symmetric monoidal (∞,1)-category of spectra
Given two coalgebras , , a coalgebra map is a morphism which respects the coalgebra structures:
\theta \circ f = F(f) \circ \eta
See also terminal coalgebra.
See coalgebra for examples on categories of modules.
Let be the category of posets. Consider the endofunctor
F_1 : Pos \to Pos
that acts by ordinal product? with
F_1 : X \mapsto X \cdot \omega \,,
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.
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.
This is discussed in
More information may be found at coalgebra of the real interval.
There are important connections beteen the theory of coalgebras and modal logic, for which see
Here are two blog discussions of coalgebra theory: