symmetric monoidal (∞,1)-category of spectra
A coalgebra over an endofunctor is like a coalgebra over a comonad, but without a notion of associativity.
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.
For a category $C$ and endofunctor $F$, a coalgebra of $F$ is an object $X$ in $C$ together with a morphism $\alpha: X \to F(X)$.
Given two coalgebras $(x, \eta: x \to F x)$, $(y, \theta: y \to F y)$, a coalgebra homomorphism is a morphism $f: x \to y$ which respects the coalgebra structures:
(The object $X$ is sometimes called the carrier of the coalgebra.)
The dual concept is an algebra for an endofunctor. Both algebras and coalgebras for endofunctors on $C$ are special cases of algebras for C-C bimodules.
If $F$ 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 $Pos$ be the category of posets. Consider the endofunctor
that acts by ordinal product? with $\omega$
where the right side is given the dictionary order, not the usual product order.
The terminal coalgebra of $F_1$ is order isomorphic to the non-negative real line $\mathbb{R}^+$, with its standard order.
The real interval $[0, 1]$ may be characterized, as a topological space, as the terminal coalgebra for the functor on two-pointed topological spaces which takes a space $X$ to the space $X \vee X$. Here, $X \vee Y$, for $(X, x_-, x_+)$ and $(Y, y_-, y_+)$, is the disjoint union of $X$ and $Y$ with $x_+$ and $y_-$ identified, and $x_-$ and $y_+$ as the two base points.
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: