Re: Simplicial versus (cubical) with connections)

Re: Simplicial versus (cubical) with connections)

[F William Lawvere]

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Dear Todd, Ross, Vaughn, et al

Trivial objects should NOT be admitted to categories A whose
set-valued functors are intended to form a combinatorial topos
to serve as a surrogate for some sort of continuous spaces.
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 CAT, has precisely all monads as its actions). Similarly,
the basic cubical sets are functors on the part A (of the algebraic
category involving two nullary operations) which consists of
finitely-presented STRICT algebras; thus this is the classifying
topos for those bi-pointed objects (in arbitrary toposes) that
satisfy the non-equational entailment

front = back entails false.

The reason is this: if a site C = Aop has an initial object, then
the left adjoint 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 with non-trivial connectivity
are constructed as equalizers of maps between connected spaces!

(Note that restrictions (on the structures classified) involving
falsity, disjunction, or existential quantification typically give
sheaf toposes, but exceptionally may just give smaller presheaf
categories; another related example is in algebraic geometry,
where classifying the algebras subject to the disjunctive
condition

x^2=x entails x=0 or x=1

merely involves the topos of all functors on those
finitely-presented algebras that satisfy 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, achieving functoriality with respect to all
continuous changes of space; "realization" refers to a process
analogous to the passage from blueprints to actual buildings
of beton and steel). As emphasized by Gabriel & Zisman, the
exactness of realization forces us to refine the default notion of
space itself, in the direction proposed by Hurewicz in the late 40s
and 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 helps the
combinatorial topos to describe the continuous category as
closely as possible. In the same spirit, the finite products of
combinatorial intervals could be required to admit the diagonal
maps that their realizations will have. There is one point
however where perfect agreement cannot be achieved: the
contrast between continuous and combinatorial forced 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 seen
in my list of axioms for Cohesion (TAC): the reasonable
combinatorial toposes satisfy all but one of the axioms, but only
the continuous examples satisfy that one. This Continuity axiom
(preservation of infinite products by pizero) I introduced in order
to obtain homotopy types that are "qualities" in an intuitive sense
(as they are automatically in suitable continuous cases).

I hope these remarks are useful.

Bill


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Re: Simplicial versus (cubical with connections)

[F. William Lawvere]

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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Re: Simplicial versus (cubical with connections)

[Todd Trimble]

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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Re: Simplicial versus (cubical with connections)

[Ross Street]

> 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

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Re: Simplicial versus (cubical with connections)

[Ronnie Brown]

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.  ;)
>

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Re: Simplicial versus (cubical with connections)

[Vaughan Pratt]

(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.  ;)


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Re: Simplicial versus (cubical with connections)

[Marco Grandis]

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

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Re: Simplicial versus (cubical with connections)

[Urs Schreiber]

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
>

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Re: Simplicial versus (cubical with connections)

[Fernando Muro]

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


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Re: Simplicial versus (cubical with connections)

[Urs Schreiber]

> 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.


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Re: Simplicial versus (cubical with connections)

[Ronnie Brown]

The result of Maltsiniotis referred to by Jonathan is very welcome.  But 
I wonder if there is still a problem with cubical sets with connection:

the geometric realisation of a simplicial group is, in a convenient 
category, a topological group, because of the homeomorphism

f: |K \times Y| \to |K| \times |Y| .

However in the case of cubical sets with connections this map f is a 
homotopy equivalence but it seems is not a homeomorphism (?).  As 
Grothendieck wrote: `homotopically speaking' that is not a problem!

For homotopies and higher homotopies cubes are nice and easy because of 
the basic formula

I^m \times I^n = I^{m+n}.

This leads to monoidal closed structures on strict cubical higher 
categories and groupoids.

For a basic discussion of other issues such as algebraic inverses to 
subdivision and commutative cubes  I refer to my 2009  Liverpool seminar 
on`What is and what should be `Higher dimensional group theory'?'

http://pages.bangor.ac.uk/~mas010/pdffiles/liverpool-beamer-handout.pdf

Ronnie




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Re: Simplicial versus (cubical with connections)

[Jonathan CHICHE 齊正航]

There is another way to state that the cube category with connections  
behaves "as well as" the simplex category. Both are strict test  
categories (as defined by Grothendieck in "Pursuing Stacks"). See  
http://www.math.jussieu.fr/~maltsin/ps/cubique.pdf. Without  
connections, the cube category is a test category, but not a strict  
one, so that the product in the cube category does not reflect the  
product of homotopy types. This issue vanishes if connections are  
allowed. Grothendieck explicitly wrote in "Pursuing Stacks" that he  
believed that, homotopically speaking, any strict test category was  
"as good as" the simplex category. For instance, he conjectured there  
that an analog of the Dold-Kan correspondence (which he called Dold- 
Puppe) holds for every strict test category. (As regards the  
existence of a Quillen model structure the cofibrations of which are  
monomorphisms on the presheaf category, and so on, see the  
introduction to Astérisque 301 by Maltsiniotis and Astérisque 308 by  
Cisinski.)

