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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeAug 31st 2017
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   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
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   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
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   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
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   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
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   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.