nLab well-pointed endofunctor

Definition

Definition

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.)

Definition

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.

Properties

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

Related concepts

• A monad is idempotent iff its underlying pointed endofunctor is well-pointed.

References

Well-pointed endofunctors were studied already under the name “symmetric unads” in:

• K. A. Hardie: Instances and ramifications of the semi-adjoint situation I. Prestable reflection and symmetric unads, Quaestiones Mathematicae 2 1-3 (1977) 147-158 [doi:10.1080/16073606.1977.9632539]

The modern terminology is due to:

• 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]