I was recently reading about Kan extensions and the codensity monad. Apparently you can think of the codensity monad as a bit like an endomorphism monoid internal to a functor category.

The right Kan extension is a bit like an exponential in a functor category.

$$[F, G](a) = \forall b, (a \rightarrow F(b)) \rightarrow G(b) $$

And you can "eval" and "curry" things as well.

$$\text{eval} : [F, G] \circ F \Rightarrow G$$

Left kan extension would be a "coexponential" like object. You can think of it as a little like a tuple of a value and a continuation.

$$(F - G)(a) = \Sigma_b\, F(b) \times (G(b) \rightarrow a) $$

It strikes me "exponential" isn't quite the right terminology. Composition certainly isn't a Cartesian product.

So left and right Kan extensions would be coresiduals and residuals?

I'm interested in exploring further so I want to know the precise and accurate terminology and notation instead of the hodgepodge of "its a bit like an exponential" I have right now.

  • 1
    $\begingroup$ Essentially the same question was asked here on MathOverflow. $\endgroup$
    – varkor
    Commented Sep 10, 2021 at 12:00
  • $\begingroup$ @varkor thank you that's a great place to start looking up things from. $\endgroup$ Commented Sep 10, 2021 at 18:05


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