nLab pointed endofunctor




An endofunctor on a category 𝒜\mathcal{A} is pointed if it is equipped with a natural transformation from the identity functor.



An endofunctor S:𝒜→𝒜S \colon \mathcal{A}\to \mathcal{A} is called pointed if it is equipped with a natural transformation σ:Id 𝒜→S\sigma \colon Id_\mathcal{A} \to S from the identity functor on 𝒜\mathcal{A}.


Def. is not the usual classical notion of a pointed object in the endo-functor category, since the identity functor is not in general a terminal object there. It is however pointed in the sense of a pointed object in a monoidal category, involving a morphism out of the monoidal unit, since the identity functor is the tensor unit for the canonical monoidal category-structure on the endofunctor category (given by horizontal composition). In this sense, a pointed endofunctor may be regarded as being equipped with a “monoidal point”. A monad has a canonical such point (see Exp. below), usually called the unit.


The notion of a pointed endofunctors, Def , naturally extends to any 2-category, where we can define a pointed endomorphism to be an endo-1-morphism s:a→as \colon a \to a equipped with a 2-morphism σ:id a⇒s\sigma \colon id_a \Rightarrow s from the identity morphism.


A pointed endofunctor (S,σ)(S, \sigma) (Def. ) is called well-pointed if Sσ=σSS\sigma = \sigma S as natural transformations S⟶S∘SS \longrightarrow S \circ S.


Relation to free algebras

The pointed algebras over an endofunctor on a pointed endofunctor induce a theory of transfinite construction of free algebras over general endofunctors.



A monad can be regarded as a pointed endofunctor where σ\sigma is its unit.


The pointed endofunctor (T,η)(T,\eta) underlying a monad (T:C→C,η:1→T,μ:T 2→T)(T : C \to C, \eta: 1\to T, \mu : T^2\to T) is well-pointed if and only if TT is idempotent, i.e. μ\mu is an isomorphism.


If μ\mu is an isomorphism then Tη=ηTT\eta=\eta T since both are sections of μ\mu.

Conversely, if Tη=ηTT\eta=\eta T, then TηT\eta is an inverse for μ\mu. Indeed,

Tη A∘μ A=μ TA∘T 2η A=μ TA∘Tη TA=1 T 2A, T\eta_A\circ\mu_A = \mu_{TA}\circ T^2\eta_A = \mu_{TA}\circ T\eta_{TA} = 1_{T^2A},

and μ A∘Tη A=1 TA\mu_A\circ T\eta_A = 1_{TA} by the right unit law.

For pointed object in the endofunctor category

Notice that the statement which one might expect, that a pointed endofunctor is a pointed object in the endofunctor category is not quite right in general.

The terminal object of the category of endofunctors on 𝒜\mathcal{A} is the functor TT which sends all objects to *\ast and all morphisms to the unique morphism *→*\ast \to \ast, where *\ast is the terminal object of the category 𝒜\mathcal{A}. So a pointed object in the endofunctor category should be an endofunctor S:𝒜→𝒜S:\mathcal{A} \to \mathcal{A} equipped with a natural transformation σ:T→S\sigma:T \to S.

Rather, a pointed endofunctor is equipped with a map from the unit object for the monoidal structure on the endofunctor category.


  • Harvey Wolff, p. 234 of: Free monads and the orthogonal subcategory problem, Journal of Pure and Applied Algebra 13 3 (1978) 233-242 [doi:10.1016/0022-4049(78)90010-5]

  • 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 11:31:12. See the history of this page for a list of all contributions to it.