Re: Simplicial versus (cubical with connections)
Dear Colleagues, I have found the `very good' properties of cubical sets essential for many aspects of my work, and are used in our new book `Nonabelian algebraic topology', (downloadable pdf from my web pages) which puts in one place the work with Higgins et al; with regard to the monoidal closed structures this is especially seen in Chapter 15, whose Introduction p. 513 mentions a crucial natural transformation \eta: \rho (X_*) \otimes \rho(Y_*) \to \rho(X_* \otimes Y_*) where X_*, Y_* are filtered spaces, and \rho gives the cubical higher homotopy groupoid, which is almost `obvious' in this cubical context, and so can be translated to other contexts. Whether these `very good properties' are `excellent' in a formal sense I do not know, but it would be very surprising if that failed! I do not find analogous properties simplicially. I got the impression from John Moore that although Dan Kan's initial work on combinatorial homotopy was cubical (see his first PNAS Note) the problems of the homotopy type of the realisation of the cartesian product discouraged any further cubical work, as the simplicial method went smoothly, although often technical. So nobody tried to fix the cubical theory. Our `fix', i.e. the connections, was forced on us for other reasons, initially to relate crossed modules and double groupoids; they were crucial for the 2-d van Kampen theorem (all done in the 1970s) and then for higher analogues. I doubt if much of this local-to-global stuff could have been conjectured, let alone proved, simplicially. On the other hand, there are two papers on classifying spaces for equivariant crossed complexes with Golasinski, Porter and Tonks where simplicial methods are used, because they link nicely with work on coherence of Cordier and Porter. A recent paper using cubical methods which could be relevant for Dmitry is Faria~Martins, J. and Picken, R. Surface {H}olonomy for {N}on-{A}belian 2-{B}undles via {D}ouble {G}roupoids}.{Adv. Math.} 226(2011) 3309–3366. Ronnie On 15/09/2011 12:56, Dmitry Roytenberg wrote:
Dear colleagues,
This raises the following question. In homotopy theory one almost always considers categories enriched in simplicial sets, endowed with the classical Kan-Quillen model structure, but are cubical sets (with or without connections) equally good for that purpose? In his "Higher Topos Theory" Lurie considers monoidal model categories S enjoying certain additional properties and proves that for each such "excellent model category" S the category Cat(S) of small S-enriched categories admits a model structure with a particularly nice description of fibrations and weak equivalences (that S = simplicial sets with the classical model structure and the Cartesian product is excellent is a result of Dwyer and Kan; the corresponding model structure on Cat(S) is due to Bergner).
So my question is, is it known whether the category of cubical sets, with or without connections, admits an excellent model structure? The reason I am asking is because in geometric applications I have in mind cubes are often easier to work with than simplices (e.g. it is easier to compute integrals of differential forms over cubes).
Thank you in advance.
Dmitry
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So my question is, is it known whether the category of cubical sets, with or without connections, admits an excellent model structure?
The only model structure on cubical sets that I am aware of is that given by Jardine: http://ncatlab.org/nlab/show/model+structure+on+cubical+sets Its cofibrations are the monos, but the other axioms of "excellent" (HTT A.3.2.16) seem problematic. But I haven't really thought about it. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
For a published reference see: MR2591923 (2010k:18022) Maltsiniotis, Georges(F-PARIS7-IMJ) La catégorie cubique avec connexions est une catégorie test stricte. (French. English summary) [The category of cubes with connections is a strict test category] Homology, Homotopy Appl. 11 (2009), no. 2, 309–326. Best, Fernando On Thu, 15 Sep 2011 21:06:32 +0200, Urs Schreiber wrote:
So my question is, is it known whether the category of cubical sets, with or without connections, admits an excellent model structure?
The only model structure on cubical sets that I am aware of is that given by Jardine:
http://ncatlab.org/nlab/show/model+structure+on+cubical+sets
Its cofibrations are the monos, but the other axioms of "excellent" (HTT A.3.2.16) seem problematic. But I haven't really thought about it.
