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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 19th 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItexFor the purposes of this question, let's say that a [[site]] S has _unions_ if for any cover {f_i: X_i → A | i∈I} of an object A and any subset J⊂I of the indexing set we can find h: Y → A and a cover {g_j → Y | j∈J} of Y such that f_j = h∘g_j for all j∈J. The reason why such a property is interesting is that it allows us to reduce the [[descent]] property for arbitrary finite covers to descent for covers with two elements and empty covers. It is satisfied for the site of smooth manifolds, for example. Is such a concept known in the literature?

   For the purposes of this question, let’s say that a site S has unions if for any cover {f_i: X_i → A | i∈I} of an object A and any subset J⊂I of the indexing set we can find h: Y → A and a cover {g_j → Y | j∈J} of Y such that f_j = h∘g_j for all j∈J.

   The reason why such a property is interesting is that it allows us to reduce the descent property for arbitrary finite covers to descent for covers with two elements and empty covers.

   It is satisfied for the site of smooth manifolds, for example.

   Is such a concept known in the literature?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeDec 20th 2015
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   Author: Zhen Lin
   Format: MarkdownItexThis looks like a generalisation of the notion of superextensive topology.

   This looks like a generalisation of the notion of superextensive topology.