Nils Baas has been emphasizing for many years, in print and in private communication, the conviction that the usual notions of n-category, infinity-category omega-category in higher category theory are not naturally suited for describing
extended cobordisms such as appearing in
hierarchical systems such as appearing in
The point is essentially that the directedness of morphisms and — related to that — the binary notion of source and target in categories and higher categories are notions alien to these contexts, which in applications have to and are essentially removed again in a second step by adding extra structure and requiring further properties, such as various monoidal structures and dualities, which allow to change the direction of morphisms, to collect objects together, etc.
In contrast to that, Baas pointed out that more naturally the above situations are thought of from the beginning in terms of hierarchies of what he calls bonds, where, quite generally, a bond is an object equipped with information of how a collection of sub-bonds sits inside it, bound by the bond.
A sketch of a generic such a situation of hierarchical bonds is a diagram
where a bond binds sub-bonds and which in turn bind sub-bonds .
For instance might be an extended cobordism with three boundary components which in turn have pieces of boundary components .
Nils Baas has made, in print and in private communication, suggestions for a formalization of such systems of hierarchical bonds and coined the term hyperstructures for these. Some references are
Nils Baas, Introducing abstract matter (pdf)
N.A.Baas; Hyperstructures as Abstract Matter , Advances in Complex Systems, vol.9 , no 3(2006), 157-182.
N.A.Baas, Hyperstructures, Topology and Datasets , Axiomathes 19(2009), 215-221.
N.A.Baas, New Structures in Complex Systems (to appear in The European Physical Journal).
The notion of replacing morphisms by bonds is familiar and concretely realized at least in a hierarchy of depth 2 in the context of groupoidification and geometric function theory, where morphisms are replaced by spans. Regarding these as cospans in the opposite category produces diagrams like the above sketch of a generic bond system.
David Corfield, Category Theory and Biology
Mike: Shouldn’t we allow “oriented bonds” as well? I am thinking of the case of, say, rings and modules, where a module that has right actions by rings and and a left action by should have as “sub-bonds” but with different “orientations.” And the “gluing” operation is tensor product, which only works if one module is a left module and the other is a right module. This example also goes up in dimension, for instance modules over algebras over rings. However, it seems that one would also needs this for cobordisms; aren’t you only allowed to glue cobordisms along boundaries whose orientations match?
Concerning the cobordisms: in the cospan picture, every boundary component explicitly comes with its embedding into the cobordism. If two boundary components are equal as unembedded objects, then the pushout will automatically glue the corresponding cobordisms in the right way by identifying the components as embedded pieces. Do you see what I mean. Maybe I am mixed up about this.
Mike: Yes, the point is that a lot of these examples aren’t really dagger at all, only compact closed (which I would prefer to call “autonomous”). My point about cobordisms has to do with orientations. If and are two manifolds sharing a boundary component , then I can glue them along it, but if they are oriented then the gluing will only inherit an orientation if ’s boundary component is and ’s boundary component is . So it’s just like rings; the category of oriented cobordisms is not dagger, only compact closed. An oriented cobordism from to is not the same as an oriented cobordism from to , but it is an oriented cobordism from to .
Urs: True. I am not sure about the best answer yet, but have thought about the following:
Suppose we have a span
and want to think of its index objects and to carry an orientation which gets reversed when we flip the legs around to
We might be able to emulate this within the setting described below by supposing that there are two different dummy objects and around which behave at least to some extent like initial objects and that the original span was secretly really the multispan:
Now, if in this multispan we want to exchange the original two legs, we could for instance rotate the entire multispan by a half turn to get
This would introduce the desired orientation on objects, as the second multispan behaves like:
For this to work out as desired, we would of course have to concentrate on planar multispans and disallow identification of multispans that can be turned into each other by reflections at lines in the plane.
Mike: It seems as though restricting to planar multispans would be quite a serious thing to do.
Urs: I am thinking about a way to formalize the idea of hyperstructure with a concrete application to extended quantum field theory and the Baez-Dolan hypothesis in mind. I would like to develop this in the following here on this page. If things work out as hoped for, this should eventually become an entry in its own right. If not, this should eventually be discarded.
Ronnie In this connection I would like to draw attention here to
The idea here was to give expression to the notion of: what is wrong with pentagons? Or rhombic dodecahedra? as part of the basic models. Two major problems were solved:
What should be the basic cells, and the polyhedral category?
How should one orient or more the basic cells?
The answer to the first question is given in terms of cells with polyhedral boundary and in which there is a shellability condition. The answer to the second question is in terms of each cell has a marked face. This turns out to imply orientation, but is more of a homotopy condition than a homology condition, i.e. is stronger than the latter. There a relation to certain posets studied by Bjorner.
On the other hand this theory is probably more group than category oriented, there is a tendency to imply inverses.
