The goal of this post is to give an alternative perspective on making a sheaf over X=Ran⩽n(M)×R⩾0, alternative to that of a previous post ("Artin gluing a sheaf 3: the Ran space," 2018-02-05). We will have one unique sheaf on all of X, valued either in simplicial complexes or simplicial sets.
Remark: Here we straddle the geometric category SC of simplicial complexes and the algebraic category sSet of simplicial sets. There is a functor [ ⋅ ]:SC→sSet for which every n-simplex in S gets (n+1)! elements in [S], representing all the ways of ordering the vertices of S (which we would like to view as unordered, to begin with).
Recall from previous posts:
Lemma: A descending path γ:I→X induces a unique morphism h(γ(0))→h(γ(1)).
Proof: Write γ(0)={P1,…,Pn} and γ(1)={Q1,…,Qm}, with m⩽n. Since the path is descending, points can only collide, not split. Hence γ induces n paths γi:I→M for i=1,…,n, with γi the path based at Pi. This induces a map h(γ(0))0→h(γ(1))0 on 0-cells (vertices or 0-objects), which completely defines a map h(γ(0))→h(γ(1)) in the desired category. ◻
Our sheaves will be defined using colimits. Fortunately, both SC and sSet have (small) colimits. Finally, we also need an auxiliary function σ:Op(X)→SC that finds the correct simplicial complex. Define it by σ(U)={Sk,ℓ if U≠∅, for k=max{1⩽k′⩽n : U∩Rank(M)×R⩾0≠∅},ℓ=max{1⩽ℓ′⩽Nk : U∩Bk,ℓ′≠∅},∗ if U=∅.
Proposition 1: Let F be the function Op(X)op→SC on objects given by F(U)=colim(σ(U)⇉ This is a functor and satisfies the sheaf gluing conditions.
Proof: We have a well-defined function, so we have to describe the restriction maps and show gluing works. Since V\subseteq U\subseteq X, every S in the directed system defining \mathcal F(V) is contained in the directed system defining \mathcal F(U). As there are maps \sigma(V)\to \mathcal F(V) and S\to \mathcal F(V), for every S in the directed system of V, precomposing with any descending path we get maps \sigma (U)\to \mathcal F(V) and S\to \mathcal F(V), for every S in the directed system of U. Then universality of the colimit gives us a unique map \mathcal F(U)\to \mathcal F(V). Note that if there are no paths (decending or otherwise) from U to V, then the colimit over an empty diagram still exists, it is just the initial object \emptyset of SC.
To check the gluing condition, first note that every open U\subseteq X must nontrivially intersect \Ran^n(M)\times \R_{\geqslant 0}, the top stratum (in the point-counting stratification). So for W = U\cap V, if we have \alpha\in \mathcal F(U) and \beta \in \mathcal F(V) such that \alpha|_W = \beta|_W is a k-simplex, then \alpha and \beta must have been k-simplices as well. This is because a simplicial takes a simplex to a simplex, and we cannot collide points while remaining in the top stratum. Hence the pullback of S\owns \alpha and T\owns \beta via some induced maps (by descending paths) from U to W and V to W, respectively, will restrict to the identity on the chosen k-simplex. Hence the gluing condition holds, and \mathcal F is a sheaf. \square
Functoriality of [\ \cdot\ ] allows us to extend the proof to build a sheaf valued in simplicial sets.
Proposition 2: Let \mathcal G be the function \Op(X)^{op}\to \sSet on objects given by \mathcal G(U) = \colim\left([\sigma(U)]\rightrightarrows S\ :\ \text{every }[\sigma(U)]\to S \text{ is induced by a descending }\gamma:I\to U\right). This is a functor and satisfies the sheaf gluing conditions.
Remark: The sheaf \mathcal G is non-trivial on more sets. For example, any path contained within one stratum of X induces the identity map on simplicial sets (though not on simplicial complexes). Hence \mathcal G is non-trivial on every open set contained within a single stratum.
References: nLab (article "Simplicial complexes"), n-category Cafe (post "Simplicial Sets vs. Simplicial Complexes," 2017-08-19)
Remark: Here we straddle the geometric category SC of simplicial complexes and the algebraic category sSet of simplicial sets. There is a functor [ ⋅ ]:SC→sSet for which every n-simplex in S gets (n+1)! elements in [S], representing all the ways of ordering the vertices of S (which we would like to view as unordered, to begin with).