Best regards,

Jonathan Chiche

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Re: Simplicial versus (cubical with connections)

[Ronnie Brown]

In response to Marco's interesting points, there is a related  way of
expressing this: degeneracies in the simplicial theory give simplices
with some adjacent faces equal; in the cubical theory, degeneracies give
cubes with some opposite faces equal, and never the twain shall meet!
The connections \Gamma_i (which arise from the monoid structure max on
the unit interval) restore the analogy with simplices, since \Gamma_i x
has two adjacent faces the same.

The advantage of cubes for our ideas was always the *easy*  expression
of `algebraic inverses to subdivision' (not so easy simplicially)  and
the application of this to local-to-global problems. The connections
were found  from trying to express the notion of `commutative cube'; an
account of this search is in the Introduction to `Nonabelian algebraic
topology'.  The nice surprise was that this extra structure was also
what was needed to get  equivalences of some algebraic categories (e.g.
crossed modules versus double groupoids with connections)  so it all
fitted together amazingly.

For more on these ideas, see

Grandis, M. and Mauri, L. Cubical sets and their site. Theory Appl.
Categ. {11} (2003) 185--201.

Higgins, P.~J. Thin elements and commutative shells in cubical
{$\omega$}-categories.  Theory Appl. Categ. {14} (2005)  60--74.

I have never tried cubical sets without degeneracies but with connections!

Ronnie





On 13/09/2011 16:12, Marco Grandis wrote:
> Dear categorists,
>
> I would like to comment on Ronnie Brown's message, copied below,
> insisting on a parallelism that is not often acknowledged, and may
> 'clarify'
> - for instance - why simplicial groups somehow behave as
> 'cubical groups with connections' (see Tonks' paper cited by RB),
> rather than as 'ordinary cubical groups'.
>
>    The degeneracies of a simplicial object correspond to the connections
>    (or higher degeneracies) of a cubical one, introduced by Brown and
> Higgins,
>    more than to the ordinary degeneracies.
>
> Formally, this fact can be motivated as follows.
>
> Let us start from the cylinder endofunctor  I(X) = X x [0, 1]  of
> topological spaces.
> Its main structure consists of natural transformations of powers of I,
> derived from
> (part of) the lattice structure of [0, 1]:
>
> - two faces  1 --> I,   sending x to (x, 0) OR (x, 1),
> - a degeneracy  I --> 1,    sending (x, t) to x,
> - two connections  I^2 --> I,    sending (x, t, t') to (x, max(t, t'))
> OR (x, min(t, t')).
>
> Then we collapse the higher face of I (for instance), and we get a
> cone functor C, with
> a monad structure:
>
> - the lower face of I gives the unit  1 --> C,
> - the lower connection gives the multiplication C^2 --> C,
> - the other transformations (including the degeneracy of I) induce
> nothing.
>
> Now the cylinder I, with the above structure (which i [myself, not the
> cylinder] call a 'diad'),
> operating on any space, gives a cocubical object with connections,
> while the monad C gives an augmented cosimplicial object.
>
> [[ Addendum.
> If one wants to take on the parallelism to the singular
> cubical/simplicial set of a space X,
> the construction becomes more involved. One should start from:
>
> - the cocubical space I* (with connections) of all standard cubes,
> produced by the cylinder I
>   on the singleton space;
>
> - the augmented cosimplicial space Delta* produced by C on the empty
> space 0
>   (taking care that C(0), defined as a pushout, is the singleton, and
> C^n(0) is the
>   standard simplex of dimension n-1).
>
> Then one applies to these structures the contravariant functor Top(-,
> X) and gets the
> singular cubical set of X (with connections) OR the singular
> simplicial set of X (augmented).
> ]]
>
> With best regards
>
> Marco Grandis
>


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Simplicial versus (cubical with connections)

[Marco Grandis]

Dear categorists,

I would like to comment on Ronnie Brown's message, copied below,
insisting on a parallelism that is not often acknowledged, and may  
'clarify'
- for instance - why simplicial groups somehow behave as
'cubical groups with connections' (see Tonks' paper cited by RB),
rather than as 'ordinary cubical groups'.

