nLab
pointed endofunctor

Contents

Contents

Definition

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

Definition

An endofunctor S:𝒜𝒜S:\mathcal{A}\to \mathcal{A} is called pointed if it is equipped with a natural transformation σ:Id 𝒜S\sigma:Id_\mathcal{A} \to S. It is called well-pointed if Sσ=σSS\sigma = \sigma S (as natural transformations SS 2S\to S^2).

The definition of a pointed endofunctor naturally extends to any 2-category, where we can define a pointed endomorphism as an endomorphism s:aas : a \to a equipped with a 2-cell σ:1 as\sigma : 1_a \Rightarrow s from the identity.

Properties

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.

Examples

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

Proposition

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

Proof

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 Aμ A=μ TAT 2η A=μ TATη TA=1, T\eta_A\circ\mu_A = \mu_{TA} \circ T^2\eta_A = \mu_{TA}\circ T\eta_{TA} = 1,

and μ ATη A=1\mu_A\circ T\eta_A = 1 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:𝒜𝒜S:\mathcal{A} \to \mathcal{A} equipped with a natural transformation σ:TS\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.

Last revised on May 25, 2021 at 15:12:44. See the history of this page for a list of all contributions to it.