Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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
   • PermaLink
   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
   • PermaLink
   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
   • PermaLink
   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
   • PermaLink
   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.