     The degeneracies of a simplicial object correspond to the  
connections
     (or higher degeneracies) of a cubical one, introduced by Brown  
and Higgins,
     more than to the ordinary degeneracies.

Formally, this fact can be motivated as follows.

Let us start from the cylinder endofunctor  I(X) = X x [0, 1]  of  
topological spaces.
Its main structure consists of natural transformations of powers of  
I, derived from
(part of) the lattice structure of [0, 1]:

- two faces  1 --> I,   sending x to (x, 0) OR (x, 1),
- a degeneracy  I --> 1,    sending (x, t) to x,
- two connections  I^2 --> I,    sending (x, t, t') to (x, max(t,  
t')) OR (x, min(t, t')).

Then we collapse the higher face of I (for instance), and we get a  
cone functor C, with
a monad structure:

- the lower face of I gives the unit  1 --> C,
- the lower connection gives the multiplication C^2 --> C,
- the other transformations (including the degeneracy of I) induce  
nothing.

Now the cylinder I, with the above structure (which i [myself, not  
the cylinder] call a 'diad'),
operating on any space, gives a cocubical object with connections,
while the monad C gives an augmented cosimplicial object.

[[ Addendum.
If one wants to take on the parallelism to the singular cubical/ 
simplicial set of a space X,
the construction becomes more involved. One should start from:

- the cocubical space I* (with connections) of all standard cubes,  
produced by the cylinder I
    on the singleton space;

- the augmented cosimplicial space Delta* produced by C on the empty  
space 0
    (taking care that C(0), defined as a pushout, is the singleton,  
and C^n(0) is the
    standard simplex of dimension n-1).

Then one applies to these structures the contravariant functor Top(-,  
X) and gets the
singular cubical set of X (with connections) OR the singular  
simplicial set of X (augmented).
]]

With best regards

Marco Grandis


On 12 Sep 2011, at 11:35, Ronnie Brown wrote:

> The reference is included in this review *MR1173825 *of the cubical  
> case.
>
> Tonks, A. P. <http://0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/ 
> mathscinet/search/publications.html?pg1=IID&s1=325533>(4-NWAL)  
> <http://0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet// 
> search/institution.html?code=4_NWAL>
> Cubical groups which are Kan.
> /J. Pure Appl. Algebra/ <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/journaldoc.html? 
> cn=J_Pure_Appl_Algebra> 81 <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? 
> pg1=ISSI&s1=118323>(1992), <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? 
> pg1=ISSI&s1=118323>no. 1, <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? 
> pg1=ISSI&s1=118323> 83–87. <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/mscdoc.html? 
> code=55U10,%2818D35,18G30%29><javascript:openWin('http:// 
> unicat.bangor.ac.uk:4550/resserv', 'AMS:MathSciNet', 'atitle=Cubical 
> %20groups%20which%20are% 
> 20Kan&aufirst=A.&auinit=AP&auinit1=A&auinitm=P&aulast=Tonks&coden=JPAA 
> A2&date=1992&epage=87&genre=article&issn=0022-4049&issue=1&pages=83-87 
> &spage=83&stitle=J.%20Pure%20Appl.%20Algebra&title=Journal%20of% 
> 20Pure%20and%20Applied%20Algebra&volume=81')>
>
> The author shows that group objects in the category of cubical sets  
> with connections [R. Brown and P. J. Higgins, J. Pure Appl. Algebra  
> 21 (1981), no. 3, 233--260; MR0617135 (82m:55015a) <http://0- 
> ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/ 
> publdoc.html?r=1&pg1=CNO&s1=617135&loc=fromrevtext>] satisfy the  
> Kan extension condition. This is a very nice correspondence with  
> the simplicial case [J. C. Moore, in Séminaire Henri Cartan de  
> l'Ecole Normale Supérieure, 1954/1955, Exp. No. 18, Secrétariat  
> Math., Paris, 1955; see MR0087934 (19,438e) <http://0-ams.mpim- 
> bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publdoc.html? 
> r=1&pg1=CNO&s1=87934&loc=fromrevtext>].

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