom Leinster’s general formula for the Cat-enriched case, and hence as a special case of the formula |BAUT(G)|=n 0(n 1) 1n 2=1|aut(G)| 1|G| |\mathbf{B} AUT(G)| = n_0 \cdot (n_1)^{-1} \cdot n_2 = 1 \cdot |aut(G)|^{-1} \cdot |G| that we were talking about # and yes, it seems to make good sense.

What deserves attention here is how that measure differs from the one we’d get by regarding the group just as a monoid GG instead of as a one-object groupoid BG\mathbf{B}G. In the first case the cardinality of the automorphism thing would be |aut(G)| 1|aut(G)|^{-1}, of course.

Another interesting question from my point of view would be: what might the one-object 2-groupoid BAUT(G)\mathbf{B} AUT(G) be the “configuration space of fields” of?

In other words, what are examples of 2-groupoids CC and DD such that BAUT(G)hom(C,D) \mathbf{B} AUT(G) \simeq hom(C,D) ?

The only solution I can see right now is the tautological one, where C=ptC = pt and D=BAUT(G)D = \mathbf{B} AUT(G).

That would be the AUT(G)AUT(G) Yetter-model over the point.

Posted by: Urs Schreiber on April 16, 2008 12:08 PM | Permalink | Reply to this

Re: 2-Structure Types

Tom, so I gather we can easily generalize the statement about cardinalities of colimits over SetSet-valued functors to cardinalities of (non-pseudo, I suppose) colimits over CatCat-valued 2-functors (so with everything regarded as being CatCat-enriched)?

The analog of prop. 3.1 in your article should read:

Let AA be a finite CatCat-enriched category and k k^\bullet a weighting on AA. If X:ACatX : A \to Cat is finite and the sum of representables, then |colimX|= ak a|X(a)||colim X| = \sum_a k^a |X(a)|, where in the right we have the ordinary cadrinality of categories.

Supposedly we can go on like this to colimits over 2-, 3-, … -category valued functors?

Posted by: Urs Schreiber on April 6, 2008 10:10 AM | Permalink | Reply to this

Re: 2-Structure Types

I am preparing some notes on the Leinster measure for 1- and 2-groupoids and their relation to the path integral measure for the Dijkgraaf-Witten and the Yetter model, respectively:

On Σ\Sigma-models and nonabelian differential cohomology (pdf)

I’d be grateful if you, Tom, could have a look at section 2.1.1, where I review the necessary tools and results from your article. Especially if you could check that what I write about the 2-groupoid case confirms with what you indicated above.

Posted by: Urs Schreiber on April 6, 2008 12:35 PM | Permalink | Reply to this

Re: 2-Structure Types

I noticed that I actually did use the formula |X|=|π 1(X)| 1|π 2(X)||X| = |\pi_1(X)|^{-1} |\pi_2(X)| for 2-groupoid cardinality here, without really appreciating it myself:

I say that there are |G| ν|H| λ|G|^\nu |H|^\lambda 1-morphisms emanating at each object, each of which have, in turn |H| ν|H|^\nu 2-morphisms emanating from them.

Then the cardinality is supposed to be the sum over objects of 1|G| ν|H| λ|H| ν\frac{1}{|G|^\nu |H|^\lambda} |H|^\nu. I had thought of this of the ordinary cardinality of the 1-groupoid obtained by quotienting out 2-isomorphisms. But that’s wrong thinking. It is only correct if there happen to be no 2-automorphisms. If there are, then they are taken into account by the homotopy cardinaity formula.

Posted by: Urs Schreiber on April 6, 2008 7:55 AM | Permalink | Reply to this

Re: 2-Structure Types

David wrote:

Jim and I wrote:

The cardinality of a category CC equals that of its underlying groupoid C 0C_0.

Just to be utterly clear, this sentence is obsolete. It’s not true of Tom Leinster’s cardinality of a category; we wrote this back when we didn’t have any definition of the cardinality of a category other than that of its underlying groupoid!

Posted by: John Baez on April 6, 2008 7:28 PM | Permalink | Reply to this

Re: 2-Structure Types

If we are interested in multisets, in the context of multisets of finite groups, it might be useful to know about the stuff types corresponding to them.

Has this been done? On the fact of it, it looks like given a set XX, we want to look at all colourings of elements by non-empty finite sets. That would be represented

e (e1)X. e^{(e - 1) X}.

