simplicial vs (cubical with connections) Ronnie Brown
categories: Re: Simplicial versus (cubical with connections) Ronnie Brown ronnie.profbrown@btinternet.com One basic intuition is that cubes are products. Yet almost none of the combinatorial toposes commonly called “cubical sets” have that feature. Is there some profound disadvantage in allowing projection maps to have their universal property (yielding diagonal maps, etc)? If so, I have never seen it spelled out. An advantage to having finite products in a site is that the (iterated) pathspace functor has a right adjoint, leading to a very natural construction of Eilenberg-Mac Lane spaces. Dan Kan told me that the reason for his switch was that cubical groups do not have the extension property (i.e., the Kan property), as simplicial groups do. But later I realized that there is ambiguity about what “cubical” means. Since these combinatorial categories are usually toposes, some light is shed on their particularity by determining what kind of structure they classify (in the established categorical sense, e.g.,the simplicial topos classifies total orders with distinct endpoints, and a simple cubical example classifies strictly bipointed objects). Concretely, there are many different theories of algebraic structure for which the unit interval is a model, and having chosen one, this structure should be preserved by geometric realization. Bill Lawvere [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I have somewhere a preprint Evrard, M. \newblock \enquote{Homotopie des complexes simpliciaux et cubiques}. \newblock Preprint. 1976 which considers the category say CP with objects {0,1}^n considered as posets and morphisms the order preserving maps. This of course includes diagonal maps. I am not sure if anyone has considered their realisation properties. Cubical sets with connections are determined by a subcategory of CP. I don't have an ideological reason fro considering them, except that it works for what we need, and mentioned before, Andy Tonks proved that cubical groups with connections satisfy the Kan condition. The paper Grandis, M. and Mauri, L. {Cubical sets and their site}. {Theory Applic. Categories} \textbf{11} (2003) 185--201. develops normal forms for some kinds of cubical sets. Grandis, M. and Mauri, L. \newblock \enquote{Cubical sets and their site}. \newblock \emph{Theory Applic. Categories} \textbf{11} (2003) 185--201. looks at normal forms for cubical sets with connections. Ronnie On 30/09/2011 23:17, F William Lawvere wrote:
categories: Re: Simplicial versus (cubical with connections)
Ronnie Brown ronnie.profbrown@btinternet.com
One basic intuition is that cubes are products. Yet almost none of the combinatorial toposes commonly called “cubical sets” have that feature. Is there some profound disadvantage in allowing projection maps to have their universal property (yielding diagonal maps, etc)? If so, I have never seen it spelled out.
An advantage to having finite products in a site is that the (iterated) pathspace functor has a right adjoint, leading to a very natural construction of Eilenberg-Mac Lane spaces.
Dan Kan told me that the reason for his switch was that cubical groups do not have the extension property (i.e., the Kan property), as simplicial groups do. But later I realized that there is ambiguity about what “cubical” means.
Since these combinatorial categories are usually toposes, some light is shed on their particularity by determining what kind of structure they classify (in the established categorical sense, e.g.,the simplicial topos classifies total orders with distinct endpoints, and a simple cubical example classifies strictly bipointed objects). Concretely, there are many different theories of algebraic structure for which the unit interval is a model, and having chosen one, this structure should be preserved by geometric realization.
Bill Lawvere
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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F William Lawvere -
Ronnie Brown