-- Fernando Muro Universidad de Sevilla, Departamento de Álgebra http://personal.us.es/fmuro [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear category theorists, a while back we had a discussion here about model structures on cubical sets. My colleague Dmitry Roytenberg sent the following message to the list, which, for some reason, did not seem to have gone through. Since it might be of interest to some members of the list I would like to repost it hereby. ---------- Forwarded message ---------- From: Dmitry Roytenberg <starrgazerr@gmail.com> Date: Thu, Sep 22, 2011 at 12:18 PM Subject: Re: categories: Re: Simplicial versus (cubical with connections) To: categories <categories@mta.ca> Dear colleagues, Thanks to everyone who has replied, either privately or on the list. Having read up on the subject a bit I've discovered that quite a lot is known by now about the homotopy theory of cubical sets, thanks mainly to the work of Cisinski and Jardine (Jardine's "Categorical homotopy theory" contains a good exposition of Cisinski's methods, especially useful for non-French speakers). The spatial model structure mentioned by Urs has inclusions as cofibrations and cubical Kan fibrations as fibrations; it is proper, combinatorial and monoidal with respect to the tensor product described by Ronnie, with weak equivalences stable under filtered colimits. There is a monoidal left Quillen equivalence from cubical to simplcial sets mapping the n-dimensional cube to the n-th power of the 1-simplex (the simplicial realization). This is enough to conclude that cubical sets form an excellent model category, in view of Lemma A.3.2.20 and Remark A.3.2.21 in HTT. It then follows from (HTT, Theorem A.3.2.24) that a cubical category is fibrant iff all its function complexes are Kan. As for presheaves on other cubical sites (i.e. with more than just the classical faces and degeneracies), Isaacson in his thesis http://www.ma.utexas.edu/users/isaacson/PDFs/diss.pdf constructs a cubical site containing Brown-Higgins connections (there called conjunctions and disjunctions, as in logic) as well as symmetries of hypercubes, closely related but different from Grandis and Mauri's site !K. Isaacson constructs a _symmetric_ monoidal models tructure on the resulting category of symmetric cubical sets, and equips it with a monoidla left Quillen equivalence from the ordinary cubical sets. However, this model category is not excellent, as not all monomorphisms are cofibrations. As far as I can tell, it is not known if there is an excellent model structure on cubical sets with connections (but without symmetries). In any case, using these connections in a differential-geometric context is problematic, not (just) because of a clash with established terminology, but because the max and min maps are only piecewise smooth. Finally, to my astonishment I have not been able to find an abstract description of any of the cubical sites, in the spirit of the simplex category being the category of non-empty finite ordinals. Clearly the cubes are to be viewed as finite power sets, but which structure on the power sets is preserved by the morphisms in each case? Best, Dmitry On Fri, Sep 16, 2011 at 3:24 PM, Fernando Muro <fmuro@us.es> wrote:
For a published reference see:
MR2591923 (2010k:18022) Maltsiniotis, Georges(F-PARIS7-IMJ) La catégorie cubique avec connexions est une catégorie test stricte. (French. English summary) [The category of cubes with connections is a strict test category] Homology, Homotopy Appl. 11 (2009), no. 2, 309–326.
Best,
Fernando
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
This is about two points of a recent message of Dmitry Roytenberg, forwarded by Urs Schreiber.
I have not been able to find an abstract description of any of the cubical sites, in the spirit of the simplex category being the category of non-empty finite ordinals
I do not know of any such description. But there is a nice abstract description of the site of cubical sets with connections, parallel to a well-known characterisation of the simplicial site: - the free strict monoidal category with an assigned dioid. See [GM], Thm. 5.2. (There are analogous results for the other cubical sites.) A `dioid' is a set with two monoid operations, where the unit of each operation is absorbant for the other. Typically, an abstract interval has such a structure, and a cylinder functor has the structure of a 'diad'. See [Gr]. (I was also using the terms 'cubical monoid' and 'cubical monad', for an obvious analogy; later I abandoned them because they could be misleading - obviously again). Every lattice is an idempotent dioid, but idempotency is - apparently - of no interest in homotopy. This leads us to the second point: smooth homotopy.
