## October 6, 2009

### Borisov on Shapes of Cells in n-Categories

#### Posted by John Baez

I got an email from the author of this paper:

asking for comments. Unfortunately I’m completely busy with work on other subjects — and some of you reading this are much better than me at comparing various definitions of $n$-category! So maybe you can say something?

A category is a graph with extra structure, so we may hope that an $n$-category is some sort of generalized graph with extra structure. The details depend, among other things, on what sort of ‘shapes of cells’ we use. Borisov claims that his concept of ‘nested graph’ gives the ‘minimal generalization’ of the concept of graph needed to handle lots of example. If this is correct, it would be good for more people to know about it! If not, it would be good for Borisov to know about it!

His theory of nested graphs is vaguely reminiscent of Berger and Moerdijk’s theory of Geometric Reedy categories. A geometric Reedy category $R$ has ‘shapes of cells’ as objects and ‘face and degeneracy maps’ as morphisms. Every shape of cell has some natural number called its ‘dimension’, any ‘face’ of some shape of cell has lower dimension, while any ‘degeneracy’ of some shape cell has higher dimension. Various axioms are imposed so that the presheaf category $Set^{R^{op}}$ has the structure of a model category; this allows us to do homotopy theory in there. The classic example is $R = \Delta$, where $Set^{R^{op}}$ is the category of simplicial sets.

But this relation between Borisov’s work and the theory of geometric Reedy categories may or may not be deceptive! I don’t know.

Posted at October 6, 2009 9:26 PM UTC

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### Re: Borisov on Shapes of Cells in n-Categories

For those to whom John’s remarks about Reedy model categories were news: more context, more explanation, more details, more examples and and more references may be found at

On the $n$Lab there is also an entry that goes in the direction of the general topic of differing shapes for higher categorties: $n$Lab:geometric shapes for higher structures. But that could use more details.

I notice that Borisov’s general strategy of using “nested graphs” as encoding arbitrary higher shapes seems to be the one we played around with at $n$Lab:hyperstructure a bit.

Posted by: Urs Schreiber on October 7, 2009 1:54 PM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

The nLab also has a pretty good discussion of various graphs:

$n$Lab: graph

Posted by: Eric Forgy on October 7, 2009 11:42 PM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

I think I am going to like this paper a lot. I’m sure I will have many questions and comments, so instead of posting multiple short interspersed (possibly distracting) questions and comments here, I created:

Nested Graph

Posted by: Eric Forgy on October 8, 2009 12:18 AM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

I am excited by this paper! I’m looking forward to understanding the relationship with geometric Reedy categories, and whether these ideas can fix the problems we encountered at hyperstructure. I’m not very far in yet, but I already have some questions. I’d be inclined to have the discussion here, since there seem to be people who follow and comment on the blog but never show up at the lab.

I think the main thing I’m having trouble with right now is wrapping my head around functors between nested graphs and what they mean “geometrically.” Probably working out some examples would help. For instance, regarding the remark after Definition 4, what is an example of a functor between plain graphs which can be factored into a merger and a contraction in two inequivalent ways, and what do those factorizations “mean”?

Posted by: Mike Shulman on October 8, 2009 9:51 PM | Permalink | PGP Sig | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

Consider a usual corolla with 4 legs, and suppose we contract all of the legs, i.e. the result is just a vertex. What is the combinatorics of this operation?

This situation might seem strange: to contract legs attached to the same vertex, but this is common for cyclic operads and algebras.

The answer, of course, is to first choose which pairs of legs are joined to make loops, and then contract the loops, i.e. we first merge the ends of legs, and then contract.

Obviously there are 3 different ways to produce the loops, and therefore there are different ways to factor the unique map from the corolla to the one-vertex graph.

In general, we allow not only edges connecting 2 nodes, but also “stars” connecting more than 2 nodes, and then there are more than 3 choices.

Posted by: Dennis Borisov on October 9, 2009 1:47 PM | Permalink | Reply to this

### functors of graphs

Great, thanks, that helps. Is this also an example? Consider the disjoint union of two corollas with 2 legs each and map it to the graph consisting only of a vertex. Then we could join the legs up in two different ways (or three, if we allow making them into loops as well) before contracting them.

A terminological comment: I think the term “epi-functor” is a bit misleading, since not all epimorphisms in $Cat$ have that property. (For instance, the inclusion of the walking arrow into the walking isomorphism is an epimorphism such that $\phi(\mathcal{G}_1)$ does not generate $\mathcal{G}_2$.) It seems fairly likely that all epimorphisms between direct categories do have that property, but if so it might be helpful to remark on it.

Posted by: Mike Shulman on October 9, 2009 3:54 PM | Permalink | PGP Sig | Reply to this

### Re: functors of graphs

Yes that is also an example. However, in the paper on plain graphs and operads, Manin and I treated morphisms from disjoint unions of graphs not as operations, but as atomizations, that are important for representations. This does not work for contracting loops on one vertex, and hence the “virtual edges” in that paper.

If one wants morphisms, having disjoint union as domains, to represent operations and not just atomizations, there is no way around factorizing into a merger/grafting and a contraction.

Good point on the terminology, thank you. For finite direct categories the two notions coincide, since there are canonical generators – irreducible morphisms – and by adding another copy of one of the generators, one can easily see that all of them have to be in the image of an epi-morphism.

Posted by: Dennis Borisov on October 9, 2009 9:15 PM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

In the last bullet point of Definition 6, what does it mean to consider a vertex “as a graph on its own”? A vertex is an object; are we supposed to think of the corresponding one-object subcategory? That feels unlikely because it isn’t a “full subgraph” in the sense of the next Definition.

Posted by: Mike Shulman on October 8, 2009 9:52 PM | Permalink | PGP Sig | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

It is the fiber that is considered as a graph on its own. Probably I should reformulate this.

Posted by: Dennis Borisov on October 9, 2009 1:53 PM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

Ah, I understand! You mean something like “a $\phi$-vertex is an object of $\mathcal{G}_1$ which is a vertex of the fiber in which it lies.”

Posted by: Mike Shulman on October 9, 2009 3:25 PM | Permalink | PGP Sig | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

Exactly. I produced this somewhat confusing formulation because there are two ways for a node to be a vertex in the fiber: it can be a vertex of the ambient graph, that happens to be in the particular fiber, or it can be vertex of the fiber itself, when we consider this fiber as a stand alone graph. Of course I mean the latter.

Posted by: Dennis Borisov on October 9, 2009 9:25 PM | Permalink | Reply to this

### Earlier Paper

Borisov makes numerous references to his earlier paper [BoM08] D.Borisov, Yu.I.Manin. Generalized operads and their inner cohomomorphisms. In Geometry and dynamics of groups and spaces, Progress in Mathematics 265, pp. 247-308, Birkäuser, Basel (2008)., but doesn’t mention that it is available on arxiv.

Posted by: Rod McGuire on October 8, 2009 10:47 PM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

I think what this paper really needs is lots of examples, all the way through. I’ve tried several times to make my way through it, but I always end up getting bogged down in all the definitions. If we had two or three good classes of examples (maybe globes, simplices, and cubes) which would follow us through the maze of definitions as we go (instead of having all the examples pushed off until the end), I think the paper would be a lot easier to read. Examples to illustrate all the “corner cases” of the definitions would also be really helpful in understanding why those particular definitions were chosen.

Posted by: Mike Shulman on October 13, 2009 3:38 AM | Permalink | PGP Sig | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

Good point. I will do it in the second version. Until then you can ask me here why I chose the definitions the way I did.

Posted by: Dennis Borisov on October 13, 2009 7:44 PM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

More from Borisov on a very Schreiber-esque topic – What is the higher dimensional infinitesimal groupoid of a manifold?.

Posted by: David Corfield on October 30, 2009 4:44 PM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

Has this blog or the nLab already addressed the issue of how Kapranov’s infintesimal paths compare to Schreiber’s?

