Re: Simplicial versus (cubical with connections)

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

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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