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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 7th 2010

    I have quietly submitted the beginning of an article on "surface diagrams" on my web. There is still quite a lot left to write up, and it needs to be formatted more prettily, but I thought I'd throw what I have (so far) out there.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeApr 8th 2010

    Awesome! I'll take a look at it as soon as I get a chance.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 8th 2010
    • (edited Apr 8th 2010)

    I came across a little bit of the complexity of surface diagrams in early versions of my thesis work (e.g. the slides here Edit: the slides have gone missing! The post is still there to read, of course - but you have to imagine the pictures, unfortunately), before I switched to a cubical approach. In particular, serious calculations were pretty much impossible, because I had no control over how surfaces were built up - I didn't have a handy decomposition theorem or the like. Then of course higher dimensional generalisations were beyond me, but are a doddle (comparatively speaking) in the cubical case.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 8th 2010

    Thanks, guys -- a few quick comments from me:

    (1) As you can see, I am for the moment considering just "progressive" surface diagrams, which may seem to be a heavy restriction, leaving out the fun stuff about duals in Gray-categories and the like. But I think it's an important theoretical first step to understand well. (For one thing, I think it should be possible to "progressify" a general diagram by refining the stratification with more strata along the sets where certain coordinate projections are nonregular.)

    (2) It seemed to me while I was writing this up that the geometric globularity conditions are technically rather awkward to work with, and that it might behoove one to drop them and adopt a cubical approach well-adapted to (semi-strict) n-fold categories. I've honestly not thought much about such algebraic structures, but Mike's most recent post at the Cafe, and his general enthusiasm for working with n-fold structures, encourages me to begin thinking in that direction.

    (3) I do have a general notion of n-dimensional surface diagram, both progressive and non-progressive, but much work remains in analyzing these. My plan is to get some stuff written down on this in the next few days.

    (4) I seriously need to learn how to draw pictures using whatever gizmos were being discussed at Modelling Surface Diagrams at the Cafe.

    • CommentRowNumber5.
    • CommentAuthorEric
    • CommentTimeApr 8th 2010

    (2) It seemed to me while I was writing this up that the geometric globularity conditions are technically rather awkward to work with, and that it might behoove one to drop them and adopt a cubical approach well-adapted to (semi-strict) n-fold categories. I've honestly not thought much about such algebraic structures, but Mike's most recent post at the Cafe, and his general enthusiasm for working with n-fold structures, encourages me to begin thinking in that direction.

    My love for cubes is no secret so I'm intrigued (if not totally lost) by your comment. You won't be surprised if I make a random statement and hope it might be relevant...

    Every smooth manifold can be triangulated, right? (I hope!)

    If so, I'm pretty sure not every manifold can be cubulated (?) :)

    However, I've guessed ("conjectured" for what I do is too strong) that any "extruded" manifold can be diamonated. Basically, given any smooth manifold, you first triangulate it. Then, with this triangulation you "extrude" the edges into an orthogonal dimension (I like to think of it as time, but it doesn't matter). If you do this correctly, you end up with directed cubes. This process I call diamonation (ericforgy).

    So cubes are actually a very natural shape for modeling the evolution of geometries just as simplices are good at modeling static geometries.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 8th 2010

    Eric -- interesting. I guess I'm a little confused, though, because I would have thought simplices can be cubulated (can't they?), so couldn't you just cubulate any triangulation in order to cubulate any manifold? Maybe I'm being stupid here, but I'm tempted just to write down how this might go.

    So far triangulations (or cubulations) of manifolds haven't entered my thinking much here, but that's not to say they won't at some stage. What's nice about cubes here is that their pastings model algebraic compositions so nicely. The types of cubes I'm using here are stratified into strata which represent cells in pasting diagrams, hence their relevance to algebraic pastings.

    • CommentRowNumber7.
    • CommentAuthorEric
    • CommentTimeApr 8th 2010

    Hi Todd. I hope I don't distract you too much.

    so couldn't you just cubulate any triangulation in order to cubulate any manifold?

    It is possible (even likely!) I am misremembering things. I haven't seriously thought about this stuff in more than 8 years and you can tell I never reached too high a level of sophistication even at my peek of nerdiness, err.. I mean mathematical prowess.

    If you can non-degenerately cubulate a smooth manifold, then I'd be a little surprised, but not extremely surprised. If it is possible, I'd like to see how for sure.

    I thought I remembered reading it somewhere.

    So far triangulations (or cubulations) of manifolds haven't entered my thinking much here

    I warned you my comment was probably random :)

    What's nice about cubes here is that their pastings model algebraic compositions so nicely.

