nLab list of notable initial algebras and terminal coalgebras

Example

A list of notable endofunctors, and their initial algebras/terminal coalgebras.

Nonexistent (co)algebras are labelled with ‘/’, and unknown ones with ‘?’.

Base category	Endofunctor	Initial Algebra	Final Coalgebra	Note, reference
Set	Const $A$	$A$	$A$
Set	$X \mapsto X$	$\varnothing$	$1$
Set	$X \mapsto 1 + X$	$\mathbb{N}$	Conatural numbers $\mathbb{N}^\infty$	extended natural number
Set	$X \mapsto A + X$	$A \times \mathbb{N}$	$A \times \mathbb{N} + 1$, ie conatural numbers “terminated” (when they aren’t $\infty$) with $A$	partial map classifier
Set	$X \mapsto X + X$ or $X \mapsto 2 \times X$	$\emptyset$	$2^\mathbb{N}$ (i.e. one definition of Stream $2$)
Set	$X \mapsto A \times X$	$\emptyset$	$A^\mathbb{N}$ (i.e. one definition of Stream $A$)	sequence, writer monad, stream
Set	$X \mapsto X \times X$ or $X \mapsto [2, X]$	$\emptyset \simeq [2, \emptyset]$	1 (the unique infinite unlabelled binary tree)
Set	$X \mapsto [A, X]$	$[A, \emptyset]$	1	reader monad
Set	$X \mapsto 1 + A \times X$	List $A$	another definition of Stream $A$; i.e. potentially infinite List $A$	list, stream
Set	$X \mapsto 1 + A \times X^2$	Finite binary tree with $A$-labelled nodes	Potentially infinite binary tree with $A$-labelled nodes	tree
Set	$X \mapsto B + A \times X^n$	Finite $n$-ary tree with $A$-labelled nodes and $B$-labelled leaves	Potentially infinite $n$-ary tree with $A$-labelled nodes with and $B$-labelled leaves
Set	$X \mapsto B + A \times \text{List}(X)$	Finite tree with $A$-labelled nodes and $B$-labelled leaves	Potentially infinite tree with $A$-labelled nodes with and $B$-labelled leaves	The number of subtrees is not fixed to a particular $n$, it could be any number
Set	$X \mapsto O \times [I, X]$	$O \times [I, \emptyset]$	Potentially infinite Moore machine
Set	$X \mapsto [I, O \times X]$	$[I, \emptyset]$	Potentially infinite Mealy machine
Set	$X \mapsto \mathcal{P}(X)$	/	/
Set	$X \mapsto \mathcal{P}_{\text{fin}}(X)$	Finite rooted forests	Potentially infinite finitely-branching rooted forests
Set	polynomial endofunctor	W-type	M-type
Bipointed Sets	$X \mapsto X \vee X$	dyadic rational numbers in the interval $[0,1]$	The closed interval $[0,1] \subseteq \mathbb{R}$	coalgebra of the real interval
linearly ordered sets	$X \mapsto \omega \cdot X$, where $\omega \cdot X$ is the cartesian product of the natural numbers with the underlying set of $X$, equipped with the lexicographic order.	$\emptyset$	The non-negative real numbers $\mathbb{R}_{\geq 0}$	real number
Archimedean ordered fields	$X \mapsto X$ the identity functor	The rational numbers $\mathbb{Q}$	The real numbers $\mathbb{R}$
Archimedean ordered fields	$X \mapsto \mathcal{D}(X)$ where $\mathcal{D}(X)$ is the Archimedean ordered field of two-sided Dedekind cuts of $X$	The real numbers $\mathbb{R}$	The real numbers $\mathbb{R}$
Archimedean ordered fields	$X \mapsto \mathcal{C}(X)$ where $\mathcal{C}(X)$ is the quotient of Cauchy sequences in the Archimedean ordered field $X$	The HoTT book real numbers $\mathbb{R}_H$	The Dedekind real numbers $\mathbb{R}$	These are the same objects in the presence of excluded middle or countable choice.
Any category	The constant functor $X \mapsto A$ given object $A$	$A$	$A$
Any category	The identity functor $X \mapsto X$	initial object	terminal object
Any extensive category	$X \mapsto 1 + X$ given terminal object $1$ and coproduct $+$	natural numbers object	?
Any closed abelian category	$X \mapsto I \sqcup (A \otimes X)$ given tensor unit $I$, tensor product $\otimes$, coproduct $\sqcup$, and object $A$	tensor algebra of $A$	cofree coalgebra over $A$	tensor algebra, cofree coalgebra
$\infty$-Grpd	$X \mapsto \Sigma X$	sphere spectrum $\mathbb{S}$	?	suspension