An endofunctor $S \colon \mathcal{A}\to \mathcal{A}$ is called pointed if it is equipped with a natural transformation $\sigma \colon Id_\mathcal{A} \to S$ from the identity functor.

(Beware that is *not* quite the notion of a pointed object in the endo-functor category, since the identity functor is not in general a terminal object there. See the remark there.)

A pointed endofunctor $(S, \sigma)$ (Def. ) is called **well-pointed** if $S\sigma = \sigma S$ as natural transformations $S \longrightarrow S \circ S$.

The dual notion is known as a **well-copointed** endofunctor.

Well-pointed endofunctors have a particularly well-behaved free algebra sequence; see *transfinite construction of free algebras*.

- Max Kelly,
*A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on.*Bull. Austral. Math. Soc. 22 (1980), 1–83. doi:10.1017/S0004972700006353

Last revised on June 5, 2023 at 09:50:37. See the history of this page for a list of all contributions to it.