In any case, using these connections in a differential-geometric context is problematic, not (just) because of a clash with established terminology, but because the max and min maps are only piecewise smooth.
For smooth homotopy one should use a different (non-idempotent) dioid, still commutative and involutive: NOT the standard interval with min, max, linked by the involution t' = 1 - t, BUT the standard interval with multiplication and *, linked by the same involution: x*y = (x'.y')' = x + y - xy. See [Gr]. [Gr] M. Grandis, Cubical monads and their symmetries, in: Proc. of the Eleventh Intern. Conf. on Topology, Trieste 1993, Rend. Ist. Mat. Univ. Trieste 25 (1993), 223-262. http://www.dmi.units.it/~rimut/volumi/25/index.html [GM] M. Grandis - L. Mauri, Cubical sets and their site, Theory Appl. Categ. 11 (2003), No. 8, 185-211. Best regards Marco Grandis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
(In the context of a presheaf category Set^J\op I'll follow the reasonably common practice of calling J the base and J\op the theory in the following.) On 10/19/2011 1:35 AM, Marco Grandis wrote:
This is about two points of a recent message of Dmitry Roytenberg, forwarded by Urs Schreiber.
I have not been able to find an abstract description of any of the cubical sites, in the spirit of the simplex category being the category of non-empty finite ordinals
I do not know of any such description. But there is a nice abstract description of the site of cubical sets with connections, parallel to a well-known characterisation of the simplicial site:
- the free strict monoidal category with an assigned dioid. See [GM], Thm. 5.2. (There are analogous results for the other cubical sites.)
I'm not sure what Marco means by "abstract" here, which may be the root of any confusion I may have concerning the following, which however seems to me to be worth saying anyway, however well known it may be, as it receives less attention on this mailing list than it deserves. One striking difference between simplicial and cubical sets is the difference between the base and the theory, which is much less for the former (simplicial sets) than the latter. Taking the finite ordinals for the base of simplicial sets as per Marco (but including the empty ordinal so that face lattices really are lattices, having a bottom face of dimension -1), the theory is representable as the duals 2^n of the finite ordinals n, whose elements are the monotone functions from n to the two-element ordinal. Like n, 2^n is linearly ordered, but unlike n it is has a top and a bottom, namely the constantly 1 and 0 functions respectively. These are constant both semantically and syntactically, the latter by virtue of being preserved by the homomorphisms of the theory thus represented. Furthermore the constants are distinct except for the dual of the empty ordinal. The underlying poset of a dual ordinal 2^n is that of the ordinal n+1, likewise |2^n| = |n + 1| (= |n| + 1 in this case) for the underlying sets. As usual with Stone duality, n is recovered from 2^n as 2^{2^n} where the first 2 is the dual ordinal 2^1 consisting of just a top and a bottom and the morphisms are the constant-preserving monotone functions. Cubical sets can be characterized very simply by their theory, which is representable as the finite *free* bipointed sets, those with distinct distinguished elements, together with the singleton bipointed set as its only non-free object (again for the sake of the face lattices being true lattices). The base can then be understood as the finite complemented distributive lattices, which are not quite the same thing as Boolean algebras by virtue of omission of "bounded" before "lattice," though they have the same underlying poset as a finite Boolean algebra and as such are clearly recognizable geometrically as primordial cubical sets. Unlike Boolean algebras, the empty CDL exists (unless you follow McKenzie, McNulty and Taylor in disallowing empty algebras on the ground that "no gods are clean-shaven" contradicts "all gods are clean-shaven") and as for simplicial sets ensures that face lattices are lattices (though not complemented ones). The underlying posets of CDLs as representing the objects of the base are therefore very different from those of the theory, which are discrete, in striking contrast to the situation with simplicial sets where they are same, give or take an element. Incidentally, unless I'm overlooking something it seems to me that the base of cubical sets must be a variety on FinSet, since the usual axioms x v ~x = 1 and x & ~x = 0 defining complement can be rendered as x v ~x = y v ~y and dually. This should suffice to rule out non-cubical CDLs. The theory of cubical sets is clearly a quasivariety on FinSet, being axiomatizable by the universal Horn formula 0 = 1 --> x = y, but I don't see any representation that makes it a variety on FinSet. Vaughan Pratt PS. Please don't take any of this as an endorsement of one over the other. I'm just the messenger. ;) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I very much like all this discussion of the abstract basis for these various cubical theories! It may be useful to comment on the intuitive origin of the connections. The first idea was to use double groupoids in some way in higher homotopy theory. Then came the question of examples: were there any interesting double groupoids? We went through various generalisations of C.Ehresmann's double groupoid of commutative squares in a group or groupoid, finally ending up with a functor (crossed modules) \to (double groupoids) which gave lots of good example of the latter. But which double groupoids arose in this way? Also for the idea of a proof of a van Kampen theorem, we needed the notion of `commutative cube in a double groupoid'. Fortunately, the notion of connection in a double groupoid (called special double groupoid with special connection in our paper) satisfied both properties! The key `transport law' was borrowed from a paper of Virsik on path connections, hence the name `connection'. In more intuitive terms, the transport law represents: `turning left with your arm outstretched is the same as turning left'. Another law says that `turning left and then right leaves you where you were'. (etc). Amazingly, it all fitted together. Chris once remarked about how well it worked, once one had got things sorted. It still took another 3 years to get with Philip the functor (pointed pairs of spaces) \to (double groupoids with connections) and then things fell into place. This functor seems generally ignored in algebraic topology. For us, it allowed `algebraic inverses to subdivision', unlike relative homotopy groups; it is this idea of `multiple compositions' which seems to lack an abstract study, though for us it was a key use of cubical structures. The aim of a van Kampen theorem is to get some algebraic control, expressed as colimits rather than exact sequences, of the gluing of complex hierarchical structures, with interactions between low and high dimensions. It was then not too hard to generalise connections to all dimensions, but there was a lot of hard work (the hand of PJH!) in the paper `The algebra of cubes' to make it all work. Ronnie On 19/10/2011 18:09, Vaughan Pratt wrote:
(In the context of a presheaf category Set^J\op I'll follow the reasonably common practice of calling J the base and J\op the theory in the following.)
On 10/19/2011 1:35 AM, Marco Grandis wrote:
This is about two points of a recent message of Dmitry Roytenberg, forwarded by Urs Schreiber.
I have not been able to find an abstract description of any of the cubical sites, in the spirit of the simplex category being the category of non-empty finite ordinals
I do not know of any such description. But there is a nice abstract description of the site of cubical sets with connections, parallel to a well-known characterisation of the simplicial site:
- the free strict monoidal category with an assigned dioid. See [GM], Thm. 5.2. (There are analogous results for the other cubical sites.)
I'm not sure what Marco means by "abstract" here, which may be the root of any confusion I may have concerning the following, which however seems to me to be worth saying anyway, however well known it may be, as it receives less attention on this mailing list than it deserves.
One striking difference between simplicial and cubical sets is the difference between the base and the theory, which is much less for the former (simplicial sets) than the latter. Taking the finite ordinals for the base of simplicial sets as per Marco (but including the empty ordinal so that face lattices really are lattices, having a bottom face of dimension -1), the theory is representable as the duals 2^n of the finite ordinals n, whose elements are the monotone functions from n to the two-element ordinal.
Like n, 2^n is linearly ordered, but unlike n it is has a top and a bottom, namely the constantly 1 and 0 functions respectively. These are constant both semantically and syntactically, the latter by virtue of being preserved by the homomorphisms of the theory thus represented. Furthermore the constants are distinct except for the dual of the empty ordinal. The underlying poset of a dual ordinal 2^n is that of the ordinal n+1, likewise |2^n| = |n + 1| (= |n| + 1 in this case) for the underlying sets.
