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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