    I can understand, in cartoon fashion at least, what it means to paste things together, but I imagine you are talking about something more sophisticated. I should clarify that I'm not talking about any old cubes, but "directed" cubes. For example, pick up a 3-cube (it helps to demonstrate if you have a Rubik's cube laying around) and pick a corner and call it x. Call the opposite corner y. A directed cube (as I mean it) has directed edges flowing from x to y with no "turning back". These directed cubes paste together nicely to form directed spaces.

    I like to think of the direction as a flow in time. The physicist in me likes to draw time "upward" on the paper, so my directed cubes tend to stand up on their corners. Hence, they tend to look more like diamonds than cubes, which is why Urs and I called a directed n-cube an n-diamond.

    Now, if you look at a Rubik's cube down the "time axis" i.e. along the diagonal of the cube, you will see a triangulation and its dual. Something like what I drew at diamonation (ericforgy).

    So when you project a diamond complex down through time, you get a simplicial complex. Or something...

    I'm sure what I just said is not completely right, but I'm also sure there is some truth to it. It might just take some effort to make it precise.

    PS: Whoa! Nice! Math rendering in preview! Andrew rocks :)

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeApr 8th 2010

    Hooray for n-fold things! (-: Unfortunately I don’t know if anyone has even written down a notion of “Gray triple category,” but presumably one could do it if required.

    Blender is an awesome tool. I recommend reading through at least some of the user manual, and then maybe looking at some of the .blend files that were being posted at those discussions.

    • CommentRowNumber9.
    • CommentAuthorEric
    • CommentTimeApr 8th 2010
    • (edited Apr 8th 2010)

    By the way Mike, I printed out your new preprint and read it on the way home. I didn’t absorb everything, but some things are starting to click. I also recently stumbled onto another paper of yours after googling “quintet” where you talked about Sq(C)Sq(C), but I can’t find it now.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 8th 2010
    • (edited Apr 8th 2010)

    Thanks for the tip, Mike – I’ll look into this.

    Eric: don’t worry – if you’re distracting me, then it’s because I want to be distracted. :-)

    Indeed, these are standard geometric cubes [0,1] n[0, 1]^n I’m talking about, and so they have directedness built in by increasing coordinates. (What I’m doing is not very mysterious: if you know what string diagrams in the standard square are, then I am just doing the analogues in higher dimension.) And indeed, the time dimension is intuitively important, as one of the most important constructions in what I’m doing is modeling deformations of 2-d string diagrams through time as surface diagrams in 2+1 dimensions, and more generally deformations of (hyper)surface diagrams in the n-cube as (hyper)surface diagrams in the (n+1)-cube. Algebraically, these deformations correspond to higher-order interchangers.

    The pasting operations are not at all sophisticated; it’s just like pasting two string diagrams together by pasting their ambient squares along horizontal or vertical edges. One just has to add enough conditions so that the pastings are smooth on submanifolds (e.g., that edges inside square string diagrams meet up smoothly, without creating angles).

    Regarding the cubulation problem, here is what I’m thinking, in more detail:

    (1) Recall that a triangulation of a topological space YY is a simplicial set XX together with a homeomorphism h:RXYh: R X \to Y, where RR denotes the geometric realization functor. (I guess most people use simplicial complexes instead of simplicial sets, but let’s not worry about this for now.) Explicitly, RXR X is given by a coend formula

    nΔX(n)σ(n)\int^{n \in \Delta} X(n) \cdot \sigma(n)

    where σ:ΔTop\sigma: \Delta \to Top is the standard affine simplex functor.

    (2) Similarly, a cubulation of YY is a cubical set CC together with a homeomorphism h:R cubCYh: R_{cub}C \to Y where R cubR_{cub} denotes the realization functor for cubical sets Set Cube opSet^{Cube^{op}}. Explicitly, R cubCR_{cub}C is given by a coend formula

    R cubC= mCubeC(m)(m)R_{cub}C = \int^{m \in Cube} C(m) \cdot \Box(m)

    where :CubeTop\Box: Cube \to Top is the standard geometric cube functor.

    (3) There should be “cubulation” functor for standard simplices, Σ:ΔSet Cube op\Sigma: \Delta \to Set^{Cube^{op}}, such that the affine simplex functor σ:ΔTop\sigma: \Delta \to Top is naturally isomorphic to the composite

    ΔΣSet Cube opR cubTop\Delta \stackrel{\Sigma}{\to} Set^{Cube^{op}} \stackrel{R_{cub}}{\to} Top

    In the informal pictures I’ve been drawing, the nn-dimensional simplex is cubulated by n+1n+1 many cubes. Let Σ n\Sigma_n denote the cubical set that is the value of Σ\Sigma at nΔn \in \Delta.

