David Corfield
coalgebra

An algebra for an endofunctor $F$ on a category $C$ is an object $c$ together with a morphism $\alpha :Fc\to c$.

A coalgebra for an endofunctor $F$ on a category $C$ is an object $c$ together with a morphism $\alpha :c\to Fc$.

Examples

• The endofunctor on Set, $F(X)=1+X$ has as algebras sets with a designated element and unary function. Initial algebra is the natural numbers with $0$ and successor. Coalgebras are partially defined unary functions. Terminal coalgebra is $\mathbb{N}\cup \{\mathrm{\infty }\}$, with function undefined at $0$, predecessor at a finite natural number, and $f(\mathrm{\infty })=\mathrm{\infty }$.

• For $X\mapsto Ax+1$, the initial algebra consists of lists (finite sequences) of elements of $A$; the final coalgebra consists of streams (finite or infinite sequences) of elements of $A$.

• For $X\mapsto Ax$, the inital algebra is empty; the final coalgebra consists of (infinite) sequences of elements of A.

• The endofunctor on Classes of sets which sends a class $A$ to the class of subclasses of $A$ which are sets has as initial algebra the class of sets and as final coalgebra the class of non-well-founded sets.

Induction & Coinduction

There appears to be an imbalance in the amount of algebraic to coalgebraic thinking within mathematics. Some options:

1. It’s not a distinction worth making –- a coalgebra for $(C,F)$ is an algebra for $(C^{\mathrm{op}},F^{\mathrm{op}})$.

2. It is a distinction worth making, but there’s plenty of coalgebraic thinking going on –- it’s just not flagged as such.

3. Coalgebra is a small industry providing a few tools for specific situations, largely in computer science, but with occasional uses in topology, etc.

Blog posts

• Coalgebraically Thinking

• The Status of Coalgebra