nLab pointed endofunctor

Idea

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

Definition

Definition

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

Remark

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.

Remark

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 $s \colon a \to a$ equipped with a 2-morphism $\sigma \colon id_a \Rightarrow s$ from the identity morphism.

Definition

A pointed endofunctor $(S, \sigma)$ (Def. ) is called well-pointed if $S\sigma = \sigma S$ as natural transformations $S \longrightarrow S \circ S$.

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

Example

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 if and only if $T$ 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,

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

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

Related concepts

• pointed object

• well-pointed endofunctor

• copointed endofunctor

References

• 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