The algebra comes in by defining poly--complexes, i.e. poly sets with thin elements.
A bond system or hyperstructure as intended in the following should be
a conglomerate of cells each of which is equipped with a prescribed collection of sub-cells, which in turn have their prescribed collection of subcells, etc;
such that whenever two such cell complexes coincide on any one part of their sub-cell complex, there exists a cell which can be regarded as gluing the two cells along this sub-cell complex.
The idea is hence akin to the geometric definition of higher category, but differs in two crucial aspects:
there is no fixed choice of geometric shapes for higher structures (such as globes, simplices, cubes, etc.) out of which the entire structure consists, instead all possible sub-cell complexes are admitted;
there is no notion of directionality imposed on the cell complexes, and in particular no ordering on the sub-cells of a given cell, neither binary into source and target cells as in globular higher categories, nor linearly into indexed faces as in simplicial higher categories.
The idea is that a hyperstructure is like a -dagger infinity-category but capturing this notion more directly than an -category equipped with an -dagger-operation would (which amounts to first introducing directionality only to remove it in a second step). Notice that in this context that categories of ordinary spans are always dagger-categories, where the dagger operation is nothing but the re-labelling of the legs of the span into source and target. Hyperstructures in particular generalize categories of spans (as described below) and their special nature is supposed to intrinsically realize the dagger-structure.
To formalize this, the idea is to observe that every cell with its complex of sub-cells should naturally form a poset under inclusions of sub-cells. Since for any collection of sub-cells which share a common collection of sub-sub-cells the result of gluing these sub-cells along their common sub-sub-cells should again be a sub-cell, this poset should be closed under colimits.
Therefore a bond system or hyperstructure should in particular assign to every poset with colimits a collection to be interpreted as the collection of cells whose sub-cell structure is of the form given by the poset. Such an assignment would naturally allow to restrict any cell to any of its sub-cells along an inclusion of posets to obtain another cell, and to extend any cell to a cell with richer but degenerate sub-cell structure along a surjection of posets. This clearly suggests that the bond system or hyperstructure is a presheaf on the category of posets with colimits.
Given this, the gluing condition or sewing condition that for any collection of cells which match on a part of their sub-cell complex there exists a cell obtained by gluing the cells along their common sub-cells can be formulated in a way analogous to the horn-filling condition in a Kan complex by requiring that the obvious map from the collection of cells which have the right shape of being glued cells to the collection of cells that admit gluing is an epimorphism.
The archetypical example of a bond system or hyperstructure as intended in the following is supposed to be the collection of multi co-spans in a category with colimits. Since a multi co-span in is supposed to be nothing but the image of a poset in , this means that the bond structure of multi co-spans should be the presheaf on represented by . This does satisfy the gluing condition in that for all multi-cospans which coincide in parts there exists in the colimit over their joint diagram. For ordinary cospans this is supposed to reproduce the ordinary composition of cospans by pushout over a joint leg.
The hyperstructure of hyperstructures
As there is a canonical notion of homomorphisms of presheaves we naturally obtain a category of bond systems or hyperstructures. By the above example this in turn naturally induces a hyperstructure of hyperstructures consisting of the multi-cospans in the category of presheaves on posets.
This allows then finally to formalize one of the situations motivating the notion of hyperstructure in the first place, that of extended quantum field theory: for instance for topological QFT this is expected to be a morphism (or more general span) of hyperstructures
where multi-cospans in the category Top are thought of as modelling extended cobordisms as described for instance at Cospans in Algebraic Topology, while multi-spans in some suitable category (multi co-spans in the opposite category ) are thought of as modelling morphisms between higher vector spaces as in groupoidification.
First recall some basics of posets to fix our notation.
We write for the category with two objects and a single nontrivial morphism between them. Recall that a poset is a category enriched over , where the canonical tensor product on is has tensor unit and .
which sends posets to their free cocompletion.
A bond system or hyperstructure is
which satisfies the following gluing condition or sewing condition: for every diagram
in the unique morphism
is an epimorphism.
For a hyperstructure and we call the collection of cells of of shape .
For a category with colimits let
be the presheaf which assigns to any poset the set of functors from to .
For a category with colimits, is a bond structure.
We have to check the sewing condition. First notice that for we have a natural bijection
in that . This says that is epi.
For a category with colimits we call the cells of the multi co-spans in , call all of the hyperstructure of multi-cospans in . Dually, for a category with colimits we write
for the hyperstructure of multi-spans in .
for the hyperstructure of hyperstructures.
The above general definition in particular reproduces the ordinary composition of cospans.
Let be a category with colimits. The cells of the hyperstructure of shape the pullback poset
are the ordinary cospans in . Let and consider the functors and in the diagram
and is the collection of pullback-diagrams in which share one leg. The cocompletion of looks in parts like
where the important point is that it contains the terminal object . Given any two cospans in that coincide on one foot
they induce the element
whose lift to is given on by
This is indeed the ordinary composite of the two cospans and .
