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]

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]

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

]]>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?

]]>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.

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

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

]]>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:

**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)

**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

left and right Kan extension

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.