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Monday, February 5, 2018

Artin gluing a sheaf 3: the Ran space

The goal of this post is to extend earlier ideas, of a sheaf defined on Confn(M)×R0, to a family of sheaves defined on nk=1Confn(M)×R0=Rann(M)×R0.

Recall our main map f:Confn(M)×R0VR()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{1Nk : UB} 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)×R0 in sSet, and by the product order, a total order on all of sSet:=f(Rann(M)×R0).

A small example

Let n=3, so X=Ran3(M)×R0. 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 UX2X3 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 X3X2, we have two other sheaves, based on where the single point maps to. However, we note that for UX good and UX1, ((iFd0×jF1)(U) defined by d0)  =  ((iFd1×jF1)(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=Rann(M)×R0. We reverse the indexation of the Fk and Xk above to make notation less cumbersome (so now Fk is Fnk+1 from (1), over Xk=Rannk+1(M)×R0). Define pullback sheaves Fd1 for 1=0,,n on X2X2 by the diagram
At the kth step, for 1<k<n, we have sheaves Fd1dk1 over km=1Xm, defined by sequences of face maps dk1 when going from Xk to Xk1 and so on, where m{0,,nm+1}. Define pullback sheaves Fd1dk1dk, for k=0,,nk+1 on k+1m=1Xk by the diagram
At the end of this inductive process, we have n! distinct sheaves Fd1dn1 on all of X. Note there is a sheaf map Fd1didn1Fd1didn1, given on U good by Fd1didn1(U)=S{S if |S0|ni,(i i)(S) else, where (i i)Sn (the symmetric group on the numbers 0,,n1) is the transposition swaps the i and i indices of S0, the 0-cells of S, inducing a map of simplicial sets. If the two sheaves differ in only two indices ii and jj, with i<j, then we get S(j j)di1dj(i i)(S). Here (j j)di1dj is the element of Sni found by taking (j j) from Snj to Sni by the sequence of group inclusion maps induced by the face maps dj,,di1.

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.
Both will hopefully be remedied in a later post.

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