Re: Simplicial versus (cubical) with connections)

[F William Lawvere <wlawvere@buffalo.edu>]

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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


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