Marco Grandis wrote:

It is the 2-truncation of "symmetric simplicial sets" as presheaves on finite cardinals, cf (*).

Marco, thanks for that, this is really nice. It hadn't occurred to me to extend undirected graphs to higher dimensions but ... of course! While "symmetric" is technically correct terminology here (and indeed graph theorists often define undirected graphs as the symmetric case of directed graphs), "undirected" conveys the appropriate intuition that the edges and higher-dimensional cells have no specific orientation. Whereas the automorphism group of a directed n-cell is the trivial group, that of an undirected n-cell is S_N where N=n+1, i.e. undirected n-cells are permitted to "flop around" in all N! possible ways. Moreover the group as a whole behaves like a single cell with regard to identification: if one of the N! copies of an undirected edge is identified with a copy of another undirected edge, all copies are identified bijectively, i.e. the two undirected cells are identified. So without taking issue with Marco's terminology "symmetric" here, since it is correct and natural, I would nevertheless like to suggest that in the context of simplicial complexes, and with ordinary graphs as a precedent, that "undirected" be considered an acceptable synonym for "symmetric". But that connection leads to another that hadn't previously occurred to me (though again this is unlikely to be news to at least some). This is the question of an appropriate language for the signature of simplicial complexes in general. Each operation can be named with a lambda-calculus term of the form \xyz.xyzzy, that is, a string of (distinct of course) variables followed by another string of the same variables with repetitions or omissions allowed. Dually to undirected simplicial complexes being a special case of (directed) simplicial complexes, the language for the latter is the special case of that for the former in which the body of the lambda term preserves the order of the formal parameter list; the smallest term thus disallowed is \xy.yx. In particular s and t (source and target) arise as respectively \xy.x and \xy.y: given an edge, bind x and y to its source and target respectively and return the designated vertex. Similarly \x.xx denotes the distinguished self-loop at a given vertex x (these being reflexive graphs since we allow contraction). The lambda terms with N=n+1 parameters have as domain the set of n-cells. The one operation that undirected graphs have that is absent in the general directed case is \xy.yx, which names the other member of the group of automorphic copies of an undirected edge. These two copies always travel together (literally as a group), justifying the intuition that the group of both of them constitutes a single edge (or n-cell). For general n these copies of a given cell are named by the linear lambda terms, those with exactly one occurrence of each formal parameter. Any given cell of a graph attaches to the rest of the graph at various points around that cell, but graph homomorphisms cannot disturb those points of attachment or incidence, though it can certainly map the cell to any of the N! isomorphic copies of itself. It should be pointed out that "undirected graph" as a "special case" of "directed graph" has its syntactic rather than semantic meaning here, in the sense that UGraph (undirected graphs) does not embed in DGraph (directed graphs), at least not in the expected way. Consider a graph with two vertices x,y, two edges from x to y, and two edges from y to x. If a graph homomorphism identifies the two edges from x to y, it need not identify the other two edges in DGraph, but it does need to identify them in UGraph. Unless I've overlooked some subtlety, 2-UGraph does however embed in the expected way in 2-DGraph, where 2 = {0,1} (= V in enriched parlance) are the possible cardinalities of "homsets", i.e. at most one edge in each direction. This is because the implicit pairing in 2-DGraph perfectly mimics the explicit pairing in 2-UGraph. This would explain why graph theorists, who usually work in 2-DGraph, encounter no ambiguity of the Set-UGraph < Set-DGraph kind when they define an undirected graph as simply a symmetric graph, one with no one-way streets. Vaughan Pratt Marco Grandis wrote:

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

wlawvere@buffalo.edu wrote:

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.

Quoting Marco Grandis <grandis@dima.unige.it>:

...

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

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.