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Saturday, February 10, 2018

Artin gluing a sheaf 4: a single sheaf in two ways

The goal of this post is to give an alternative perspective on making a sheaf over X=Rann(M)×R0, 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 [  ]:SCsSet 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:XSC and g=[f]:XsSet,
  • the SCk-stratification of Rank(M)×R0,
  • the point-counting stratification of Rann(M),
  • the combined (via the product order) SCn-stratification of Rann(M)×R0,
  • an induced (by the SCk-stratification) cover by nested open sets Bk,1,,Bk,Nk of Rank(M)×R0,
  • a corresponding induced total order Sk,1,,Sk,Nk on f(Rank(M)×R0).
The product order also induces a cover by nested opens of all of X and a total order on f(X) and g(X). We call a path γ:IX a descending path if t1<t2I implies h(γ(t1))h(γ(t2)) in any stratified space h:XA. Below, h is either f or g.

Lemma: A descending path γ:IX induces a unique morphism h(γ(0))h(γ(1)).

Proof: Write γ(0)={P1,,Pn} and γ(1)={Q1,,Qm}, with mn. Since the path is descending, points can only collide, not split. Hence γ induces n paths γi:IM for i=1,,n, with γi the path based at Pi. This induces a map h(γ(0))0h(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, if U, for k=max{1kn : URank(M)×R0},=max{1Nk : UBk,}, if U=.

Proposition 1: Let F be the function Op(X)opSC on objects given by F(U)=colim(σ(U)S : every σ(U)S is induced by a descending γ:IU). 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 VUX, every S in the directed system defining F(V) is contained in the directed system defining F(U). As there are maps σ(V)F(V) and SF(V), for every S in the directed system of V, precomposing with any descending path we get maps σ(U)F(V) and SF(V), for every S in the directed system of U. Then universality of the colimit gives us a unique map F(U)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 of SC.

To check the gluing condition, first note that every open UX must nontrivially intersect Rann(M)×R0, the top stratum (in the point-counting stratification). So for W=UV, if we have αF(U) and βF(V) such that α|W=β|W is a k-simplex, then α and β 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 Sα and Tβ 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 F is a sheaf.

Functoriality of [  ] allows us to extend the proof to build a sheaf valued in simplicial sets.

Proposition 2: Let G be the function Op(X)opsSet on objects given by G(U)=colim([σ(U)]S : every [σ(U)]S is induced by a descending γ:IU). This is a functor and satisfies the sheaf gluing conditions.

Remark: The sheaf 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 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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