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Does the following exactness property have a name?

Consider a category that has pullbacks, and colimits of countable sequences of monomorphisms. Suppose given a diagram

pair of countable unions

such that each $A_n \to A_{n+1}$ is monic, the bottom row is a colimit, and all the squares

partial square

are pullbacks (hence each $B_n \to B_{n+1}$ is also monic). Then the exactness property says that the top row is a colimit if and only if all the squares

final square

are pullbacks.

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Yes - the category is called 'exhaustive'. See ncatlab.org/nlab/show/exhaustive%20category ;-P – David Roberts May 2 '12 at 3:49
    
For everyone else, check the references at that nLab page. – David Roberts May 2 '12 at 3:49
    
@David: Very funny. – Mike Shulman May 2 '12 at 17:20
    
Also: according to the definition I put on the nLab, what I described in the above question is technically an $\omega$-exhaustive (or "countably exhaustive") category. – Mike Shulman May 3 '12 at 10:18

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