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category theory

# Contents

## Definition

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

###### Definition

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

The definition of a pointed endofunctor naturally extends to any 2-category, where we can define a pointed endomorphism as an endomorphism $s : a \to a$ equipped with a 2-cell $\sigma : 1_a \Rightarrow s$ 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 $\sigma$ 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 a pointed object in the endofunctor category is not quite right in general.

The terminal object of the category of endofunctors on $\mathcal{A}$ is the functor $T$ 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 $\mathcal{A}$. So a pointed object in the endofunctor category should be an endofunctor $S:\mathcal{A} \to \mathcal{A}$ equipped with a natural transformation $\sigma:T \to S$.

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

Last revised on March 8, 2020 at 13:02:48. See the history of this page for a list of all contributions to it.