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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 27th 2014
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   Author: Urs
   Format: MarkdownItexI happened to look at the old entry _[[duality]]_ again and found it a bit thin. So I have expanded it a bit now. This is not meant to be definite, please feel invited to further expand and/or fine-tune: *** Instances of "dualities" relating two different, maybe opposing, but to some extent equivalent concepts or phenomena are ubiquitous in [[mathematics]] (and in [[mathematical physics]], see at _[[dualities in physics]]_). In terms of general abstract concepts in [[category theory]] instances of dualities might be (and have been) organized as follows: 1. **involution** -- any [[automorphism]] which is an [[involution]], hence which squares to the [[identity]] may be though of as exhibiting two dual perspectives on the objects that it acts on; * **abstract duality** -- the operation of sending a [[category]] to its [[opposite category]] is such an involution on [[Cat]] itself (and in fact this is the only non-trivial automorphism of [[Cat]], see [here]((infinity,n)Cat#automorphisms)). This has been called _abstract dualty_. While the construction is a priori tautologous, any given [[opposite category]] often is equivalent to a [[category]] known by other means, which makes abstract duality interesting. * **concrete duality** -- given a [[closed category]] $\mathcal{C}$ and any [[object]] $D$ of it, then the operation $[-,D] : \mathcal{C} \to \mathcal{C}^{op}$ obtained by forming the [[internal hom]] into $D$ sends each object to something like a a $D$-dual object. This is particularly so if $D$ is indeed a [[dualizing object in a closed category]] in that applying this operation twice yields an [[equivalence of categories]] $[[-,D],D] : \mathcal{C} \stackrel{\simeq}{\to} \mathcal{C}$ (so that $[-,D]$ is a (contravariant) [[involution]] on $\mathcal{C}$). If $\mathcal{C}$ is in addition a [[closed monoidal category]] then under some conditions on $D$ (but not in general) this kind of concrete dualization coincides with the concept of forming [[dual objects]] in [[monoidal categories]]. From ([Lawvere-Rosebrugh, chapter 7](#LawvereRosebrugh))): > Not every statement will be taken into its formal dual by the process of dualizing with respect to $V$, and indeed a large part of the study of mathematics >>space vs. quantity > and of logic >>theory vs. example >may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond. (p. 122) 1. **adjunction** -- another categorical concept of duality is that of [[adjunction]], as in pairs of [[adjoint functors]]. Via the many incarnations of [[universal constructions]] in [[category theory]] this accounts for all dualities that arise as instances as the dual pairs * [[limit]] and [[colimit]] * left and right [[Kan extension]] * [[end]] and [[coend]] * [[dependent sum]] and [[dependent product]] * [[existential quantification]] and [[universal quantification]] (Given that the saying has it that "Everything in mathematics is a Kan extension", this goes some way in explaining the ubiquity of duality in mathematics.) When the adjoint functors are [[monads]] and hence [[modalities]], then adjointness between them has been argued to specifically express the concept of [[duality of opposites]]. Adjunctions and specifically [[dual adjunctions]] ("[[Galois connection]]") may be thought of as a generalized version of the above abstract duality: every [[dual adjunction]] induces a _maximal dual equivalence_ between [[subcategories]].

   I happened to look at the old entry duality again and found it a bit thin.

   So I have expanded it a bit now. This is not meant to be definite, please feel invited to further expand and/or fine-tune:

   ---

   Instances of “dualities” relating two different, maybe opposing, but to some extent equivalent concepts or phenomena are ubiquitous in mathematics (and in mathematical physics, see at dualities in physics).

   In terms of general abstract concepts in category theory instances of dualities might be (and have been) organized as follows:

   1. involution – any automorphism which is an involution, hence which squares to the identity may be though of as exhibiting two dual perspectives on the objects that it acts on;

   • abstract duality – the operation of sending a category to its opposite category is such an involution on Cat itself (and in fact this is the only non-trivial automorphism of Cat, see here). This has been called abstract dualty. While the construction is a priori tautologous, any given opposite category often is equivalent to a category known by other means, which makes abstract duality interesting.

