cotensor

The cotensor product, denoted M ☐_C N, is a construction in the theory of coalgebras and comodules. It takes a right C-comodule M and a left C-comodule N over a coalgebra C (typically over a field k) and produces a new object that encodes the compatibility between their coactions.

Definition and construction: Let ρ_M: M → M ⊗ C be the coaction on M and ρ_N: N → C ⊗ N be the coaction on N. The cotensor product M ☐_C N is the equalizer of the two maps ρ_M ⊗ id_N and id_M ⊗ ρ_N from M ⊗ N to M ⊗ C ⊗ N. Equivalently, it is the subspace of M ⊗ N consisting of finite sums ∑ m_i ⊗ n_i such that (ρ_M(m_i)) ⊗ n_i = m_i ⊗ (ρ_N(n_i)) in M ⊗ C ⊗ N. In this sense, the cotensor is the largest subobject of M ⊗ N on which the left and right coactions agree.

Universal property and functoriality: The cotensor product is a subobject of M ⊗ N and is functorial in both arguments. It satisfies the universal property of an equalizer: any map from M ⊗ N to a C-comodule T that makes the two compositions with the coactions equal factors uniquely through M ☐_C N.

Special cases and relations: If C is the base field k equipped with its trivial coalgebra structure and M, N are just k-vector spaces with the corresponding trivial comodule structures, then M ☐_C N reduces to the ordinary tensor product M ⊗_k N. More generally, cotensoring generalizes to corings over rings and appears naturally in the study of comodules, relative Hopf modules, and descent theory.

See also: tensor product of comodules, corings, coalgebras, Hopf algebras.