Has either of those sources discussed the Nash `stable tangent bundle’ of a _topological_ manifold?

Posted by: jim stasheff on October 30, 2009 6:02 PM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

More from Borisov on a very Schreiber-esque topic – What is the higher dimensional infinitesimal groupoid of a manifold?.

Thanks for the reference! I missed that. Will read it.

I should say that for an ordinary manifold, the infinitesimal path $\infty$-groupoid that I am talking about is presented by the infinitesimal singular simplicial complex that seems to go back to Joyal (as far as I understad) and which was especially studied and understood in its relevance by Anders Kock. All I am adding at this point is that I observe that we are entitled to think of this simplicial object as presenting an $\infty$-Lie groupoid, an object in the smooth $(\infty,1)$-topos. The claim is that this simple moves makes a wealth of classical results suddenly fall into place and produce a nice and useful picture of $\infty$-Lie theory.

But then I show in addition that the construction of the infinitesimal path $\infty$-groupoid left-Kan extends from ordinary manifolds to smooth $\infty$-groupoids themselves. So we may speak of the ininitesimal path $\infty$-groupoid $\Pi^{inf}(X)$ of an $\infty$-Lie groupoid $X$ (long-time readers of this blog will recognize this as the final answer to the old thread of ideas posted here under the headline “paths in categories”).

Again presenting this concretely using simplicial methods, this is always equivalent to the simplicial realization of the bisimplicial object of infinitesimal simplices in a simplicial manifold. This in turn makes then all the classical results on the simplicial deRham complex fall into place. (Compare the current (beginning of a) discussion on the AlgTop mailing list about the Chevalley-Eilenberg algebra of these $\infty$-Lie groupoids).

That just to put David’s attribution of this idea into perspective. Will have to read Borisov now…

Posted by: Urs Schreiber on October 31, 2009 12:56 AM | Permalink | Reply to this

### Re: Borisov on Shapes of Cells in n-Categories

Has this blog or the $n$Lab already addressed the issue of how Kapranov’s infintesimal paths compare to Schreiber’s?

I should point out that in the context that I am thinking about an $\infty$-Lie algebroid is literally an $\infty$-Lie groupoid all whose $k$-morphism in the given presentation have infinitesimal extension.

Under this identification of $\infty$-Lie algebroids

(by the way: Jim, I saw your query box there only right now, have replied now)

…with special $\infty$-Lie groupoids I show that the infinitesimal path $\infty$-groupoid is the ordinary tangent Lie algebroid.

So, as above, part of the point here is that various classical structures fall into their proper place in a more unified general nonsense.

Posted by: Urs Schreiber on October 31, 2009 1:20 AM | Permalink | Reply to this

### Borisov on infinitesimal paths

I had now a first look at

The impression I get is that the structure that is being described here for a given manifold $X$ is that corresponding to the simplicial object which in degree $k$ is the space of those $k$-simplices in $X$ each of whose edges is a first order infinitesimal, i.e.

$\{ (x_0, \cdots, x_k) \in X^{k+1} | \forall 0 \leq i \lt k : x_i \sim_1 x_{i+1}; \} \,.$

That makes $x_0$ and $x_k$ infinitesimal neigbours of order $k$ and hence implies that functions on this in degree $k$ see the $k$-jets on $X$. Which is what Dennis Borisov has in his construction.

I will have to have another look at Kapranov’s construction that motivates this. What are the motivating constructions and examples for this?

Posted by: Urs Schreiber on October 31, 2009 1:58 AM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

Sections of iterated tangent bundles are used in synthetic geometry for quite a while. There they are called micro-cubes. One of their uses (with addition of markings) is in definition of differential forms (e.g. in the books “Basic concepts of synthetic differential geometry” by R.Lavendhomme, “Models for smooth infinitesimal analysis” by I.Moerdijk and G.Reyes).

