Exactness for sequential unions of monomorphisms?

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

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

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

are pullbacks.

• Yes - the category is called 'exhaustive'. See ncatlab.org/nlab/show/exhaustive%20category ;-P May 2 '12 at 3:49
• For everyone else, check the references at that nLab page. May 2 '12 at 3:49
• @David: Very funny. 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. May 3 '12 at 10:18
• @DavidWhite I think answering with a community wiki is probably the better answer, as I think of closing as saying that the question is not fit for the site (and this question is fit); I put the comment answer as a community wiki instead. Mar 28 '19 at 15:22