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In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.

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Stacks in modern number theory/arithmetic geometry

Perhaps this doesn't count as "modern" but stacks are ubiquitous in the 1972 Antwerp paper of Deligne and Rapoport. … Recall that the $\Gamma_0(N)$ moduli problem is not representable, and so they must frequently work directly with stacks before moving to the coarse moduli scheme we all know and love. …
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