However, A.Kock in “Synthetic geometry of manifolds” argues that, as far as differential forms are concerned, marked micro-cubes have redundant information, indeed, a differential form is defined as an alternating form on the micro-cubes, with respect to the natural action of the symmetric groups.

In my paper I define the full infinitesimal groupoid, not the differential forms, and hence I need all parts of micro-cubes, also the symmetric one. This is the motivation for using sections of iterated tangent bundles.

The motivation for Kapranov’s construction is given at length in his paper.

Posted by: Dennis Borisov on November 1, 2009 2:26 PM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

However, A.Kock in “Synthetic geometry of manifolds” argues that, as far as differential forms are concerned, marked micro-cubes have redundant information

Yes. Already in his first book he makes the point that functions on infinitesimal simplices give a more non-redundant encoding of differential forms.

Therefore my previous question: I think one can see that Anders Kock’s results show that there is a canonical isomorphism from the deRham complex of $X$ to the normalized Moore complex of the cosimplicial algebra of degreewise functions on the simplicial object that in degree $k$ is the space of $(k+1)$-tuples of points in $X$ that are pairwise first order infinitesimal neighbours.

Your construction reminds me of what one would get along these lines if instead of $(k+1)$-tuples that are pairwise infinitesimal neighbours one considers the larger space of $(k+1)$-tuples where only each subsequent pair is required to be a first order infinitesimal neighbour pair.

That is of course a natural simplicial object in a smooth topos to study. Do you know how your construction would relate to that?

There are various other simplicial objects on infinitesimal paths here. For instance the Kan-fibrant replacement of the one considered by A. Kock is of interest. That has in degree $k$ the $k$-fold $\infty$-jets.

The motivation for Kapranov’s construction is given at length in his paper.

I meant the motivating examples for your construction. I thought that knowing them would help me understand your construtcion.

Posted by: Urs Schreiber on November 2, 2009 11:13 AM | Permalink | Reply to this

### Kapranov’s construction

Kapranov constructs a Lie-Rinehart algebra with A the algeera of smooth functions and L the space of vector fields, but Lie-Rinehart algebras work for a general commutative A and L = Der(A).

Has anyone tried A, the algebra of continuous functions? is there an obvious black hole defeating any usefulness?

Posted by: jim stasheff on November 2, 2009 2:04 PM | Permalink | Reply to this

### Re: Kapranov’s construction

I thought the algebra of continuous functions on a manifold didn’t have any nonzero derivations.

What’s a ‘Lie-Rinehart algebra’? I’m imagining it’s a commutative ring A and an A-module L with a bracket that’s antisymmetric, obeys the Jacobi identity and has

[x, a y] = x(a) y + a [x,y]

where we also assume that L acts as derivations of A to make sense of x(a).

Posted by: John Baez on November 2, 2009 5:44 PM | Permalink | Reply to this

### Re: Kapranov’s construction

According to this:

Let $R$ be a unital commutative algebra over $k$ (Note: a field of characteristic 0). A Lie-Rinehart algebra over $R$ is a Lie algebra $L$ over $k$, equipped with a structure of a unital left $R$-module and a homomorphism of Lie algebras $\rho: L \to Der_k(R)$ into the Lie algebra of derivations on $R$, which is a map of left $R$-modules satisfying the following Leibniz rule

$[X,rY] = r[X,Y] + \rho(X)(r)Y$

for any $X,Y\in L$ and $r\in R$. We shall write $\rho(X)(r) = X(r)$.

Posted by: Eric Forgy on November 2, 2009 6:17 PM | Permalink | Reply to this

### Re: Kapranov’s construction

Thanks, Eric! Great, that’s exactly what I guessed! – except that I was a bit more general: my A (your $R$) could be any commutative ring, not just a commutative algebra over a field of characteristic zero.

(If it’s not obvious that my definition matches yours except for this, well, that just goes to show how many different ways there are to say the same thing.)

Restricting to characteristic zero is probably smart, since I think the concept of Lie algebra requires (or at least wants) a few extra subtleties when we leave characteristic zero.

