nLab
pointed endofunctor

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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. Such an endofunctor is well-pointed precisely when the monad is idempotent.

For pointed object in the endofunctor category

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

The terminal object of the category of endofunctors on CC is the functor TT which sends all objects to *\ast and all morphisms to the unique morphism **\ast \to \ast, where *\ast is terminal object of category CC. So the pointed object in the endofunctor category should be a endofunctor S:CCS:C \to C equipped with a natural transformation σ:TS\sigma:T \to S.

Last revised on September 3, 2018 at 10:07:27. See the history of this page for a list of all contributions to it.