Posted by: David Corfield on April 6, 2008 4:10 PM | Permalink | Reply to this

Re: 2-Structure Types

Let’s see. For any groupoid GG we have the stuff type ‘being a GG-colored finite set’, and the generating function of this is

e |G|xe^{|G| x }

Anyone who finds this mysterious should see the week 9 notes from the Winter 2004 seminar. There we do the example of ‘being a 1/21/2-colored finite set’ (G=/2G = \mathbb{Z}/2), which has generating function

e z/2e^{z/2}

We come close to talking about ‘being a finite-set colored finite set’ (G=FinSet 0G = FinSet_0), but don’t seem to quite mention it. Anyway, this should have generating function

e eze^{e z}

so ‘being a nonempty-finite-set-colored finite set’ should have generating function

e (e1)ze^{(e-1) z}

Okay, I guess that’s right. I was getting this stuff mixed up with ‘being a finite set partitioned into possibly empty parts,’ with generating function

e e z e^{e^z}

and ‘being a finite set partitioned into nonempty parts’, with generating function

e e z1e^{e^z - 1}

But anyway: the groupoid of ‘nonempty-finite-set-colored finite sets’ — that is, finite sets whose elements are labelled by nonempty finite sets — isn’t the same as the usual groupoid of ‘multisets’, is it? I thought a multiset was just a set whose elements are labelled by natural numbers. A natural number, unlike a finite set, doesn’t have automorphisms…

However, it’s quite possible that you’re using the word ‘multiset’, not in the standard way, but in the way that’s actually suited to the problem you’re thinking about! In which case all these elaborate comments could be summarized as:

“Yup.”

(By the way, I’m going to Shanghai tomorrow. Being there will remind me of the birth of this blog — and the good old days, shortly before this blog was born, when I’d sit in Starbucks working out stuff about Klein 2-geometry to post on your blog! One big difference is that it won’t be sweltering hot at this time of year. So, I may walk around more and blog less.)

Posted by: John Baez on April 6, 2008 11:59 PM | Permalink | Reply to this

Re: 2-Structure Types

So what is the standard convention for the symmetries of a multiset? Let’s say we’re dealing with

{a,a,b,b,c}. \{a, a, b, b, c\}.

I was thinking of this as a 2 element set assigned to aa, another to bb, and a 1 element set to cc. Symmetries would allow permutations of the copies of a letter, and also between letters assigned sets of the same cardinality, i.e., the aas and bbs here.

You say people think of my multiset as the map which assigns 2 to aa and bb and 1 to cc. Then as you say,

A natural number, unlike a finite set, doesn’t have automorphisms…

So, again, what are the symmetries of a multiset? Just permutations of the letter types of the same cardinality, so here the cyclic group of order 2?

But then, as you say, maybe I don’t want multisets. So should I say that a finite groupoid is equivalent to a sum of a finite number of groups, possible with repeats?

Being there will remind me of the birth of this blog – and the good old days, shortly before this blog was born, when I’d sit in Starbucks working out stuff about Klein 2-geometry to post on your blog!

Yes, the good old days and our first steps towards Mathematics 2.0 blogging. Tell us if you have further profound thoughts on the Nine Zigzag Bridge (August 14, 2006 2:52 AM entry). Hmm, maybe we didn’t go on to use that stabilizer idea all that well.

Posted by: David Corfield on April 7, 2008 8:59 AM | Permalink | Reply to this

Re: 2-Structure Types

Here is a comment on a relation between structure types and colimits, attemtping to relate the definition of Baez-Dolan generating functions for structure types to Tom Leinster’s formula for the relation between colimits and cardinality:

Given a bimonoidal category VV with cardinality

||:(V,,)(,+,.) |\cdot | : (V,\oplus,\otimes) \to (\mathbb{R}, + , .)

and given any category Σ\Sigma, it would be natural to equip

hom(Σ,V) hom(\Sigma,V)

with a cardinality. Motivated by thinking in terms of functions from sets to numbers instead of functors between categories, it is natural to expect this to come from the integral. So we could try ||: hom(Σ,V) (ΣFV)| ΣF|. \begin{aligned} |\cdot| :& hom(\Sigma,V) \to \mathbb{R} \\ &(\Sigma \stackrel{F}{\to}V) \mapsto |\int^\Sigma F| \end{aligned} \,.

Where the integral denotes the coend, which is just the colimit here.

