Coverages that are not pretopologies - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T16:42:17Z http://mathoverflow.net/feeds/question/91266 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91266/coverages-that-are-not-pretopologies Coverages that are not pretopologies David Roberts 2012-03-15T10:42:05Z 2012-03-17T00:51:40Z <p>A <a href="http://ncatlab.org/nlab/show/coverage" rel="nofollow">coverage</a> on a category $C$ is a collection of covering families <code>$\{u_i \to a\}$</code> for each object $a$ of $C$ such that for each arrow $b\to a$ there is a covering family for $b$ which fits into a commutative square with the original covering family.</p> <p>A <a href="http://ncatlab.org/nlab/show/Grothendieck+pretopology" rel="nofollow">pretopology</a> is an example of a coverage, but much stronger, in that the weak pullbacks are required to be pullbacks, covering families need to be closed under composition, and families consisting of a single isomorphism are covering.</p> <p>In order to define sheaves, it is enough to start with a coverage, and indeed there is a very useful coverage on the category $Diff$ of finite dimensional smooth manifolds consisting of <a href="http://ncatlab.org/nlab/show/good+open+cover" rel="nofollow">good open covers</a>. Also, given a topological space, and a basis for the topology on it, one can define a coverage which is not a pretopology using these subsets.</p> <p>However, the coverage of good open covers on $Diff$ does satisfy the 'covers compose' axiom of a pretopology, but not the other two, so is not in a sense completely general. What I would like to see is examples of coverages which satisfy zero, one or two but not all of the pretopology axioms, in various combinations. And, most importantly, are not contrived finite examples, but on categories which can be considered 'geometric' (probably should be <a href="http://ncatlab.org/nlab/show/extensive+category" rel="nofollow">extensive</a>, or similar), like $Ring^{op}$, $Top$, $Diff$ and so on.</p>