In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.

Let K be a field, and C be a coalgebra over K. A (right) comodule over C is a K-vector space M together with a linear map

such that

where Δ is the comultiplication for C, and ε is the counit.

Note that in the second rule we have identified ${\displaystyle M\otimes K}$ with ${\displaystyle M\,}$.

One important result in algebraic topology is the fact that homology ${\displaystyle H_{*}(X)}$ over the dual Steenrod algebra ${\displaystyle {\mathcal {A}}^{*}}$ forms a comodule. This comes from the fact the Steenrod algebra ${\displaystyle {\mathcal {A}}}$ has a canonical action on the cohomology

${\displaystyle \mu :{\mathcal {A}}\otimes H^{*}(X)\to H^{*}(X)}$

When we dualize to the dual Steenrod algebra, this gives a comodule structure