    (4) Assuming (3) works out, then given a triangulation (X,h:RXY)(X, h: R X \to Y) of a space YY, we have isomorphisms

    Y nX(n)σ(n) nX(n)( mΣ n(m)(m)) m( nX(n)Σ n(m))(m)\array{ Y & \cong & \int^n X(n) \cdot \sigma(n) \\ & \cong & \int^n X(n) \cdot (\int^m \Sigma_n(m) \cdot \Box(m)) \\ & \cong & \int^m (\int^n X(n) \cdot \Sigma_n(m)) \cdot \Box(m) }

    where in the last line we used a “Fubini theorem” for interchange of coends. Thus, defining the cubical set CC by

    C(m)= nX(n)Σ n(m)C(m) = \int^n X(n) \cdot \Sigma_n(m)

    we have a homeomorphism Y mC(m)(m)=R cubCY \cong \int^m C(m) \cdot \Box(m) = R_{cub} C, i.e., we obtain a cubulation of YY.

    So it’s all on (3), and assuming it pans out, I’ll put some things down in the Lab.

    • CommentRowNumber11.
    • CommentAuthorEric
    • CommentTimeApr 8th 2010

    If the Gods of math granted me two wishes, my second wish would be to have enough knowledge to throw around coend formulas like that :) gulp!

    I don’t know if it matters, but I think what I vaguely remember was in relation to “smooth manifolds” and not just topological spaces. Does that make any difference? The problem had to do with the fact the cubes had to become degenerate or something like that.

    Speaking of Fubini and cubes and simplices. I had an appendix in my dissertation “Integration over the Join of Simplices” (pdf page 109) and a subsection “A Fubini-Type Integral Identity”. I wonder if that could be related to your step (4).

    • CommentRowNumber12.
    • CommentAuthorEric
    • CommentTimeApr 8th 2010
    • (edited Apr 8th 2010)

    Oh. Here is a challenge. Try to 2-cubulate S 2S^2. You can’t without using a degenerate square, i.e. a triangle. That is a simple coordinate mapping problem (edit: which is related to “you can’t comb the hair of a bowling ball” :))

    My diamonation (ericforgy) conjecture would go along the lines of, “Although you cannot 2-cubulate S 2S^2, you can 3-cubulate S 2×IS^2\times I.”

    More generally, “Although you cannot, in general, nn-cubulate any smooth nn-dimensional manifold \mathcal{M}, you can always (n+1)(n+1)-cubulate ×I\mathcal{M}\times I.”

    PS: A proof of the diamonation conjecture would actually be VERY significant for computational geometry and computational physics and likely open up entirely new branches of study.

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 8th 2010
    • (edited Apr 8th 2010)

    The 2-sphere? But that’s easy: that’s homeomorphic to the boundary of a cube, which consists of squares. Thus it’s homeomorphic to the realization of a cubical set.

    Maybe you have something more refined in mind, but the reason why I carefully laid out the definitions as I did was to avoid confusion and be clear, and not talk past each other.

    • CommentRowNumber14.
    • CommentAuthorEric
    • CommentTimeApr 8th 2010

    Maybe you have something more refined in mind

    No. I’m just being dense :)

    It’s past my bedtime and I had a REALLY long day. I was thinking in terms of trying to put smooth coordinates on S 2S^2 for some reason.

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeApr 9th 2010

    @Eric #9, were you talking about this paper? (A list of all my papers and preprints can be found on my home page.)

    • CommentRowNumber16.
    • CommentAuthorEric
    • CommentTimeApr 9th 2010

    @Mike: Nope. It was this one. In particular, the discussion about Sq()Sq(\mathcal{B}) in the section on “Uniqueness of the bicategorical trace” resonated with things I’ve been thinking about (but at a “engineering” level). tileorder is also very interesting.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2010

    Todd wrote:

    Maybe you have something more refined in mind
    

    Eric replied:

    No. I'm just being dense :)

    Actually, no, you have something else in mind, Eric! :-)

    What you used to be looking for was a way to cubulate Lorentzian spaces such that all edges of all cubes are lightlike.

    • CommentRowNumber18.
    • CommentAuthorEric
    • CommentTimeApr 9th 2010

    Actually, no, you have something else in mind, Eric! :-)

    Yeah yeah. What Urs said. That’s what I meant :)

    What you used to be looking for was a way to cubulate Lorentzian spaces such that all edges of all cubes are lightlike.

    Yes yes. BUT… I want to model stuff sitting in space too and evolving in time. For example, let’s say I want to simulate the radar scattering from a metallic aircraft. From a differential geometry point of view, we can think of a metallic objects as a region of a topological space “removed” from the manifold representing space. The physical boundary conditions are equivalent to just removing that region of space.

    It would be hard to model an aircraft using cubes, but reletalively straightforward, e.g. using CAD software, to model it with triangles. So given a triangulated model of some space, I want to extrude it into a Lorenztian spacetime in such a way that I end up with cubes. This is diamonation (ericforgy)

    So yeah, we want a cubulation of some Lorentzian spacetime, but we want to obtain it from some kind of discretization of space, e.g. a triangulation. If we could do that, engineers would take note of our stuff. Modeling Minkowski space is not very fun from an engineer’s perspective. You have to put SOMETHING in your space :)

    But yeah, I almost responded saying that we want to cubulate stuff in a way that we end up with a directed space. I don’t know if the cube model of S 2S^2 would be a directed space.

    The fact that any smooth manifold can be cubulated is cool though. Todd’s coend kung fu was a neat show :)

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 9th 2010

    @ Todd #10

    is there some way of cooking up (3) by looking at \Delta vs \Box and some functor between them? Urs did some magic with cubical and simplicial sets on his web at some point, and the relation to orientals, I think.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2010

    Urs did some magic with cubical and simplicial sets on his web at some point, and the relation to orientals, I think.

    Hm, not sure what you are thinking of here.

    • CommentRowNumber21.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 9th 2010

    Maybe it wasn't magic then :), and I can't remember where it was either. I'll have a look later.

    • CommentRowNumber22.
    • CommentAuthorEric
    • CommentTimeApr 9th 2010

    I thin David is referring to the page where I drew some simplices inside a cube (?) I don’t remember where it was either.

    Btw, is there a way to see a list of files uploaded to the nLab?

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2010
    • (edited Apr 9th 2010)

    Oh, that, yes. That’s at interval object. and used at my differential cohomology in an oo-topos – survey.

    So there is the proposition that given a cartesian interval object 0I1 0 \to I \leftarrow 1 in a category CC one can naturally turn the assignment [n]I n [n] \mapsto I^n (i.e. nn-fold carteian product on the right) into a cosimplicial object

    Δ I:ΔC \Delta_I : \Delta \to C

    by taking face maps that identify inside each nn-cube I nI^n an nn-simplex. A “collared n-simplex”.

    That construction gives then for each cartesian interval object a notion of fundamental oo-groupoid of any object XCX \in C, by setting Π(X):=Hom C(Δ I ,X)sSet \Pi(X) := Hom_C(\Delta_I^\bullet, X) \in sSet . With a bit of cofibrant replacement thrown in, this construction extends to an \infty-functor that sends every object in the oo-topos over CC to is path oo-groupoid. That’s described at path oo-groupoid.

    In fact, the same construction also refines to produce the “path (oo,1)-category” of an object in an (oo,2)-topos. Some comments on this are at Lorentzian manifold in the section on the path 2-category of a causal Lorentzian space.

    • CommentRowNumber24.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 9th 2010

    David #19: I think it’s pretty easy to produce a functor Σ:ΔSet Cube op\Sigma: \Delta \to Set^{Cube^{op}} that does what I asked it to; it’s not the cubulation of the simplex I originally had in mind though. But while mulling that over, I had some some technical questions about triangulations, which I’ll ask in a separate discussion.

    (Oh, yes, and this will be all about surface diagrams. Kidding.)

    So, let’s recall that one way of defining the affine simplex functor σ:ΔTop\sigma: \Delta \to Top is by taking advantage of the fact that Δ\Delta (with augmentation, which is the version of Δ\Delta I normally use) is the walking monoid, so all we need to do is cook up an appropriate monoidal structure on TopTop and a monoid therein, and the walking monoid property takes care of the rest: we get an induced monoidal functor σ:Δ:Top\sigma: \Delta: \to Top. So: take the monoidal product on TopTop to be “topological simplicial join”: the join XYX \star Y of two spaces XX, YY may be defined for our purposes by pushout of the diagram

    Xπ 1X×Y1 X×{0}×1 YX×I×Y1 X×{1}×1 YX×Yπ 2YX \stackrel{\pi_1}{\leftarrow} X \times Y \stackrel{1_X \times \{0\} \times 1_Y}{\to} X \times I \times Y \stackrel{1_X \times \{1\} \times 1_Y}{\leftarrow} X \times Y \stackrel{\pi_2}{\to} Y

    and now take the monoid in TopTop to be the 1-point space with its unique monoid structure.