Now, if FF is nondegenerate as in Tom’s definition 3.2, p. 16 we know that | ΣF|= nSet |F(n)|n!. |\int^\Sigma F| = \sum_{n \in Set_\sim} \frac{|F(n)|}{n!} \,.

But of course in general FF will not be nondegenerate.

However, given any FF, we can turn it into a nondegenerate functor without really losing information, by artifically adding degrees of freedom such as to lift any non-freedom of the action of FF: we choose a finite set XX and replace FF by F X:nF(n)×X nF_X : n \mapsto F(n)\times X^n and let automorphisms of nn act by the obvious permutation rep on the X nX^n-factor.

Then Tom’s formula does apply to F XF_X | ΣF X|= nSet |F(n)×X n|n!. |\int^\Sigma F_X| = \sum_{n \in Set_\sim} \frac{|F(n) \times X^n|}{n!} \,.

While nice, this trick depends on a choice of XX. So we decide to keep track of that and not send FF to a number, but to a function from classes of XXs to numbers:

||:F(|X| ΣF X= n|F(n)×X n|n!)), |\cdot| : F \mapsto \left( |X| \mapsto \int^{\Sigma} F_X = \sum_{n \in \mathbb{N}} \frac{|F(n)\times X^n|}{n!} \right)) \,,

where now on the right the generating function for the structure type FF has appeared.

The point being: the categorical integral aka colimit is a natural candidate for equiping hom-categories of maps into things that have a cardinality with a cardinality themselves. To make Tom’s formula for the colimit work one may have to add degrees of freedom. Doing so for maps from Set to Set reproduces the idea of generating functions for structure types.

Posted by: Urs Schreiber on April 10, 2008 3:55 PM | Permalink | Reply to this

Re: 2-Structure Types

Does your sum over nSet ~n \in Set_~ mean the domain of FF, i.e., Σ\Sigma, is FinSet 0FinSet_0?

But then you had said “given any category Σ\Sigma”.

Posted by: David Corfield on April 11, 2008 9:35 AM | Permalink | Reply to this

Re: 2-Structure Types

Yes. I said that given any category Σ\Sigma, it seems that it would be natural to equip hom(Σ,V)hom(\Sigma,V) with the cardinality obtained from integration.

Then I said: look what becomes of this statement when we take Σ=FinSet 0\Sigma = FinSet_0: we reproduce the notion of generating functions for structure types!

Posted by: Urs Schreiber on April 11, 2008 9:43 AM | Permalink | Reply to this

Re: 2-Structure Types

Oh, now I see what you mean. I left out a sentence or two in my comment!

Before I say “Now, if…” I mean to have restricted attention to Σ=FinSet 0\Sigma = FinSet_0. Sorry. I think I was interrupted in the middle of typing this comment.

Posted by: Urs Schreiber on April 11, 2008 9:46 AM | Permalink | Reply to this

Re: 2-Structure Types

Hm, only that I am wrong that the action can be made free this way. Hm…

Posted by: Urs Schreiber on April 11, 2008 11:11 AM | Permalink | Reply to this
Read the post Klein 2-Geometry X
Weblog: The n-Category Café
Excerpt: Acting on groups
Tracked: April 15, 2008 11:27 AM

Re: 2-Structure Types

There’s the issue with these 2-structure types that equivalence classes are not so easy to represent as the series associated with ordinary structure types.

But we ought to remember that the representation of the latter as a natural number indexed series is not faithful. E.g., the species being either an unordered blue pair or an unordered red pair is represented the same way as being an ordered pair.

A faithful representation of the isomorphism classes of species is the semi-ring

[X,E 2,E 3,C 3,E 4,E 4,E 2E 2,E 2X 2,...], \mathbb{N}[X, E_2, E_3, C_3, E_4, E_4, E_2 \circ E_2, E_2 \circ X^2,...],

i.e., sums of molecules, which are products of atoms, see p.2 of this. Molecules are associated with subgroups of S nS_n.

So we shouldn’t expect anything too straightforward from 2-structure types. Molecular ones being associated to sub-2-groups of automorphism 2-groups of finite categories.

Posted by: David Corfield on April 21, 2008 9:34 PM | Permalink | Reply to this
Read the post cp cerebral
Weblog: cp cerebral
Excerpt: I have been on Stan\'s sight for almost 2 years. He is a wonderful father who stuck around in the Autism crusade even after he recovered his own son. Started a school where they juice, have an HBOT chamber and an on sight Chef for the special diets. Wo...
Tracked: November 9, 2008 8:09 PM

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