Re: Simplicial versus (cubical with connections)

["F. William Lawvere" <wlawvere@hotmail.com>]

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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