Given a monoidal category and a coalgebra in denote by () the category of right (resp. left) -comodules; similarly for an algebra , denote by (resp. ) the category of left -modules (right -modules); if the monoidal category is symmetric or there is instead an appropriate distributive law, then there are extensions of this notation to bimodules, bicomodules, relative Hopf modules, entwined modules etc. e.g. for left-right relative -Hopf modules where is a -comodule algebra over a bialgebra .
Let be a commutative unital ring, and let be -linear with (in particular it has zero morphisms). Given a coalgebra in , a left -comodule , a right -comodule , their cotensor product is an object in given by
If equalizers exist in , this formula extends to a bifunctor
If is a bialgebra in and is a right -comodule algebra then the same formula defines a bifunctor
Let now be the symmetric monoidal category of -modules.
Let be another -coalgebra, with coproduct . If is flat as a -module (e.g. is a field), and a left - right -bicomodule, then the cotensor product is a -subcomodule of . In particular, under the flatness assumption, if is a surjection of coalgebras then is a left - right -bicomodule via and respectively, hence is a functor from left - to left -comodules called the induction functor for left comodules from to .
Cotensor products in noncommutative geometry appear in the role of space of sections of a associated vector bundles of quantum principal bundles (which in affine case correspond to Hopf-Galois extensions). See e.g.
For a nonaffine extension of the sections of associated quantum vector bundle, using localization theory see
In Hopf algebra theory, cotensor products appear as early as in
The authors mention that they learned the notion from Mac Lane who knew it earlier in more general contexts. An important problem is that the cotensor product of bicomodules is in general (even for ) not associative, even up to an isomorphism. Cotensor products play a prominent role in various treatments of Galois theory in noncommutative geometry; a particularly geometric approach is within a version of noncommutative algebraic geometry based on usage of monoidal categories, as sketched in