---
tag: 6BS0300CMV
kind: concept
name: F-coalgebra
msc: [18C15, 68Q85]
sources:
  - url: https://en.wikipedia.org/wiki/F-coalgebra
same_as:
  wikidata: Q3062935
  wikipedia: F-coalgebra
---

## Summary

For an endofunctor $F$ of a category, an $F$-coalgebra is an object $A$ with a morphism $A \to F(A)$.
It is the dual of an $F$-algebra.
Coalgebras model state-based systems such as automata, streams and transition systems.
The map describes what one can observe of a state and where it goes next.
Terminal coalgebras give infinite data such as streams, and the uniqueness of maps into them gives proofs by coinduction.

## Explanations

### L1 human

Think of $A$ as a set of states of a machine.
The coalgebra map tells, for each state, what the machine shows and what state comes next.
For example, with $F(X) = \N \times X$ on sets, take $A = \N$ and send $n$ to $(n, n + 1)$.
Starting at $0$, the machine shows $0, 1, 2, \dots$ forever, which is an infinite stream.