    Now mimic this construction for cubical sets in the following way: there is a promonoidal structure on the site CubeCube where, if XX, YY are objects of CubeCube, we define XYX \star Y to be the pushout (in cubical sets) of the same diagram as above (but replacing ×\times by the monoidal product \otimes of CubeCube). Note that even though \otimes is not cartesian product in CubeCube (because CubeCube lacks diagonals), the monoidal unit 11 of Cube is terminal, so we do have “projection maps” to work with to produce the analogous diagram in CubeCube (the II is the monoidal generator of CubeCube). This promonoidal product on CubeCube induces a monoidal product on cubical sets which I’ll call “cubical simplicial join”. As before, the terminal cubical set is a monoid with respect to this monoidal product, so by the walking monoid property we obtain a monoidal functor

    Σ:ΔSet Cube op\Sigma: \Delta \to Set^{Cube^{op}}

    which plays a role analogous to the affine simplex functor into TopTop.

    The crucial observation is that geometric realization R cub:Set Cube opTopR_{cub}: Set^{Cube^{op}} \to Top takes cubical simplicial joins to topological simplicial joins, when restricted to the full subcategory given by the essential image of Σ\Sigma. By definition, R cubR_{cub} takes \otimes on representable objects to cartesian products of geometric cubes, and preserves colimits and in particular pushouts. So it takes cubical simplicial joins of representable cubical sets to topological simplicial joins. It is not unfortunately not generally true that

    R cub(XY)R cub(X)R cubY(1)R_{cub}(X \star Y) \cong R_{cub}(X) \star R_{cub} Y \qquad (1)

    for general cubical sets XX, YY; for example, if X=0X = 0 is initial, then the left side is initial but the right side is isomorphic to R cub(Y)R_{cub}(Y). Basically, the problem is that topological simplicial join \star does not preserve all colimits in each of its separate arguments. It does however preserve connected colimits in each of its separate arguments. Therefore, the isomorphism (1) obtains if XX and YY are connected colimits of representables, and in particular applies when XX and YY belong to the essential image of Σ\Sigma.

    We conclude that σ:ΔTop\sigma: \Delta \to Top and R cubΣ:ΔTopR_{cub} \circ \Sigma: \Delta \to Top take monoidal products in Δ\Delta to topological simplicial joins, and both take the walking monoid of Δ\Delta to the one-point space. By the universal property of Δ\Delta, it follows that these functors are naturally isomorphic (as monoidal functors, even), which is what we (or I, anyway) wanted to show.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeApr 9th 2010
    • (edited Apr 10th 2010)

    To return to the original topic of this thread, I have read through what you wrote at the link you gave, Todd. It’s very clear and well-written, although I had trouble following some of it without the pictures. (-: A few questions:

    1. Can the Joyal-Street work you mentioned in terms of 2-computads be found written anywhere? I’m only familiar with their work on various kinds of monoidal categories and I don’t remember computads appearing. But my memory could be failing me.

    2. What is your opinion on the philosophical relationship between geometry and “semi-strictness”? At one point you seemed to be saying that the right notions of semi-strictness could be suggested by the geometry. But then later on, it felt to me as though you were putting the Gray kind of semi-strictness into the geometry “by hand” by restricting the domain and codomain to be “generic.”

    3. Have you thought about whether an “(,n)(\infty,n) version” could be done without needing to do any quotienting by deformation, then later quotienting out the higher cells to obtain the corresponding result for the nn version? IIRC Lurie’s approach to the cobordism hypothesis does something like this, saying that it’s actually easier that way; could something like that be “computadified” to work in this situation too? (Am I right that the “cobordism hypothesis” is essentially the same idea, but in the context with duals, and only for the computad having one generating object and no generating k-morphisms for k>0k\gt 0?)

    Regarding double and nn-fold categories, after thinking about it a bit more, I’m not convinced that they would actually be any easier. Just thinking about double categories for the moment, they won’t have the “globularity” condition as stated, but I think they’ll have their own “cubicality” condition which is just as complicated. I would expect string diagrams for double categories to have two kinds of 1-strata, corresponding to vertical and horizontal arrows, and we would have to require that the “horizontal” 1-strata meet only the “vertical” boundary and vice versa. Moreover there are more complicated “net conditions” saying in exactly what ways the 0-strata can be connected by 1-strata of the two types, such as those at tileorder. And if you want to model double categories according to the usual definition, you’d also have to somehow exclude pinwheels, which might turn out to be a huge pain. (Alternately, one could modify the definition of double category to allow pinwheel composition, since pinwheels can actually be composed in all naturally-occurring double categories that I know of, but that itself would take some work and thought.)

    So, while I do definitely want to have string diagrams for nn-fold things and (n×k)(n\times k)-things, I’m not sure that we should expect them to be any easier than the globular case; rather the reverse, in fact.

