nLab well-pointed endofunctor




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.)


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.


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.