Re: Simplicial versus (cubical with connections)

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

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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