    • CommentRowNumber26.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 10th 2010
    • (edited Apr 10th 2010)

    Thanks Mike – I really appreciate the feedback; it motivates me to press on. I’ll just try to respond to the first two queries; the third I’ll need time to think about.

    If I recall correctly, Joyal-Street talk instead about things like the free monoidal category generated by a “tensor scheme”, and this author-pair never mentions computads. I probably picked up the idea that there’s a natural extension to computads either by talking with Street in his office or by reading things he wrote himself. I’d have to dig around to recall where, but there’s a good chance he discusses this in his 1993 lectures given at Buffalo. Could it be in his Handbook of Algebra article, reference 45 in his publication list? My memory is that he discusses both computads and string diagrams in that article.

    As for (2): we probably agree that the general idea of semi-strictness has many potential implementations (cf. the “unit conjecture” that n-categories are equivalent to structures where all the data except unit data are strict), and philosophically I’m not particularly wedded to the Gray kind. But I am envisioning the Gray kind of semi-strictness as generally amenable to geometric description, in terms of stratifying the space of surface diagrams by “degrees of coincidence”. So that if (for example) one isn’t sure what the Gray kind of semi-strict 4-categories should be exactly, one might have a good chance of finding out by performing a geometric analysis of the space of 4d-diagrams as stratified by coincidence-degree (in other words, studying how regions where diagrams are generic are separated by “walls” of greater coincidence). Needless to say, coincidence refers to geometry of a fundamentally extrinsic sort, fundamentally tied to coordinate embeddings, but is not less “geometric” for all that, and it’s something that I think real geometers like Scott Carter could sink their teeth into, both in terms of giving clear and useful definitions, and in terms of analysis, which would then be used to reap algebraic insights.

    As for n-fold categories – you may well be right; I admit I have not thought too hard about it. Yes, I guess one problem would be telling horizontal 1-cells apart from vertical ones (what would the progressivity conditions be to distinguish them?), and that’s just the first step. Ah well. Perhaps I was just reacting against the technical rigidity of globularity as I have defined it.

    (On a side note, I am interested in this pinwheel being interpretable in naturally occurring double categories – could I get you to elaborate?)

    • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeApr 10th 2010

    It could be that article, thanks. I’ll look at it when I get a chance; unfortunately it doesn’t seem to be available on the web even with a university library access.

    I think I see what you are getting at with semi-strictness; it’s sort of a two-way street? On the one hand, the need for semistrictness suggests to us restrictions to put on the geometry, whereas the geometry resulting from those restrictions may suggest to us a more precise version of the correct semistrict algebraic structure?

    My first guess at the progressivity conditions for double-category strict diagrams would be that the strings representing vertical arrows should be horizontally progressive (project nonsingularly to the x-direction) and the strings representing horizontal arrows should likewise be vertically progressive. But actually, I think there may be some ordering/partition condition that will need to be imposed on the edges connecting to each vertex as well, since a given square can only have a top and bottom which are horizontal arrows and a left and right which are vertical ones. With only the progressivity condition, you could have two vertices connected by arbitrarily many edges at about a 45% angle which alternate between ones labeled “vertical” and ones labeled “horizontal,” and that’s certainly impossible in a double category.

    Finally, I wrote a bit about composing pinwheels here.

    • CommentRowNumber28.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 10th 2010

    Yes, you could look at it as a two-way street. I also think that the idea of stratifying the space of surface diagrams by degrees of coincidence has a spiritual kinship to things like Cerf theory or singularity theory, which geometers are accustomed to thinking about, and that this stratification is a natural one to consider (so that things like “generic diagrams” are independently sourced in the geometry, and not put in by hand as it were by algebraic demands).

    It might already be clear to you (of course, I’m implicitly talking to others as well), but in for example the space of all planar string diagrams, the subspace of generic string diagrams is open and dense, and divided up into connected regions or “chambers”. To get from one chamber to another, one has to cross a “wall” where two 0-strata are at the same height. Deformations or paths from one generic string diagram to another which manage to stay wholly within a chamber correspond to identity cells in the free Gray-category on the terminal Gray-computad. A path which crosses a wall corresponds to a nontrivial 3-cell isomorphism.

    Now, in the singular space of nongeneric string diagrams, the string diagrams of coincidence-degree 1 constitute an open and dense subspace, again divided into chambers separated by walls consisting of diagrams of greater degree. Consider for example the space of configurations of three points in the square, and consider two paths from one configuration to another whose trajectories or graphs (from the perspective of projection to the yz-plane) look like the two sides of a Reidemeister III move. A homotopy between those two paths will have to pass through such a higher-order wall, at a string diagram where all three points are at the same height; from the point of view of the tangle projection, this would be where we have a triple crossing during the Reidemeister III move. Algebraically, it would correspond to a “Yang-Baxterator” in a free Gray-type semi-strict 4-category.

