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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 2nd 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI wonder if it would be worth gathering together some place the extra properties and structures a category may have. Along the lines of [[additive and abelian categories]], but bringing in qualities on the distributive side of things too. In the last couple of days I've noticed a couple more examples: [adhesive category](http://arxiv.org/abs/1103.0600) >Adhesive categories are categories which have pushouts with one leg a monomorphism, all pullbacks, and certain exactness conditions relating these pushouts and pullbacks, and [belian category](http://arxiv.org/abs/1105.5290)

   I wonder if it would be worth gathering together some place the extra properties and structures a category may have. Along the lines of additive and abelian categories, but bringing in qualities on the distributive side of things too.

   In the last couple of days I’ve noticed a couple more examples:

   adhesive category

   Adhesive categories are categories which have pushouts with one leg a monomorphism, all pullbacks, and certain exactness conditions relating these pushouts and pullbacks,

   and belian category

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 2nd 2011
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   Author: Mike Shulman
   Format: MarkdownItexGood idea! Perhaps [[exactness properties of categories]]?

   Good idea! Perhaps exactness properties of categories?

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 2nd 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIs there a definition of what counts as an *exact* property? It covers all of the properties in [[familial regularity and exactness]] and [[additive and abelian categories]]?

   Is there a definition of what counts as an exact property? It covers all of the properties in familial regularity and exactness and additive and abelian categories?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeNov 2nd 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI don't know a formal definition; I think it's generally used to refer to properties that relate to the interaction of limits and colimits. Steve Lack and Richard Garner's recent paper on "Lex colimits" gives a formal setting for many exactness properties; they're abstract colimits relative to the doctrine of finitely-complete categories. I don't know whether *all* properties colloquially called 'exactness properties' fit into such a framework.

   I don’t know a formal definition; I think it’s generally used to refer to properties that relate to the interaction of limits and colimits. Steve Lack and Richard Garner’s recent paper on “Lex colimits” gives a formal setting for many exactness properties; they’re abstract colimits relative to the doctrine of finitely-complete categories. I don’t know whether all properties colloquially called ’exactness properties’ fit into such a framework.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 3rd 2011
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   Author: DavidRoberts
   Format: MarkdownItexThere's also a notion of category with _structure_ (not just property) that has a form of exactness: subcanonical site.

   There’s also a notion of category with structure (not just property) that has a form of exactness: subcanonical site.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeApr 9th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexFinally got around to creating [[exactness property]]. I'm not attached to the name.

   Finally got around to creating exactness property. I’m not attached to the name.

7. • CommentRowNumber7.
   • CommentAuthorFinnLawler
   • CommentTimeApr 9th 2012
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   Author: FinnLawler
   Format: MarkdownItexAdded a link to the arXiv version of the Garner--Lack paper to [[exactness property]].

   Added a link to the arXiv version of the Garner–Lack paper to exactness property.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeApr 9th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIt's great when other people cover for my laziness.

   It’s great when other people cover for my laziness.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeApr 10th 2012
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   Author: TobyBartels
   Format: MarkdownItexWhy is ‘exactness property’ more general than ‘familial regularity and exactness’?

   Why is ‘exactness property’ more general than ‘familial regularity and exactness’?

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeApr 10th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAn exactness property is an informal term for some sort of commutativity between limits and colimits. Familial regularity and exactness is about a particular class of such conditions. Adhesive categories and extensive categories, for instance, are exactness properties that are not instances of familial regularity and exactness.

    An exactness property is an informal term for some sort of commutativity between limits and colimits. Familial regularity and exactness is about a particular class of such conditions. Adhesive categories and extensive categories, for instance, are exactness properties that are not instances of familial regularity and exactness.