The goal of this post is to extend earlier ideas, of a sheaf defined on Confn(M)×R⩾0, to a family of sheaves defined on ⋃nk=1Confn(M)×R⩾0=Ran⩽n(M)×R⩾0.
Recall our main map f:Confn(M)×R⩾0VR(−)→SCnHom(Δ∙,−)→sSet. Following Definition 1 and Proposition 2 in a previous post ("Artin gluing a sheaf 2: simplicial sets and configuration spaces," 2018-01-31), define a sheaf Fk on Xk by Fk(U)={Sk,max{1⩽ℓ⩽Nk : U∩Bℓ≠∅} if U is good,S∅ else if U≠∅,(1) for all k=1,…,n. We have assumed a total order on all simplicial complexes on k vertices, induced by a cover Uk,…,Uk,Nk of nested opens of Xk. This induces a total order Sk,1,…,Sk,Nk on the image of Rank(M)×R⩾0 in sSet, and by the product order, a total order on all of sSet′:=f(Ran⩽n(M)×R⩾0).
Recall our main map f:Confn(M)×R⩾0VR(−)→SCnHom(Δ∙,−)→sSet. Following Definition 1 and Proposition 2 in a previous post ("Artin gluing a sheaf 2: simplicial sets and configuration spaces," 2018-01-31), define a sheaf Fk on Xk by Fk(U)={Sk,max{1⩽ℓ⩽Nk : U∩Bℓ≠∅} if U is good,S∅ else if U≠∅,(1) for all k=1,…,n. We have assumed a total order on all simplicial complexes on k vertices, induced by a cover Uk,…,Uk,Nk of nested opens of Xk. This induces a total order Sk,1,…,Sk,Nk on the image of Rank(M)×R⩾0 in sSet, and by the product order, a total order on all of sSet′:=f(Ran⩽n(M)×R⩾0).
A small example
Let n=3, so X=Ran⩽3(M)×R⩾0. We already have F1,F2,F3 on X1,X2,X3, respectively, and we will extend them from the top down to sheaves over all of X, as in the diagram below.
The map i will be the inclusion of an open set into a larger one, and j the inclusion of a closed set into a larger one. Recall that the pullback of two sheaves is defined equivalently by a map of sheaves on the boundary of the open nd closed sets. With that in mind, for U⊆X2∪X3 good, the pullback square
defines Fd0, where the d0 indicates the face map that skips the 0th spot. The sheaf Fd1 is defined similarly, but by the face map d1, and Fd2 by the face map d2. For each of these three sheaves on X3∪X2, we have two other sheaves, based on where the single point maps to. However, we note that for U⊆X good and U∩X1≠∅, ((i∗Fd0×j∗F1)(U) defined by d0) = ((i∗Fd1×j∗F1)(U) defined by d0), where × denotes the pullback over the appropriate sheaf, and similarly for the other sheaves on good sets intersecting X1. We now have 6 unique shaves on all of X.
Generalizing
Now let n be any positive integer, and X=Ran⩽n(M)×R⩾0. We reverse the indexation of the Fk and Xk above to make notation less cumbersome (so now Fk is Fn−k+1 from (1), over Xk=Rann−k+1(M)×R⩾0). Define pullback sheaves Fdℓ1 for ℓ1=0,…,n on X2∪X2 by the diagram
At the kth step, for 1<k<n, we have sheaves Fdℓ1⋯dℓk−1 over ⋃km=1Xm, defined by sequences of face maps dℓk−1 when going from Xk to Xk−1 and so on, where ℓm∈{0,…,n−m+1}. Define pullback sheaves Fdℓ1⋯dℓk−1dℓk, for ℓk=0,…,n−k+1 on ⋃k+1m=1Xk by the diagram
At the end of this inductive process, we have n! distinct sheaves Fdℓ1⋯dℓn−1 on all of X. Note there is a sheaf map Fdℓ1⋯dℓi⋯dℓn−1→Fdℓ1⋯dℓ′i⋯dℓn−1, given on U good by Fdℓ1⋯dℓi⋯dℓn−1(U)=S↦{S if |S0|⩽n−i,(ℓi ℓ′i)(S) else, where (ℓi ℓ′i)∈Sn (the symmetric group on the numbers 0,…,n−1) is the transposition swaps the ℓi and ℓ′i indices of S0, the 0-cells of S, inducing a map of simplicial sets. If the two sheaves differ in only two indices ℓi≠ℓ′i and ℓj≠ℓ′j, with i<j, then we get S↦(ℓj ℓ′j)dℓi−1⋯dℓj(ℓi ℓ′i)(S). Here (ℓj ℓ′j)dℓi−1⋯dℓj is the element of Sn−i found by taking (ℓj ℓ′j) from Sn−j to Sn−i by the sequence of group inclusion maps induced by the face maps dℓj,…,dℓi−1.
Remark: This construction is not the most satisfying, for several reasons:
Remark: This construction is not the most satisfying, for several reasons:
- we do not have a single sheaf, rather a family of sheaves, and
- the use of "good" sets leaves something to be desired, as we should be able to consider larger sets.
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