MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Yes - the category is called 'exhaustive'. See ;-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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.