Extending the definition of "pure of dimension n" from simplicial complexes to simplicial sets?

2011-12-31T13:06:51Z

Recall that a (combinatorial) simplicial complex $X$ is said to be $n$-dimensional if it contains at least one face of dimension $n$ and no faces of dimension $n+1$. Further, an $n$-dimensional finite simplicial complex $X$ is said to be pure if every face $\delta$ of dimension $k\lt n$ is a face of some $n$-dimensional face of $X$.

For simplicial sets, we have a different definition for dimension given as:

A simplicial set $S$ is said to be $n$-skeletal if the inclusion $\operatorname{Sk}^n S \subseteq S$ of the $n$-skeleton of $S$ is an isomorphism. We say that $S$ is $m$-dimensional if $m$ is the smallest number for which $S$ is $m$-skeletal.

Given a simplicial complex $X$ as a simplicial set, we see that the associated simplicial set $\Delta[X]$ is $n$-dimensional if and only if $X$ is $n$-dimensional.

Is there a useful notion extending the definition of purity to those simplicial sets with only finitely many nondegenerate simplices?

The property that I'm hoping for in such an extension is sort of a generalization of the "prism decomposition" for products of simplices. That is, for the product of two simplicial sets $S$ and $T$, pure of dimension $s$ and $t$ respectively, I would like the product $S\times T$ to be pure of dimension $s+t$. It is certainly the case that the product has dimension $s+t$ (this can be seen working simplex by simplex), but without a proper definition of purity, we can't give a real generalization.

Answer by David Roberts for Extending the definition of "pure of dimension n" from simplicial complexes to simplicial sets?

2011-12-31T20:00:33Z

I [EDIT: almost] would follow the suggestion you made in your comment.

[EDIT: below the line was my original answer, which is wrong, as pointed out by Karol in the comments. Here is an answer which is less functorial, but I hope more correct.]

It would be nice to define pure dimension internal to categories other than $Set$. Unfortunately the definition so far need the internal logic to satisfy excluded middle, and I'm not sure how to get around this. My intuition is that a finitely extensive regular category should be the minimum requirement.

Assume $S$ is $n$-dimensional (i.e. $n$-skeletal). Let $ND(S_n) \subset S_n$ be the non-degenerate $n$-simplices (in Harry's original example this is a finite set), that is, the compliment of the joint image of $$ s_{i_1}\ldots s_{i_{n-k}} : \coprod_{k = 0}^{n-1} S_k \to S_n $$ (we can do this step for simplicial objects in a Boolean topos, or more generally in a regular category with complements of all subobjects and finite coproducts). This defines a presheaf on $\tilde\Delta_n$, the subcategory of $\Delta$ consisting of objects $\le n$ and only the coface maps. The $n$-simplices are $ND(S_n)$ and all lower dimensional simplices are the same as for $S$. Unfortunately this construction is not functorial, because a map of simplicial sets may send non-degenerate simplices to degenerate ones.

Definition: A presheaf $S'$ on $\tilde\Delta_n$ is of pure dimension if the collection of maps

$$ d_{i_1}\ldots d_{i_{n-k}} : S'_n \to \coprod_{k = 0}^{n-1} S'_k $$

is jointly regular epimorphic. This step works in any regular category with finite coproducts. This differs from Harry's suggestion in that we ask that all $k$-simplices are a face of a non-degenerate $n$-simplex.

I'll have a think about the proof that pure dimension, defined this way, is additive. It is here that I think the extensivity should be used.

I believe there is a functor $$ R:Set^{\Delta^{op}} \to Set^{\tilde\Delta^{op}} $$ (where $\tilde\Delta$ is the subcategory of $\Delta$ with the same objects but only coface maps) that takes a simplicial set and removes all degenerate simplices. If I am not mistaken, this should be adjoint to one of the adjoints to the restriction along the inclusion $\tilde\Delta \to \Delta$, the one which adds the smallest number of degenerate simplices. (Check this, I am in a rush to leave on an interstate drive)

There are analogues $R_n$ for the finite subcategories of $\Delta$ consisting of all objects $\le n$.

Assume $S$ is $n$-dimensional. Define it to be of pure dimension if $S' := R_n sk_n S$ satisfies:

for all $m \le n$, the face maps $d_i : S'_m \to S'_{m+1}$ are jointly surjective.

I think the functors $R,R_n$ preserve products, as do the skeleton functors, so showing that dimension is additive should be ok.

Extending the definition of "pure of dimension n" from simplicial complexes to simplicial sets? - MathOverflow

Extending the definition of "pure of dimension n" from simplicial complexes to simplicial sets?

Answer by David Roberts for Extending the definition of "pure of dimension n" from simplicial complexes to simplicial sets?