   • concrete duality – given a closed category $\mathcal{C}$ and any object $D$ of it, then the operation $[-,D] : \mathcal{C} \to \mathcal{C}^{op}$ obtained by forming the internal hom into $D$ sends each object to something like a a $D$-dual object. This is particularly so if $D$ is indeed a dualizing object in a closed category in that applying this operation twice yields an equivalence of categories $[[-,D],D] : \mathcal{C} \stackrel{\simeq}{\to} \mathcal{C}$ (so that $[-,D]$ is a (contravariant) involution on $\mathcal{C}$). If $\mathcal{C}$ is in addition a closed monoidal category then under some conditions on $D$ (but not in general) this kind of concrete dualization coincides with the concept of forming dual objects in monoidal categories.

     From (Lawvere-Rosebrugh, chapter 7)):

     Not every statement will be taken into its formal dual by the process of dualizing with respect to $V$, and indeed a large part of the study of mathematics

     space vs. quantity

     and of logic

     theory vs. example

     may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond. (p. 122)

   1. adjunction – another categorical concept of duality is that of adjunction, as in pairs of adjoint functors. Via the many incarnations of universal constructions in category theory this accounts for all dualities that arise as instances as the dual pairs

   • limit and colimit

   • left and right Kan extension

   • end and coend

   • dependent sum and dependent product

   • existential quantification and universal quantification

     (Given that the saying has it that “Everything in mathematics is a Kan extension”, this goes some way in explaining the ubiquity of duality in mathematics.)

     When the adjoint functors are monads and hence modalities, then adjointness between them has been argued to specifically express the concept of duality of opposites.

     Adjunctions and specifically dual adjunctions (“Galois connection”) may be thought of as a generalized version of the above abstract duality: every dual adjunction induces a maximal dual equivalence between subcategories.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 27th 2014
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   Author: David_Corfield
   Format: MarkdownItexGreat to have more added, in view of that workshop I mentioned.

   Great to have more added, in view of that workshop I mentioned.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeMar 1st 2014
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   Author: TobyBartels
   Format: MarkdownItexThe adjunction duality is not really different from the abstract duality under involution. A limit in $C$ is a colimit in $C^op$, etc.

   The adjunction duality is not really different from the abstract duality under involution. A limit in $C$ is a colimit in $C^op$, etc.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 1st 2014
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   Author: Urs
   Format: MarkdownItexLet's see, how should we phrase it. Having an adjoint pair is different from identifying a ctegory with an opposite category, though of course passing to opposites interchanges left and right adjoints.

   Let’s see, how should we phrase it. Having an adjoint pair is different from identifying a ctegory with an opposite category, though of course passing to opposites interchanges left and right adjoints.

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeMar 5th 2014
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   Author: TobyBartels
   Format: MarkdownItexWhen you say ‘having an adjoint pair’, you don\'t mean a [[adjoint pair|pair of adjoint functors]] but a [[adjoint triple|pair of adjunctions]], right? So an example would be $(1 \overset{const_\bot}\to C) \dashv (U\colon C \overset{!}\to 1) \dashv (G\colon 1 \overset{const_\top}\to C)$. You would want to call anything like this a duality?

   When you say ‘having an adjoint pair’, you don't mean a pair of adjoint functors but a pair of adjunctions, right? So an example would be $(1 \overset{const_\bot}\to C) \dashv (U\colon C \overset{!}\to 1) \dashv (G\colon 1 \overset{const_\top}\to C)$. You would want to call anything like this a duality?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 5th 2014
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   Author: Urs
   Format: MarkdownItexRight, i suppose i didn't say it well. Let's try to improve. Right now i am sort of offline though.

   Right, i suppose i didn’t say it well. Let’s try to improve. Right now i am sort of offline though.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMar 31st 2023
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   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Francis Borceux]], Section 1.10 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) &lbrack;[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/duality/48">diff</a>, <a href="https://ncatlab.org/nlab/revision/duality/48">v48</a>, <a href="https://ncatlab.org/nlab/show/duality">current</a>

   added pointer to:

   • Francis Borceux, Section 1.10 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) [doi:10.1017/CBO9780511525858]

   diff, v48, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 20th 2023
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   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Saunders MacLane]], §II.1 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) &lbrack;[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/duality/49">diff</a>, <a href="https://ncatlab.org/nlab/revision/duality/49">v49</a>, <a href="https://ncatlab.org/nlab/show/duality">current</a>

   added pointer to:

   • Saunders MacLane, §II.1 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

   diff, v49, current