Posted by: John Baez on November 2, 2009 6:45 PM | Permalink | Reply to this

### Re: Kapranov’s construction

What’s a ‘Lie-Rinehart algebra’?

You asked this question before here.

I am thinking of a rule that would say that anyone who asks a question a second time after having received an answer is forced to admit that archiving technical discussion on the nLab is a GoodThing™ and to create an $n$Lab page with the answer, so as to never forget it again.

Posted by: Urs Schreiber on November 2, 2009 7:16 PM | Permalink | Reply to this

### Re: Kapranov’s construction

Whoops! Unfortunately if I went along with your new rule I’d need to spend more and more time writing $n$Lab entries as I get more and more senile and forgetful. I don’t want to die while writing $n$Lab entries listing my relatives names…

Posted by: John Baez on November 2, 2009 7:30 PM | Permalink | Reply to this

### Re: Kapranov’s construction

Unfortunately if I went along with your new rule I’d need to spend more and more time writing $n$Lab entries as I get more and more senile and forgetful.

Why is that worse than keeping asking the corresponding questions? It seems that in the long run it saves you time.

But here is the entry for you:

Posted by: Urs Schreiber on November 2, 2009 7:39 PM | Permalink | Reply to this

### Lie-Rinehart and Courant-Dorfman

Thanks for adding to Lie-Rinehart pair the explicit compatibility condition between the actions.

I have now also added a pointer to Dmitry’s Courant-Dorfman algebra. Possibly of interest for that other discussion.

Posted by: Urs Schreiber on November 3, 2009 9:39 AM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

Just to be precise: in terms of synthetic geometry I use micro-cubes (on the group of diffeomorphisms of M, centered at the identity). Micro-cubes are indeed cubes, i.e. they are not just sequences of consecutive first order neighbors, but several sequences, starting at one point, and occasionally converging. But you are right, it is characteristic of them that only consecutive pairs are first order neighbors.

So why do I use these, instead of simplices, where all pairs are neighbors? Or something else? The reason is that I am constructing the full infinitesimal groupoid, and not differential forms.Here the construction by Kapranov becomes central.

Looking at his paper (and the introduction to mine) you will see the necessity of substituting the usual Lie algebroid of vector fields with the corresponding free Lie algebroid.

If you accept this free construction as the infinitesimal analog of the fundamental groupoid, the immediate question is: what is the higher dimensional part that provides the contractions?

In my paper I answer this question, and here is the (synthetic) motivation for my answer: consider the usual Lie bracket of vector fields, in terms of jets (or synthetic differential geometry) this bracket is not a first jet, but second, and it only happens so that the first order parts of it are 0, an hence we can view it as a vector field. In short: to contract an infinitesimal loop we need a 2-jet of a 2-dimensional submanifold.

What kind of infinitesimal object provides such jets? Notice that the two vector fields by themselves define a 2-dimensional object: the whiskers. In this object not every pair of points are first order neighbors, and the contracting submanifold should contain these whiskers. Micro-square is the logical conclusion.

Posted by: Dennis Borisov on November 2, 2009 2:20 PM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

consider the usual Lie bracket of vector fields, in terms of jets (or synthetic differential geometry) this bracket is not a first jet, but second, and it only happens so that the first order parts of it are 0, an hence we can view it as a vector field.

Maybe I am not sure that I am following this heuristic motivation. In your paper you say

Let $\alpha, \beta$ be tw vector fields on $\mathcal{M}$, and assume for simplicity that they commute, i.e. $[\alpha, \beta] = 0$. The left hand side of the equation represents a loop, while the right hand side stands for a constant path.

You introduce this by the words “However, this is not precise.”, suggesting that somehow there is a problem in the standard way of thinking of the Lie bracket.

But, actually, I think the standard way to think of $[\alpha,\beta]$ is in fact not as being the loop given by $\alpha$ and $\beta$, but being the straight line connecting the endpoints of a curve given by $\alpha$ and $\beta$.

