nLab well-pointed endofunctor

Contents

Definition

Definition

An endofunctor S:𝒜𝒜S \colon \mathcal{A}\to \mathcal{A} is called pointed if it is equipped with a natural transformation σ:Id 𝒜S\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,σ)(S, \sigma) (Def. ) is called well-pointed if Sσ=σSS\sigma = \sigma S as natural transformations SSSS \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.

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]

Last revised on July 8, 2025 at 15:54:36. See the history of this page for a list of all contributions to it.