symmetric monoidal (∞,1)-category of spectra
The concept plays a role in computer science for models of state-based computation (see also monad (in computer science)). The concept of the terminal coalgebra of an endofunctor is a way of encoding coinductive types.
(The object is sometimes called the carrier of the coalgebra.)
If is equipped with the structure of a monad, then a coalgebra for it is equivalently an endomorphism in the corresponding Kleisli category. In this case the canonical monoidal category structure on endomorphisms induces a tensor product on those coalgebras.
See coalgebra for examples on categories of modules.
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.
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 connections between the theory of coalgebras and modal logic for which see
and with quantum mechanics, for which see this and
Here are two blog discussions of coalgebra theory: