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Category Theory edit
Is "sieve" indeed something in category theory to glue things together, or do you mean "sheaf"? Perhaps such elaborations don't belong in a disambiguation page and we should only refer as it as "Sieve (category theory)".
I am not an expert, but what I know is from [1]. A sieve S on an object a is a collection of arrows that ends in a (so, a is the codomain) and is "closed under composition on the right". The latter means that if there is an arrow (a morphism) from node b to a in the sieve, then for all arrows leading to b, there is a corresponding composed arrow in S. So, the sieve wraps all arrows leading b and connects it to a. There can be multiple sieves on an object. It depends on how many incoming arrows the object a has.
It is possible to define a sheaf using covering sieves. But like you would be able to get from above is that a sieve is not the one defining how to glue.
[1] "Pattern matching: a sheaf-theoretic approach", Srinivas, PhD thesis, 1991, University of California