In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions (see below).

Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes).

There are also F-coalgebras, with important applications in computer science.

One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angular momentum and spin. For this purpose, one uses the Clebsch–Gordan coefficients. Given two systems ${\displaystyle A,B}$ with angular momenta ${\displaystyle j_{A}}$ and ${\displaystyle j_{B}}$, a particularly important task is to find the total angular momentum ${\displaystyle j_{A}+j_{B}}$ given the combined state ${\displaystyle |A\rangle \otimes |B\rangle }$. This is provided by the total angular momentum operator, which extracts the needed quantity from each side of the tensor product. It can be written as an "external" tensor product

The word "external" appears here, in contrast to the "internal" tensor product of a tensor algebra. A tensor algebra comes with a tensor product (the internal one); it can also be equipped with a second tensor product, the "external" one, or the coproduct, having the form above. That they are two different products is emphasized by recalling that the internal tensor product of a vector and a scalar is just simple scalar multiplication. The external product keeps them separated. In this setting, the coproduct is the map

that takes

For this example, ${\displaystyle J}$ can be taken to be one of the spin representations of the rotation group, with the fundamental representation being the common-sense choice. This coproduct can be lifted to all of the tensor algebra, by a simple lemma that applies to free objects: the tensor algebra is a free algebra, therefore, any homomorphism defined on a subset can be extended to the entire algebra. Examining the lifting in detail, one observes that the coproduct behaves as the shuffle product, essentially because the two factors above, the left and right ${\displaystyle \mathbf {j} }$ must be kept in sequential order during products of multiple angular momenta (rotations are not commutative).

The peculiar form of having the ${\displaystyle \mathbf {j} }$ appear only once in the coproduct, rather than (for example) defining ${\displaystyle \mathbf {j} \mapsto \mathbf {j} \otimes \mathbf {j} }$ is in order to maintain linearity: for this example, (and for representation theory in general), the coproduct must be linear. As a general rule, the coproduct in representation theory is reducible; the factors are given by the Littlewood–Richardson rule. (The Littlewood–Richardson rule conveys the same idea as the Clebsch–Gordan coefficients, but in a more general setting).