    (It would obviously help to be in the same room to talk about this, failing this, I’ve got to figure out how this Blender works.)

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2010
    • (edited Apr 10th 2010)

    Todd,

    concerning your functor Σ\Sigma: looks good. Do you want to add that to triangulation?

    We might want to create a page with more on the definition of join that you are using here. It currently appears in the entry join of quasi-categories, but should probably be discussed in a wider context.

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2010
    • (edited Apr 10th 2010)

    Oh, and Eric,

    in case you are wondering what the abstract machinery that Todd invokes means: it’s actually very simple, that cubical set representing the nn-simplex which Todd’s functor Σ\Sigma builds is effectively the nn-cube with one degenerate face, roughly speaking.

    For instance the triangle is regarded as a square whose top edge is collapsed to a point. This “collapsing” is precisely what that join pushout that Todd wrote down achieves.

    • CommentRowNumber31.
    • CommentAuthorEric
    • CommentTimeApr 10th 2010

    Urs said:

    Oh, and Eric,

    in case you are wondering what the abstract machinery that Todd invokes means: it’s actually very simple, that cubical set representing the nn-simplex which Todd’s functor Σ\Sigma builds is effectively the nn-cube with one degenerate face, roughly speaking.

    For instance the triangle is regarded as a square whose top edge is collapsed to a point. This “collapsing” is precisely what that join pushout that Todd wrote down achieves.

    Thanks. There is little chance I could have parsed that from what Todd wrote :)

    That sounds pretty much like what I wrote about in my dissertation I referenced above when I said:

    Speaking of Fubini and cubes and simplices. I had an appendix in my dissertation “Integration over the Join of Simplices” (pdf page 109) and a subsection “A Fubini-Type Integral Identity”. I wonder if that could be related to your step (4).

    • CommentRowNumber32.
    • CommentAuthorMike Shulman
    • CommentTimeApr 10th 2010

    @Todd: okay, I think I see what you’re getting at with genericity. This is neat stuff.

    • CommentRowNumber33.
    • CommentAuthorMike Shulman
    • CommentTimeApr 12th 2010

    @Todd #26, you’re right, it is in the Handbook of Algebra article. Thanks! He also says at the very end that “Verity and the author have developed the use of surface diagrams for tricategories generalising the use of string diagrams for bicategories.” (-:

    The use of “string-like diagrams” for parity complexes in that paper is also new to me; I don’t understand it yet, but maybe that’s because I don’t understand parity complexes. Maybe Daniel Schaeppi can explain it to me.

    • CommentRowNumber34.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 13th 2010
    • (edited Apr 13th 2010)

    There are some gorgeous string-like diagrams for simplexes (i.e., orientals) and cubes which were drawn up by Iain Aitchison a long time ago. You should ask Street to show them to you sometime (or maybe he’d mail a photocopy).

    The Parity Complexes paper is a fearsome technical work, but the orientals paper is more approachable.

    • CommentRowNumber35.
    • CommentAuthorMike Shulman
    • CommentTimeApr 20th 2010

    There are a couple of things in Street’s HA article which seem not quite right to me.

    He doesn’t require the boundary points of his graphs to be anchored at the bottom of a square or a strip, rather he just lets them float anywhere in R 2R^2 as long as each boundary point is either above or below all the non-boundary vertices. Then he defines a deformation to be any homeomorphism of R 2R^2 which preserves the aboveness order on edges. That seems to me as though it would identify graphs that should be different. E.g. if α\alpha is an endo-2-cell of the empty composite at xx and ff is an endo-1-cell of xx, then the two whisker composites αf\alpha f and fαf\alpha should be different in general, but it seems to me that their string diagrams in Street’s sense would be deformation-equivalent in his sense.

    Also, in his definition of graphs labeled by a computad, he doesn’t explicitly require any labeling of the plane regions. Presumably he intends that the necessary labeling can be recovered from the labeling of the 1-cells, but it seems to me that this way you would get labeled graphs that are impossible to compose. E.g. if α\alpha and β\beta are endo-2-cells of the empty composites at xx and yy respectively, then the graph with two inner vertices and no edges and no boundary could be labeled by α\alpha and β\beta if you don’t require that plane regions also be labeled, but of course α\alpha and β\beta can’t actually be composed.

    • CommentRowNumber36.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 26th 2010

    Hi Mike. Sorry for slowness in response; just returning to the internet after an irritating hiatus.

