The n-Category Café

It is not true that $|G(X)|$ is naturally isomorphic to $[X]$.

For example, if we take $X=\{\{1\},\{2\},\{1,2\}\}$ as in Jesse’s example, then $G(X)$ contains simplices of the form $(1,2,1,2,1,\ldots)$ that are non-degenerate (they do not have consecutive repeated vertices) and have dimension as large as you like. This means that $|G(X)|$ is infinite-dimensional, unlike $[X]$ which is 1-dimensional.

However, it sounds plausible that $|G(X)|\to [X]$ is a homotopy equivalence. For example, in $|G(X)|$ the 2-simplex $(1,2,1)$ appears as a globular 2-cell with boundary the edges $(1,2)$ and $(2,1)$, so the problematic circle formed by $(1,2)$ and $(2,1)$ can now be contracted.

The classical quick reference for some of this is in the little early survey article by Curtis: Simplicial homotopy theory, Advances in Math. 6 (1971), 107 – 209. He attributes to Barratt a result that the geometric realisation of a simplicial set can be triangulated, and thus is the geometric realisation (in the simplicial complex sense) of a simplicial complex. The proof is sketched in Curtis and involves the double barycentric subdivision of the simplicial set. (The discussion is in the last section of the paper.)

Thanks!

My own puzzle does not involve barycentric subdivisions, but Curtis’ paper (free online from the Evil Empire) is still relevant.

In Example 1.3, Curtis describes a way to get a simplicial set from a simplicial complex. This should be approximately functor I’m calling $G : SCpx \to SSet$ in my conversation with Richard and Jesse above. However, Curtis works with “geometrical” simplicial complexes rather than the “abstract” simplicial complexes I’d prefer to think about. So, I should be careful.

In Section 1.29, Curtis describes the usual geometric realization of a simplicial set $X$, which he calls $R(X)$ rather than the now-standard $|X|$. He then claims that if we take a geometrical simplicial complex $X$, and let $K$ be the simplicial set we get from it by the procedure in Example 1.3, then $R(K)$ is homeomorphic to $X$.

This seems to contradict Richard’s claim (above) that that $R(G(X))$ is not homeomorphic to $X$. Perhaps the distinction between “geometrical” and “abstract” simplicial complexes is a bigger deal than I imagined. Or conceivably Curtis is just wrong, since he doesn’t actually prove his claim. Or maybe Richard is wrong… or more likely, I’m just mired in confusion.

Nonetheless, I feel I’m making progress! So thanks!

Nice! It’s neat how a spoonful of probability theory helps the medicine go down. Has anyone used this connection to do anything interesting?

The formula for $\mu(A)$ reminds me of the formula for conditional probability, and now one of the pictures in the Wikipedia article on conditional probability reminds me of something about simplicial complexes.

Is there a nice probabilistic interpretation of your map $|N(X)| \to [X]$, or am I just imagining things?

My understanding is that the $-1$-simplices are the connected components of the augmented simplicial set (which is not the same thing as being the connected components of the un-augmented simplicial set). This is because the connected components functor is left adjoint to the discrete inclusion of sets into augmented simplicial sets, and is therefore calculated by evaluating at the terminal object of the site. Since $[-1]$ is the terminal object, the $-1$-simplices have to be the connected components. I’m not sure this is a good way to think about things – it’s more like “formal” connected components (since the $-1$-simplices are not necessarily the limit of all the other ones).

Do your phantom vertices relate to connected components in this way? I don’t immediately see a relation, but I haven’t sat down with it.

Yes, I think of the $(-1)$-simplices as something like ‘formal connected components’, with the property that any number of actual connected components can belong to each formal connected component. I think they’re pretty different from the phantoms.

For those a bit intimidated by the adjunction you mentioned, a nice lowbrow way to understand all this is to note that there’s a face map

$$\partial_0 : X_1 \to X_0$$

assigning a $(-1)$-simplex to each vertex of our augemented simplicial set, and the relations on face maps imply that if two vertices are connected by an edge, they must be assigned the same $(-1)$-simplex. So, two vertices lying in the same connected component must belong to the same ‘formal connected component’, but not conversely.

Todd Trimble puts all this to good use in his note on bar constructions. I’ll quote a bit.

B.1. Acyclic structures 

We follow the algebraists' convention, taking the simplicial category 
to mean the category of finite ordinals (including the empty ordinal) 
and order-preserving maps. It is well-known that \Delta is initial 
amongst strict monoidal categories equipped with a monoid, which in 
\Delta is 1. This induces a monad 1 + - on \Delta, called the 
translation monad. By composition, this in turn induces a pullback 
comonad P on simplicial sets; by the Kan construction, it has a left
adjoint which is a monad C on simplicial sets, called the cone monad. 

To explain this terminology, we recall the topologists' convention, 
where \Delta is restricted to the full subcategory \Delta_+ of 
non-empty ordinals.  If S-set denotes the category of simplicial 
sets under the algebraists' convention, and S_{+}-set that under the
topologists' convention, then by restriction we get a functor 

S-set --> S_{+}-set, 

which has a left adjoint. The left adjoint augments a S_{+} set X by 
its set of path components \pi_{0}(X). Starting with X, we can apply 
the left augmentation, followed by the cone monad, followed by 
restriction S-set --> S_{+} set: this gives a monad which, passing 
to geometric realization, is the mapping cone of X --> \pi_{0}(X). 
The right adjoint to this monad carries a comonad structure; the
category of algebras over the monad is equivalent to the category 
of coalgebras over the comonad. This category of algebras/coalgebras
could be called the "acyclic topos": the algebras X are acyclic as
simplicial sets.  More to the point, the algebra structure CX --> X 
witnesses this acyclicity by providing a representative basepoint 
for each path component of X, together with a well-behaved simplicial
homotopy which contracts each component down to its basepoint. 

Definition: An acyclic structure is an algebra over C (or coalgebra 
over P). An S-acyclic structure is one augmented over a set S. 

A morphism between acyclic structures is just a C-algebra map; a 
morphism between S-acyclic structures is one whose component at S is 
the identity. 

It doesn't matter whether the monad C is taken under the algebraists' 
or topologists' convention: the category of C-algebras in S-set is
equivalent to the category of C-algebras in S_{+}-set, since given a
C-algebra in S-set, 

                      --> 
            ... X_{1} --> X_{0} --> X_{-1}, 

it follows from acyclicity that this portion of the simplicial 
structure is a split coequalizer, so that the augmentation map is 
the usual augmentation to its set of path components. 

In the sequel, it will be useful to regard an acyclic structure, 
as a functor X: \Delta^{op} --> Set, as a right coalgebra XT --> X 
over the translation comonad T: \Delta^{op} --> \Delta^{op}. 

Perhaps worth mentioning is that both of these notions of simplicial complex are “order ideals” (in the boolean algebra on the ground-set; see Stanley’s Enumerative Combinatorics). The “augmented” ones correspond to squarefree monomial ideals. For more general ideals one considers order ideals of monomials (equivalenty, multi-complexes)—their study goes back to Macaulay.