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Stream: learning: questions

Topic: pseudosections?

Naso (Jun 14 2022 at 01:50):

In 'Sheaf Theory Through Examples' (Rosiak) https://arxiv.org/pdf/2012.08669.pdf in a footnote (p229) it says '...in a variety of applications of sheaves, including beyond sheaves of vector spaces on complexes, it is possible that exact equality of assignments will be unattainable or the least valuable thing to consider. There are a few ways to develop machinery to accommodate this, but it is the beyond the scope of this book to touch on them in detail."

It then cites as a simpler example some work of Michael Robinson who defines epsilon-approximate sections/pseudosections.

What is this other machinery Rosiak hints to?

Why I'm interested: I'm looking at presheaves of simplicial sets on a topological space, and I want to evaluate the sheaf condition with a relaxed compatibility condition for sections, so that two sections $s, t$ are compatible, not only if they are equal when restricted to their intersection, but if they are both a degeneracy of the same nondegenerate cell.
Then I want to compute the Cech cohomology to find out about global sections etc.

My solution atm is to postcompose the presheaf of simplicial sets $F : \mathcal{O}(X)^{op} \to \mathsf{sSet}$ with the functor $nd : \mathsf{sSet} \to \mathsf{Set}$ (https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/beginner.20questions/near/282761175 ) that sends a simplicial set to its set of nondegenerate cells, so that $nd \circ X$ is just a presheaf of sets and the ordinary compatibility condition for sections is the one I want.

So I would like to know if this approach is reasonable, or is there a better way, maybe involving the machinery Rosiak was hinting at?

(Maybe this is relevant: the presheaves $F : \mathcal{O}(X)^{op} \to \mathsf{sSet}$ I'm looking at are all subpresheaves of a fixed one $F \leq S$, so that postcomposing with $nd$ is actually reversible (since a simplicial subset is fully specified by its nondegenerate cells).)