Mike: It seems to me that this definition doesn’t contain enough information. Yes, the ordinary composite of the two cospans is one lift to , but there are plenty of other lifts, and there doesn’t seem to be anything in the presheaf you’ve described that can characterize that particular lift as the “correct” gluing (in particular, its universal property in seems to have been forgotten).
Urs: I was thinking of next defining “non-flabby” as the presheaf of co-continuous functors from a given poset into , so that the above lifts are the only ones (right?). Then I want to say that a morphism of hyperstructures from the bordism hyperstructure defined below to a non-flabby multicospan hyperstructure is fixed by its value on the point if it regards the interval as weakly equivalent to the point.
I am thinking of the difference between the above “flabby” version and the non-flabby version as the difference between general Kan complexes and those that are the nerve of an -groupoid for which above the fillers are unique.
Mike: Using cocontinuous functors will make the lifts unique, at least up to isomorphism. It doesn’t seem that you’ve included any information about isomorphisms, but perhaps that can be remedied with higher spans or with presheaves of groupoids or categories. But I don’t see the “flabby” version as very much like a Kan complex; in a Kan complex the fillers of horns are unique up to homotopy, because of the higher-dimensional horn-filling conditions, but I don’t see anything like that going on here.
Urs: But we have these higher filling conditions here automatically, too: if there are two different multispans which glue two given ones, they share these two given ones and can be glued along them. This should say that any two fillers are themselves connected by a filler.
Mike: It seems as though if you’re considering two multispans to be equivalent if they can be connected by a filler, and your horn-filling conditions stipulate that all fillers exist, then your structure is fairly trivial: everything is equivalent to everything else.
Let , the hyperstructure of extended cobordisms, be the smallest sub-hyperstructure of which contains the interval object and is closed under cartesian product of multi-cospans in .
Mike: In order for that to make sense, you need there to be a smallest sub-hyperstructure such that (blah). But I don’t see any reason for that to exist; the intersection of sub-hyperstructures need not be a hyperstructure. Even in a simplicial set the intersection of sub-Kan-complexes need not be a Kan complex; if contains one filler for a given horn and contains a different filler, then may contain no filler at all.
Urs: I agree that it is not a priori clear that some smallest hyperstructure with a given property exists. But I am not sure that I see what this has to do with intersections. Shouldn’t we we be looking at inclusions?
But in the particular case at hand, it feels that this smallest hyperstructure should certainly exist, no? I should think about formalizing it, but heuristically it should be given by the colimit over inclusions of the following steps for building it:
then include for everey multi-cospan obtained so far also its product with the interval cospans;
then form all composites of multi-cospans obtained so far
then continue with 1. .
Mike: In general, there are two ways to construct the smallest sub-widget of such that (blah):
Usually, if one works the the other does too. Occasionally one fails where the other succeeds due to set-theoretic technicalities. But here I think the problem is intrinsic and will rear its head wearing a different mask in either case.
It definitely does not feel to me as though the smallest hyperstructure should exist. The problem with your proposal is when you say “form all composites.” Since composites are not an operation, this can’t mean “for any and already in our sub-hyperstructure, add in the composite of and .” If we just arbitrarily pick some particular composite of and to add, then there might be other sub-hyperstructure that chose a different composite, so what we end up with won’t be the smallest (in the “minimum” sense) sub-hyperstructure such that (blah). We might be able to construct a minimal sub-hyperstructure, but it wouldn’t be uniquely determined, so we couldn’t talk about the hyperstructure of extended cobordisms.
Toby: And then, you might have to rely on Zorn's Lemma (the axiom of choice) to prove that even such a minimal structure exists!
This means that multi-cospans in arise from iteratively tensoring with the interval and forming pushouts, which amounts to foming co-span co-traces.
For instance with the interval there is also the circle in , arising as the composition of the interval with itself over both legs
With the circle there is then also the cylinder
With the cylinder there is also the torus
Etc. Then there are the extended cobordisms proper, which have deeper layers of sub-cells. For instance the product of the interval with itself gives the disk regarded as an extended cobordisms
with four 1-dimensional and four 0-dimensional boundary components.
Mike: I’m not sure if this is intentional or not, because I don’t know what your goal is, but even if the definition makes sense, these “extended cobordisms” include things that aren’t very manifold-like. For instance, if you glue the interval-with-endpoints to another copy of the interval-with-endpoints along one of the endpoints, you get a cell like
and now restricting along a suitable inclusion of posets, we get
and we can now glue this to another interval along the point , obtaining a wedge of three copies of the interval. I think the things you glue along should somehow be “canceled out” and no longer appear as sub-cells of the glued result (this is certainly what happens in the case of modules).
Urs: true. I need to think about a nice way to formalize such a cancellation.