Recall from previous posts:
- maps f:X→SC and g=[f]:X→sSet,
- the SCk-stratification of Rank(M)×R⩾0,
- the point-counting stratification of Ran⩽n(M),
- the combined (via the product order) SC⩽n-stratification of Ran⩽n(M)×R⩾0,
- an induced (by the SCk-stratification) cover by nested open sets Bk,1,…,Bk,Nk of Rank(M)×R⩾0,
- a corresponding induced total order Sk,1,…,Sk,Nk on f(Rank(M)×R⩾0).
Lemma: A descending path γ:I→X induces a unique morphism h(γ(0))→h(γ(1)).
Proof: Write γ(0)={P1,…,Pn} and γ(1)={Q1,…,Qm}, with m⩽n. Since the path is descending, points can only collide, not split. Hence γ induces n paths γi:I→M for i=1,…,n, with γi the path based at Pi. This induces a map h(γ(0))0→h(γ(1))0 on 0-cells (vertices or 0-objects), which completely defines a map h(γ(0))→h(γ(1)) in the desired category. ◻
Our sheaves will be defined using colimits. Fortunately, both SC and sSet have (small) colimits. Finally, we also need an auxiliary function σ:Op(X)→SC that finds the correct simplicial complex. Define it by σ(U)={Sk,ℓ if U≠∅, for k=max{1⩽k′⩽n : U∩Rank(M)×R⩾0≠∅},ℓ=max{1⩽ℓ′⩽Nk : U∩Bk,ℓ′≠∅},∗ if U=∅.
Proposition 1: Let F be the function Op(X)op→SC on objects given by F(U)=colim(σ(U)⇉ This is a functor and satisfies the sheaf gluing conditions.
Proof: We have a well-defined function, so we have to describe the restriction maps and show gluing works. Since V\subseteq U\subseteq X, every S in the directed system defining \mathcal F(V) is contained in the directed system defining \mathcal F(U). As there are maps \sigma(V)\to \mathcal F(V) and S\to \mathcal F(V), for every S in the directed system of V, precomposing with any descending path we get maps \sigma (U)\to \mathcal F(V) and S\to \mathcal F(V), for every S in the directed system of U. Then universality of the colimit gives us a unique map \mathcal F(U)\to \mathcal F(V). Note that if there are no paths (decending or otherwise) from U to V, then the colimit over an empty diagram still exists, it is just the initial object \emptyset of SC.
To check the gluing condition, first note that every open U\subseteq X must nontrivially intersect \Ran^n(M)\times \R_{\geqslant 0}, the top stratum (in the point-counting stratification). So for W = U\cap V, if we have \alpha\in \mathcal F(U) and \beta \in \mathcal F(V) such that \alpha|_W = \beta|_W is a k-simplex, then \alpha and \beta must have been k-simplices as well. This is because a simplicial takes a simplex to a simplex, and we cannot collide points while remaining in the top stratum. Hence the pullback of S\owns \alpha and T\owns \beta via some induced maps (by descending paths) from U to W and V to W, respectively, will restrict to the identity on the chosen k-simplex. Hence the gluing condition holds, and \mathcal F is a sheaf. \square
Functoriality of [\ \cdot\ ] allows us to extend the proof to build a sheaf valued in simplicial sets.
Proposition 2: Let \mathcal G be the function \Op(X)^{op}\to \sSet on objects given by \mathcal G(U) = \colim\left([\sigma(U)]\rightrightarrows S\ :\ \text{every }[\sigma(U)]\to S \text{ is induced by a descending }\gamma:I\to U\right). This is a functor and satisfies the sheaf gluing conditions.
Remark: The sheaf \mathcal G is non-trivial on more sets. For example, any path contained within one stratum of X induces the identity map on simplicial sets (though not on simplicial complexes). Hence \mathcal G is non-trivial on every open set contained within a single stratum.
References: nLab (article "Simplicial complexes"), n-category Cafe (post "Simplicial Sets vs. Simplicial Complexes," 2017-08-19)
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