**reversing the arrows**in the construction.

I'm aware that this appears bizarre, but there's nothing else it

*could*arise from, right? Duality shows up everywhere in mathematics, but the only "natural" notion of duality we get with arrows is to reverse them. So somehow, if category theory is to formalise all mathematical intuition in some way, then all the good dualities you see must somehow be nicely expressible in terms of this category theoretic duality.

To drive this point home, let's just list out a bunch of "good things appearing in twos" we've seen in mathematics (other than what we've already seen) and see how they can be expressed categorically.

**Dual order**(i.e. given a relation $\le$, the dual order is defined by $\ge$) -- this one is just trivial. Every poset is a preordered set and therefore can itself be seen as a category, so the dual order is the dual of this category.**Sup, Inf**(i.e. the lower bound of the upper bounds, the upper bound of the lower bounds in a lattice) -- this follows directly from the above.**LCM, GCD**-- special case of the above, as the integers can be organised into a lattice based on divisibility.**de Morgan duality**(i.e. the duality induced by the complement/negation operation in Boolean algebra, e.g. between unions and intersections) -- in the case of logic, propositions can be ordered by implication; in the case of set theory, sets can be ordered by inclusion.**Addition and multiplication**-- Follows from the duality of sums and products of objects by considering the category of finite sets.**A subspace and its orthogonal complement**(more generally, a subobject and a quotient by it, i.e. where $O\to A\to B\to C\to O$, $A$ and $C$).

To make this more precise:

We define theopposite categoryas the category with all arrows reversed. Where an invertible functor between these categories maps a diagram to another, those diagrams are calleddual notions.

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