dual

Duality" is a deep concept in mathematics, but an intuitive way to think about it is in terms of tables. When we lay out data in a table, we are familiar with the idea that each row represents a different "entity", and each column represents a different "property". (1/10)

If we flip it over (swapping rows for columns), now we have a new data layout in which each "entity" is really a property, and each "property" is really an entity. It's clearly the same data, just represented differently; if we flip again, we get back to where we started. (2/10).

But this flipped table represents a totally different view of the world: rather than viewing entities as being "fundamental" (and characterized by their properties), now we view properties as being "fundamental" (and characterized by their values for different entities). (3/10)

Mathematically, we can formalize "properties" as functions, mapping elements to values. So what this "flipped table" intuition is telling us is that, for any mathematical structure over which we can define functions, there exists a "flipped" version of that structure... (4/10)

...whose "elements" are really functions, and whose "functions" are really elements: this is the essence of duality. For example, we can imagine a vector space (whose elements are vectors), and then define a set of linear functionals (mapping vectors to scalars) over it. (5/10)

Its dual is then a new vector space whose "vectors" are the linear functionals of the old space, and whose "linear functionals" are the vectors of the old space. Whenever you take a dot product, you're pairing a vector with its dual (a linear functional) to yield a scalar! (6/10)

There is a deeper philosophical point here: sometimes it is useful to study a structure directly (e.g. point set topology); yet sometimes it is useful to study the functions defined over that structure instead (e.g. Morse theory). Duality relates these two perspectives. (7/10)

You can characterize a group via its set of elements, or you can characterize it via its collection of homomorphisms. Taken to its logical extreme, this motivates a fundamental "duality" between the set-theoretic perspective and the categorical perspective on structure. (8/10)

Whenever you take the (conjugate) transpose of a matrix, or reverse the arrows in a category, you are shifting between these two perspectives. Column vectors in a vector space become row vectors in its dual. Output wires in a category become input wires in its dual. (9/10)

The fact that these perspectives are so philosophically different explains why subjects like homology and cohomology theory, which are "the same" (up to arrow reversal), feel so conceptually distinct: the dual is always the same in content, but often distinct in vibe. (10/10)