As usual with Stone duality, n is recovered from 2^n as 2^{2^n} where the first 2 is the dual ordinal 2^1 consisting of just a top and a bottom and the morphisms are the constant-preserving monotone functions.
Cubical sets can be characterized very simply by their theory, which is representable as the finite *free* bipointed sets, those with distinct distinguished elements, together with the singleton bipointed set as its only non-free object (again for the sake of the face lattices being true lattices). The base can then be understood as the finite complemented distributive lattices, which are not quite the same thing as Boolean algebras by virtue of omission of "bounded" before "lattice," though they have the same underlying poset as a finite Boolean algebra and as such are clearly recognizable geometrically as primordial cubical sets. Unlike Boolean algebras, the empty CDL exists (unless you follow McKenzie, McNulty and Taylor in disallowing empty algebras on the ground that "no gods are clean-shaven" contradicts "all gods are clean-shaven") and as for simplicial sets ensures that face lattices are lattices (though not complemented ones).
The underlying posets of CDLs as representing the objects of the base are therefore very different from those of the theory, which are discrete, in striking contrast to the situation with simplicial sets where they are same, give or take an element.
Incidentally, unless I'm overlooking something it seems to me that the base of cubical sets must be a variety on FinSet, since the usual axioms x v ~x = 1 and x & ~x = 0 defining complement can be rendered as x v ~x = y v ~y and dually. This should suffice to rule out non-cubical CDLs.
The theory of cubical sets is clearly a quasivariety on FinSet, being axiomatizable by the universal Horn formula 0 = 1 --> x = y, but I don't see any representation that makes it a variety on FinSet.
Vaughan Pratt
PS. Please don't take any of this as an endorsement of one over the other. I'm just the messenger. ;)
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 19/10/2011, at 12:27 AM, Urs Schreiber quoted Dmitry Roytenberg:
Finally, to my astonishment I have not been able to find an abstract description of any of the cubical sites, in the spirit of the simplex category being the category of non-empty finite ordinals.
Dear Urs and Dmitry Perhaps the example I learned from Dominic Verity that I gave in my CT 1995 Halifax talk will suffice for this. Using strings, we derive a model for the free monoidal category containing a cointerval. Cubical sets are functors from this to Set. See pages 7, 8, 9 of http://www.mta.ca/~cat-dist/ct95.html or http://www.maths.mq.edu.au/~street/LowDTop.pdf Apologies if this model was already mentioned. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Perhaps the example I learned from Dominic Verity that I gave in my CT 1995 Halifax talk will suffice for this. Using strings, we derive a model for the free monoidal category containing a cointerval. Cubical sets are functors from this to Set.
Further to my last message: That CT1995 talk was very much part of a three-man show: Street, Verity, Trimble. In particular, for the example I am referring to above, the original combinatorial model came from Verity, the string derivation of the model was Trimble, and all I did was draw the diagrams using MacDraw. Apparently there is material on this example in the nLab http://nlab.mathforge.org/nlab/show/cube+category Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
My impression is that there are at least two distinct notions of cubical set which have entered this discussion. One version describes cubical sets as presheaves on the Lawvere theory generated by two constants or 0-ary operations; this is close to what Vaughan described. More precisely, instead of taking the category whose objects are finite sets equipped with two distinct points (which is opposite to the Lawvere theory), he adds in a terminal object (where the two constants are forced to coincide), giving a category C. Anyway, whether one takes the Lawvere theory or C^{op}, the result is a category with finite cartesian products and an interval object, and one notion of cubical set is that of presheaf on this category. Whereas cubical sets in the sense described by Ross are different: they are presheaves on the free *monoidal* category with an interval object. This category does not include diagonal maps. I expect this is the notion of cubical set that Dmitry and Urs were actually concerned with, but in any event, both the cartesian version and the monoidal version of the cubical site appear in the literature, and it is important to clarify which notion is meant. Todd Trimble [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Trivial objects should NOT be admitted to categories A whose set valued functors should form a combinatorial topos