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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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 sheaves 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.
   • CommentAuthorzskoda
   • CommentTimeAug 31st 2017
   • PermaLink
   Author: zskoda
   Format: MarkdownItex$n$Lab entries have entry [[opposite category]] which is redirected to if calling for [[dual category]]. There is also entry [[duality]]. I know that the category textbooks consider opposite and dual synonyms, as far as the definitions are concerned but in practice there is slight difference in usage which is sensible enough to take formal stance about. Everybody who uses term opposite category means literally the original category with arrows reversed. When saying dual category one sometimes means the same, the opposite category, but more often in practice one means any category __equivalent__ to the opposite category (not necessarily isomorphic to it). In other words, dual is an extension of the notion of opposite up to an equivalence. I think we should reflect this standard usage in $n$Lab somehow. But do we have a consensus among us ?

   $n$Lab entries have entry opposite category which is redirected to if calling for dual category. There is also entry duality. I know that the category textbooks consider opposite and dual synonyms, as far as the definitions are concerned but in practice there is slight difference in usage which is sensible enough to take formal stance about.

   Everybody who uses term opposite category means literally the original category with arrows reversed.

   When saying dual category one sometimes means the same, the opposite category, but more often in practice one means any category equivalent to the opposite category (not necessarily isomorphic to it). In other words, dual is an extension of the notion of opposite up to an equivalence.

   I think we should reflect this standard usage in $n$Lab somehow. But do we have a consensus among us ?

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeAug 31st 2017
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn English we could say that the opposite category is _the_ dual category, but one can also make sense of a dual category and not of an opposite category. Maybe this is not fully satisfactory as one can have some other default duality in mind when talking about certain context, when the dual category will not be the opposite category.

   In English we could say that the opposite category is the dual category, but one can also make sense of a dual category and not of an opposite category. Maybe this is not fully satisfactory as one can have some other default duality in mind when talking about certain context, when the dual category will not be the opposite category.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeAug 31st 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAccording to the [[principle of equivalence]], there is no difference...

   According to the principle of equivalence, there is no difference…

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeSep 1st 2017
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI know, but the terminology "opposite" refers to an exact construction, that is the way literature defines it. If it wants to respect the principle the definition has to change.

   I know, but the terminology “opposite” refers to an exact construction, that is the way literature defines it. If it wants to respect the principle the definition has to change.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeSep 1st 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexJust because something is defined by an exact construction doesn't mean it doesn't respect the principle of equivalence. Cartesian products of sets (under a ZFC foundation) are constructed by an exact construction, but we only ever use their universal property.

   Just because something is defined by an exact construction doesn’t mean it doesn’t respect the principle of equivalence. Cartesian products of sets (under a ZFC foundation) are constructed by an exact construction, but we only ever use their universal property.