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.

  • $\begingroup$ Yes - the category is called 'exhaustive'. See ncatlab.org/nlab/show/exhaustive%20category ;-P $\endgroup$ May 2 '12 at 3:49
  • $\begingroup$ For everyone else, check the references at that nLab page. $\endgroup$ May 2 '12 at 3:49
  • $\begingroup$ @David: Very funny. $\endgroup$ May 2 '12 at 17:20
  • $\begingroup$ 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. $\endgroup$ May 3 '12 at 10:18
  • 2
    $\begingroup$ @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. $\endgroup$
    – user44191
    Mar 28 '19 at 15:22

As said in the comments, the relevant term seems to be exhaustive; the term was defined in response to this question.


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