[Note from moderator: an earlier version of this message was badly formatted and may have been difficult to read so it is being reposted; as a reminder, posts should be sent with text-only formatting] Dear Todd, Ross, Vaughn, et al Trivial objects should NOT be admitted to categories A whose set-valued functors are intended to form a combinatorial topos to serve as a surrogate for some sort of continuous spaces. 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 CAT, has precisely all monads as its actions). Similarly, the basic cubical sets are functors on the part A (of the algebraic category involving two nullary operations) which consists of finitely-presented STRICT algebras; thus this is the classifying topos for those bi-pointed objects (in arbitrary toposes) that satisfy the non-equational entailment front = back entails false. The reason is this: if a site C = Aop has an initial object, then the left adjoint 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 with non-trivial connectivity are constructed as equalizers of maps between connected spaces! (Note that restrictions (on the structures classified) involving falsity, disjunction, or existential quantification typically give sheaf toposes, but exceptionally may just give smaller presheaf categories; another related example is in algebraic geometry, where classifying the algebras subject to the disjunctive condition x^2=x entails x=0 or x=1 merely involves the topos of all functors on those finitely-presented algebras that satisfy 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, achieving functoriality with respect to all continuous changes of space; "realization" refers to a process analogous to the passage from blueprints to actual buildings of beton and steel). As emphasized by Gabriel & Zisman, the exactness of realization forces us to refine the default notion of space itself, in the direction proposed by Hurewicz in the late 40s and 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 helps the combinatorial topos to describe the continuous category as closely as possible. In the same spirit, the finite products of combinatorial intervals could be required to admit the diagonal maps that their realizations will have. There is one point however where perfect agreement cannot be achieved: the contrast between continuous and combinatorial forced 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 seen in my list of axioms for Cohesion (TAC): the reasonable combinatorial toposes satisfy all but one of the axioms, but only the continuous examples satisfy that one. This Continuity axiom (preservation of infinite products by pizero) I introduced in order to obtain homotopy types that are "qualities" in an intuitive sense (as they are automatically in suitable continuous cases). I hope these remarks are useful. Bill [For admin and other information see: http://www.mta.ca/~cat-dist/ ]