For, let’s not assume for simplicity that $\alpha$ and $\beta$ commute. Then there is no loop in the first place.

What kind of infinitesimal object provides such jets? Notice that the two vector fields by themselves define a 2-dimensional object: the whiskers. In this object not every pair of points are first order neighbors, and the contracting submanifold should contain these whiskers. Micro-square is the logical conclusion.

I am inclined to say that it mico-squares may be one conclusion.

I mentioned another one: the $\infty$-groupoid presented by (i.e. the Kan-fibrant replacement) of the Joyal-Kock infinitesimal singular simplicial complex:

- its 1-morphisms are sequences of points, each the first order neighbour of its predecessor

- its 2-morphisms are similarly arbirary triangles obtained by gluing first order infinitesimal triangles,

- its $k$-morphisms are arbitrary $k$-simplices obtained by gluing first order $k$-simplices.

So in degree $k$ these are “$k$-fold $\infty$-jets”.

Does the simplicial object that you construct satify the Kan condition? If not, potentially its Kan fibrant replacement isn’t that different from that of the Joyal-Kock complex?

Posted by: Urs Schreiber on November 2, 2009 8:19 PM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

Posted by: Toby Bartels on November 2, 2009 8:56 PM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

Thanks, fixed. For the record: we are now talking about Dennis Borisov’s new paper titled What is the higher-dimensional infinitesimal groupoid of a manifold? with an eye towards how it relates to another definition that goes back (at least) to A. Joyal and (notably to) A. Kock in its general form and to A. Grothendieck in its realization for schemes (as now recalled at infinitesimal path $\infty$-groupoid - History).

Posted by: Urs Schreiber on November 2, 2009 9:45 PM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

If the vector fields do not commute then the loop is of course not the value of their usual bracket, but the loop obtained by going around the pentagon. If the fields commute then the pentagon is actually a square.

I do not know if my construction is fibrant in any sense.

Posted by: Dennis Borisov on November 2, 2009 10:57 PM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

I do not know if my construction is fibrant in any sense.

In which sense do you think of the construction you give as a higher groupoid? What model of higher groupoid do you have in mind?

Posted by: Urs Schreiber on November 2, 2009 11:34 PM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

If you mean combinatorics (shape) then both cubical and simplicial (in the infinitesimal world they usually come together).

If you mean operations of compositions, then they are given by a thin structure (the composition product) as explained at the end of the paper.

If you mean weakening of the structure, there is no weakening, everything is strict. In particular I do not use fibrant simplicial sets as the model. The homotopy maps do not define compositions, but equivalences, in the same sense as differentials in DGAs do not define compositions. For example: the fundamental groupoid is still the whole degree 1 part (i.e. Kapranov’s construction), and the corresponding cohomology (i.e. the usual Lie algebroid of vector fields) is a result of collapsing equivalence classes, i.e. contracting loops.

Posted by: Dennis Borisov on November 3, 2009 2:29 AM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

If you mean operations of compositions,

For instance, yes. I was thinking of everything that comes with the notion of an $n$-groupoid.

You must have some definition of $n$-groupoid or $\infty$-groupoid in mind, I suppose?

then they are given by a thin structure (the composition product) as explained at the end of the paper.

Ah, I see. I had missed that on first reading. So you are thinking of $n$-groupoids realized in terms of complicial sets?

If you mean weakening of the structure, there is no weakening, everything is strict.

In that your complicial set has unique horn fillers? Is that what you mean by it being strict? But if so, then it would of course be a Kan complex!

Posted by: Urs Schreiber on November 3, 2009 9:02 AM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

Is that pentagonal comment explicated in your paper?

Posted by: jim stasheff on November 3, 2009 1:54 PM | Permalink | Reply to this

### Re: Borisov on infinitesimal paths

Of course: this loop is the difference between the free and the usual Lie bracket.

Posted by: Dennis Borisov on November 3, 2009 4:26 PM | Permalink | Reply to this

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