Subject: categories: Re: Undirected graphs citation

From: Marco Grandis <[email-masked]>
Date: Wed, 1 Mar 2006 09:41:45 +0100

Vaughan Pratt asked about:

>  undirected graphs ...  as presheaves on the full subcategory 1 and
> 2 of Set?
 
It is the 2-truncation of "symmetric simplicial sets" as presheaves
on finite cardinals, cf (*).

Curiously, symmetric simplicial sets have been rarely considered.
Even if simplicial complexes (well-known!) are a symmetric notion and
have a natural embedding in symmetric simplicial sets.  While
simplicial sets are a directed notion, used as an undirected one in
classical Algebraic Topology.

(*) M. Grandis, Finite sets and symmetric simplicial sets, Theory
Appl. Categ. 8 (2001), No. 8, 244-252.

Marco Grandis

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From: [email-masked]
Date: Wed, 01 Mar 2006 16:29:08 -0500

Why the "curious" omission of this topos from most discussions of combinatorial topology ?

The introduction of ordered simplices by Eilenberg 60 years ago is usually explained in
subjective terms like the annoying extra degeneracies that had to be eliminated in the
previous theory . However there is a objective requirement clearly pointed out by Gabriel &
Zisman. (They spoke of Kelley spaces but that was a mistake due to Kelley's excellent
exposition of the k-spaces of Hurewicz). The requirement is that geometric realization r (left
adjoint to "singular" s) be left exact, so that products and equations on fleshed out spaces
can be the reflection simply of the same operations on the combinatorial models.

If we construe spaces as Johnstone did (or in many other possible ways) as forming
themselves a topos, the the above requirement is simply that r/s constitute a geometric
morphism of toposes. To understand the qualitative distinction between the possible
codomain combinatorial toposes, it is very helpful to note their role as CLASSIFYING toposes.
That role is not always easy  to grasp in the specific if one starts with primitives and axioms
for a first order theory, tries to present the corresponding Lindenbaum category, then takes
sheaves on that, etc. Fortunately in some cases one can bypass that presentation process
because the resulting small category is already well known.

Preheaves on the category of non-empty finite posets clearly classify arbitrary non-trivial
distributive lattices in any Grothendieck topos. In other words an s/r teory  can be based on
an "interval" object in spaces that has a DL structure. If we want that to factor through the
subtopos of simplicial sets, we note that the latter is the classifier for those special DLs that
are totally ordered, which translates geometrically to a condition on an interval that the
square is a union of two triangles; since "union" depends on the topology of space, that
condition is often not true for DLs that  at first glance look like intervals.

There are other relevant subtoposes (ie positive classes of DLs) for example those for which
the canonical map from the generic one I to the truth-object is actually a homomorphism
with respect to both lattice operations.

But the presheaves on non-empty finite sets is the subtopos that classifies Boolean algebras.
The generic BA is the obvious one. It does not really describe well its relation to the total
orderings to call it the "synmmetric version". As the one contains all small categories, this
one analogously contains all groupoids. It can still receive an r/s pair as required if only a
space with a BA structure is used as the "interval". The natural choice for that is the in finite
dimensional sphere, which indeed has a continuous BA structure that contains the usual
interval as a subDL. If only this had been known 60 years ago, we could have done without
the simplicial sets, for this singular theory reads to the same homotopy category.  Note that
any Grothendieck topos (including ss!) has a canonical BA object, hence enjoys a canonical
r/s theory valued here.

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From: "Clemens.BERGER" <[email-masked]>
Date: Thu, 02 Mar 2006 11:13:18 +0100

[email-masked] wrote:

>>> undirected graphs ...  as presheaves on the full subcategory 1
>>and 2 of Set?
>>>
A comment on combinatorial models for spaces : there is the notion of a
CW-poset (due to Bjoerner) which is a poset that can be identified with
the poset of cells of a regular CW-complex; those posets have very
special properties, but from a homotopy point of view, every space is
weakly homotopy equivalent to a regular CW-complex, so every space has a
``CW-poset-model''. Taking the nerve of this poset yields a
``simplicial-set-model'' of the same space; geometrically, the passage
from the poset to its nerve corresponds to barycentric subdivision. Now,
there is an old paper by Daniel Kan (Amer. J. Math. 79 (1957) 449-476,
section 10), where he shows that the barycentric subdivision of a
simplicial set actually has the structure of a symmetric simplicial set.
So, to some extent, symmetric simplicial sets are doubly subdivided
regular CW-complexes. What I wanted to point out is that in the
progression CW-poset->simplicial set->symmetric simplicial set, the
combinatorial information about the space is not increasing, but rather
decreasing, which is one possible argument in favour of simplicial sets
versus symmetric simplicial sets. Another argument for the category of
simplicial sets is that it is a topos which contains the category of
small categories as a full reflective monoidal subcategory. A third
argument is the ubiquitous occurence of simplicial cotriple resolutions.

In his pursuit of stacks, Grothendieck raises the question of what is
the structure on a small category A, which ensures that the presheaf
topos A^ may be endowed with a homotopy theory equivalent to the
homotopy theory of spaces (such a category is called a test-category) or
with the homotopy theory of spaces above a given space (such a category
is called a local-test-category). In the presence of an r/s adjunction
like in Lavwere's message above, the criteria of being local-test or
test are easily spelled out:

A is local-test iff for each object a of A, the projection A[a]\times
\Omega_A\to A[a] realises to a weak equivalence, where \Omega_A is the
subobject classifier of A^ and A[a] is the presheaf represented by a.

A is test iff A is local-test and the nerve of A is weakly contractible.

With best regards,

                       Clemens Berger.