Algebras and modules
Model category presentations
Geometry on formal duals of algebras
A coalgebra over an endofunctor is like a coalgebra over a comonad, but without a notion of associativity.
For a category and endofunctor , a coalgebra of is an object in and a map . (The object may be called the carrier of the coalgebra)
Given two coalgebras , , a coalgebra map is a morphism which respects the coalgebra structures:
The dual concept is an algebra for an endofunctor. Both algebras and coalgebras for endofunctors on are special cases of algebras for C-C bimodules.
See also terminal coalgebra.
Coalgebras for functors on
- , the set of probability distributions on : Markov chain on .
- , the power set on : binary relation on , and also the simplest type of Kripke frames.
- : deterministic automaton.
- : nondeterministic automaton.
- , for a set of labels, : labelled binary tree.
- , for a set of labels, : labelled transition system.
See coalgebra for examples on categories of modules.
The real line as a terminal coalgebra
Let be the category of posets. Consider the endofunctor
that acts by ordinal product? with
where the right side is given the dictionary order, not the usual product order.
The terminal coalgebra of is order isomorphic to the non-negative real line , with its standard order.
This is theorem 5.1 in
The real interval may be characterized, as a topological space, as the terminal coalgebra for the functor on two-pointed topological spaces which takes a space to the space . Here, , for and , is the disjoint union of and with and identified, and and as the two base points.
There are important connections beteen the theory of coalgebras and modal logic, for which see
Here are two blog discussions of coalgebra theory: