Re: Simplicial versus (cubical with connections)

[Vaughan Pratt <pratt@cs.stanford.edu>]

(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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