intended as a surrogate for continuous spaces in some sense.That is to say, there is a reason why the delta of simplicial sets does not have a unit object (unlike the delta that as a strict monoid in CAThas precisely all monads as its actions). Similarly the basic cubical sets are functors on the part A of the algebraic category of two nullary operations which consist of finitely presented STRICT algebras; thus this is the classifying topos for bipointed objects (in arbitrary toposes) toposes that satisfy the non-equational entailment csub0 =csub1 entails false.The reason is this: if a site C=Aop has an initial object , then the leftadjoint to the inclusion of constant functors , which should model the notion of connected components, is representable , hence preserves equalizers; but the basic intuitive examples of spaces withnon-trivial higher connectivity are constructed as equalizers betweenconnected spaces ! ( Note that restrictions (on the structures classified) involving falsity, disjunction, or existential quantification typically give sheaf toposesbut exceptionally may just give smaller presheaf categories ; another example is in algebraic geometry where classifying the algebrassubject to the disjunctive conditionx^2=x entails x=0 or x=1merely involves the topos of all presheaves on those fp algebras satisfying the same condition.) According to the paradigm set by Milnor, the relation between continuous and combinatorial is a pair of adjoint functors called traditionally singular and realization . ("Singular", as emphasized by Eilenberg, means that the figures on which the combinatorial structure of a space lives should not be required to be monomorphisms,in order that in order that that structure should be functorial wrt all continuous changes of space ; "realization" refers to a process analogous to to the passage from blueprints to actual buildings of beton and steel).As emphasized by Gabriel and Zisman, the exactness of realizationforces us to refine the default notion of space itself, in the directionproposed by Hurewicz in the late 40s and well-described by J L Kelley in 1955. Further refinements suggest that the notion of continuous could well be taken as a topos, of a cohesive (or gros) kind.The exactness of realization is a example of the striving to make the surrogate combinatorial topos (= having a site with finite homs ???) describe the continuous category as closely as possible. For example the finite products of combinatorial intervals might be required to admit the diagonal maps that their realizations have. There is one point however where perfect agreement cannot be achieved ( Is this a theorem?) : the contrast between continuous and combinatorialforced Whitehead to introduce a specific notion he called weak equivalence, as explained by Gabriel-Zisman, in order to extract the correct homotopy category. The contrast can readily be read off of my list of axioms for Cohesion (TAC) : the reasonable combinatorial toposes satisfy all but one of the axioms,but only the continuous examples satisfy it. That Continuity axiom (preservationof infinite products by pizero) was introduced in order to obtain homotopytypes that are "qualities" in an intuitive sense (as they should be automaticallyin the continuous case).
Date: Sat, 22 Oct 2011 09:07:59 -0400 From: trimble1@optonline.net Subject: categories: Re: Simplicial versus (cubical with connections) To: ross.street@mq.edu.au; pratt@cs.stanford.edu CC: categories@mta.ca
My impression is that there are at least two distinct notions of cubical set which have entered this discussion. One version describes cubical sets as presheaves on the Lawvere theory generated by two constants or 0-ary operations; this is close to what Vaughan described. More precisely, instead of taking the category whose objects are finite sets equipped with two distinct points (which is opposite to the Lawvere theory), he adds in a terminal object (where the two constants are forced to coincide), giving a category C. Anyway, whether one takes the Lawvere theory or C^{op}, the result is a category with finite cartesian products and an interval object, and one notion of cubical set is that of presheaf on this category.
Whereas cubical sets in the sense described by Ross are different: they are presheaves on the free *monoidal* category with an interval object. This category does not include diagonal maps. I expect this is the notion of cubical set that Dmitry and Urs were actually concerned with, but in any event, both the cartesian version and the monoidal version of the cubical site appear in the literature, and it is important to clarify which notion is meant.
Todd Trimble
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participants (8)
-
F. William Lawvere -
Fernando Muro -
Marco Grandis -
Ronnie Brown -
Ross Street -
Todd Trimble -
Urs Schreiber -
Vaughan Pratt