nLab
coalgebra for an endofunctor

For a category $C$ and endofunctor $F$, a coalgebra of $F$ is an object $X$ in $C$ and a map $\alpha :X\to F(X)$.

Given two coalgebras $(x,\eta :x\to Fx)$, $(y,\theta :y\to Fy)$, a coalgebra map is a morphism $f:x\to y$ which respects the coalgebra structures:

$$\theta \circ f=F(f)\circ \eta$$\theta \circ f = F(f) \circ \eta

The dual concept is an algebra for an endofunctor.

See also terminal coalgebra.

Examples of coalgebras for functors on Set:

• $X\to F(X)=D(X)$, the set of probability distributions on $X$: Markov chain on $X$.
• $X\to F(X)=P(X)$, the powerset on $X$: Binary relation on $X$.
• $X\to F(X)=X^A\times B$: Deterministic automaton.
• $X\to F(X)=P(X^A\times B)$: Nondeterministic automaton.
• $X\to F(X)=A\times X\times X$, for a set of labels $A$: Labelled binary tree.

See coalgebra for examples on categories of modules.

Blog resources

David Corfield, Coalgebraically Thinking

David Corfield, The Status of Coalgebra

References

Jiri Adamek, Introduction to coalgebras, Theory and Applications of Categories, Vol. 14 (2005), No. 8, 157-199.