    I haven’t looked at that article in a long time (and it would probably take me a while to dig it out), but from what I can tell, I suspect you are right on all counts. Certainly his notion of deformation looks very wrong to me, and you make a convincing case that he can’t get away with suppressing labels on the planar regions. It would also be weird to me if he could get away without anchoring the boundary points to the horizontal boundaries of a strip, since Joyal-Street apparently took pains over details like this, and even challenge the reader to improve on their setup!

    It’s odd, because Street is usually so careful…

    • CommentRowNumber37.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 26th 2010

    If I can throw out an overview question, will your work, Todd, help with the material we discussed on a long thread back here, which dealt with singularities, fundamental n-categories with duals of stratified spaces, and the generalized tangle hypothesis?

    • CommentRowNumber38.
    • CommentAuthorMike Shulman
    • CommentTimeJun 6th 2010

    In case anyone hasn’t been checking it, Todd has added substantially to his surface diagrams page since this thread was begun. It’s continuing to be very good! I have a couple more questions.

    1. The notion of “pseudogroup on a topological space” seems to me very much like a “continuous action” of an inverse semigroup. Is there a connection?

    2. The statement “Free(C)Free(C) may be described up to 2-equivalence as a 2-category whose 2-cells are pairs ([X],ϕ:U(X)C)([X],\phi:U(X)\to C), where ϕ\phi is a 2-computad map” looks to me very much like the general description of a parametric right adjoint functor. However, I’m pretty sure that the “free 2-category” monad on the (presheaf) category of 2-computads is not itself p.r.a., since it “includes” the “free commutative monoid” functor acting on the 2-cells with nullary source and target. My first guess at what’s being said is that “labeled string diagrams” are a p.r.a. functor, and “deformations” are a p.r.a. equivalence relation on this functor whose quotient is the free 2-category monad; thus we have a “p.r.a. presentation” of the latter monad. Does that seem right?

    3. More of a comment/suggestion: I think appendix B would be a nice start to a page on the main nLab about o-minimal structures

    I’ve also been thinking some more about string diagrams for double categories. I still don’t think they’re any easier than ones for 2-categories, but I don’t think they’re as hard either as I thought they might be back in comments 25 and 27. Right now I think what you want for the “rectangularity” condition (the counterpart of “globularity”) is that the “vertical” strings are restricted to make an angle of no more than 45 45^\circ with the vertical axis, and the “horizontal” strings make a similar angle with the horizontal axis, which isn’t too bad. I’m currently having a bit of trouble with the details, though.

    • CommentRowNumber39.
    • CommentAuthorTim_Porter
    • CommentTimeJun 6th 2010

    On pseudogroups/inverse semigroups etc., there is a result of Ehresmann and Wagner(?<- spelling is problematic) which shows how any inverse semigroup can be embedded in an ‘ordered groupoid’, There is a related and earlier result that any inverse semigroup can be embedded in one given by partial bijections on a set. This is a form of Cayley permutation representation theorem. There are good sources to this material in Mark Lawson’s book,Inverse semigroups: the theory of partial symmetries. He has put some material from his Ottawa lectures on the web at http://www.ma.hw.ac.uk/~markl/ottawa.html.

    • CommentRowNumber40.
    • CommentAuthorMike Shulman
    • CommentTimeJun 6th 2010

    Thanks, Tim. I have heard of those results; in fact they are part of what made me think that there should be a connection, a pseudogroup seeming like a sort of “ordered groupoid” with order being inclusion on open subsets of X.

    • CommentRowNumber41.
    • CommentAuthorTim_Porter
    • CommentTimeJun 6th 2010

    Have a look at Mark’s notes. He is an excellent expositor and also knows the history of the subject very well. Perhaps we need a summary of that stuff somewhere, but I am not sure where in the Lab it might best go.

    • CommentRowNumber42.
    • CommentAuthorMike Shulman
    • CommentTimeJun 6th 2010
    • (edited Jun 6th 2010)

    Well some of it could go at inverse semigroup and pseudogroup

    Edit: I started a page at inverse semigroup.

    • CommentRowNumber43.
    • CommentAuthorJohn Baez
    • CommentTimeJun 6th 2010
    So I guess I'd need to know Todd's secret password to see what he's doing on his web?

    (I forget if I have some secret superpowers that allow me to see everything on the nLab...)
    • CommentRowNumber44.
    • CommentAuthorMike Shulman
    • CommentTimeJun 6th 2010

    Todd’s page is not publically editable, but it is “published” and hence publically viewable. But it doesn’t seem to be possible to make links to the published version from here (is it?) so when I type Surface diagrams (toddtrimble) it’ll go to the place prompting you for a password. But if you manually change the “show” in the URL to “published,” then it’ll take you where you want to go.

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