A paradox:

- Goodwillie calculus considers only finitary functors.
- $TC$ isn't finitary.
- Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.

(Here, a finitary functor is one preserving filtered colimits[1]. $\partial$ denotes the first Goodwillie derivative. $TC,K,THH$ are respectively topological cyclic homology, algebraic $K$-theory, and topological Hochschild homology, regarded as functors from $E_1$-ring spectra to spectra.)

Obviously I don't fully understand that last point, which is just a rough idea I think I've read somewhere, and that's what I want to ask about.

**Questions:**

What does it mean to say that the first Goodwillie derivative of $TC$ is $THH$?

Is there a general formalism for Goodwillie calculus of non-finitary (but, say, accessible) functors? If so, how much of the usual theory goes through?

If the answer to (2) is "yes", does it specialize in the case of $TC$ to recover the answer to (1)?

[1] At any rate, in Goodwillie calculus one always requires one's functor to commute with sequential colimits. Any functor that commutes with sequential colimits and $\aleph_1$-filtered colimits commutes with all filtered colimits. $TC$ is defined from $THH$ (which commutes with filtered colimits) via a countable limit and therefore $TC$ commutes with $\aleph_1$-filtered colimits. I conclude that if $TC$ doesn't commute with filtered colimits, then it already doesn't commute with sequential colimits, and so standard Goodwillie calculus doesn't apply to it.