Contents
### Context

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# 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.

###### Proposition

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

###### Proof

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

Conversely, if $T\eta=\eta T$, then $T\eta$ is an inverse for $\mu$. Indeed,

$T\eta_A\circ\mu_A = \mu_{TA} \circ T^2\eta_A = \mu_{TA}\circ T\eta_{TA} = 1,$

and $\mu_A\circ T\eta_A = 1$ 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 $\mathcal{A}$ 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.