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A coverage on a category $C$ is a collection of covering families $\{u_i \to a\}$ 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 weak pullback diagram (that is, the existence without commutative square with the uniqueness of pullbacks)original covering family.

A pretopology 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.

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 good open covers. 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.

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 extensive, or similar), like $Ring^{op}$, $Top$, $Diff$ and so on.

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# Coverages that are not pretopologies

A coverage on a category $C$ is a collection of covering families $\{u_i \to a\}$ 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 weak pullback diagram (that is, the existence without the uniqueness of pullbacks).

A pretopology 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.

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 good open covers. 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.

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 extensive, or similar), like $Ring^{op}$, $Top$, $Diff$ and so on.