Covariant and contravariant

The terms covariant and contravariant come up in many contexts. An earlier post discussed how the terms are used in programming and category theory. The meaning in programming is an instance of the general use in category theory.

Vector fields can be covariant or contravariant too. This is also an instance of the categorical usage, except the terminology is backward.

Michael Spivak explains:

Nowadays such situations are always distinguished by calling the things which go in the same direction “covariant” and the things which go in the opposite direction “contravariant.” Classical terminology used these same words, and it just happens to have reversed this: a vector field is called a contravariant vector field, while a section of T*M is called a covariant vector field. And no one had the gall or authority to reverse terminology sanctified by years of usage. So it’s very easy to remember which kind of vector field is covariant, and which is contravariant — it’s just the opposite of what it logically ought to be.

Emphasis added.

In defense of classical nomenclature, it was established decades before category theory. And as Spivak explains immediately following the quotation above, the original terminology made sense in its original context.

From Spivak’s Differential Geometry, volume 1. I own the 2nd edition and quoted from it. But it’s out of print so I linked to the 3rd edition. I doubt the quote changed between editions, but I don’t know.

Related: Applied category theory

3 thoughts on “Covariant and contravariant

  1. Marvellous! Thank you. This resolves an unfinished investigation by Erik Meijer and me on . It so happens that I have Spivak’s magnum opus on my bucket list — I bought it when it first came out and it’s still on my bookshelf waiting for me.

  2. “It’s the opposite of what it ought to be” is a mnemonic I have to avoid at all costs. I’ve used it a few times in the past, and invariably it becomes a vicious cycle in which my intuition about “what it ought to be” keeps flipping as I internalize it, and I get it wrong half the time for the rest of my life.

  3. It’s not the opposite of what it ought to be; you’re just mistaken in thinking vector fields are fundamental! Differential forms (“co-vectors” or “covariant vector fields”) vary WITH the underlying sheaf of functions under coordinate transformations, while (contravariant) vector fields vary in the OPPOSITE direction to functions.

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