symmetric monoidal (∞,1)-category of spectra
Given a monoidal category $\mathcal{M}$ and a coalgebra $C$ in $\mathcal{M}$ denote by $\mathcal{M}^{C}$ ($resp. {}^{C}\mathcal{M}$) the category of right (resp. left) ${C}$-comodules; similarly for an algebra $E$, denote by ${}_E\mathcal{M}$ (resp. $\mathcal{M}_E$) the category of left E-modules (right $E$-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. Write ${}_E\mathcal{M}^B$ for left-right relative $(E,B)$-Hopf modules where $E$ is a $B$-comodule algebra over a bialgebra $B$.
Let $k$ be a commutative unital ring, and let $\mathcal{M}$ be $k$-linear (in particular it has zero morphisms).
Given a coalgebra $C$ in $\mathcal{M}$, a left $C$-comodule $(N,\rho_N \colon N\to N\otimes C)$, a right $C$-comodule $(M,\rho_M \colon M\to C\otimes M)$, their cotensor product is an object in $\mathcal{M}$ given by the kernel
If equalizers exist in $\mathcal{M}$, this formula extends to a bifunctor
If $B$ is a bialgebra in $\mathcal{M}$ and $E$ is a right $B$-comodule algebra then the same formula defines a bifunctor
Let now $\mathcal{M}=({}_k\mathrm{Mod},\otimes_k)$ be the symmetric monoidal category of $k$-modules.
Let $D$ be another $k$-coalgebra, with coproduct $\Delta_C$. If $D$ is flat as a $k$-module (e.g. $k$ is a field), and $N$ a left $D$- right $C$-bicomodule, then the cotensor product $N \Box M$ is a $D$-subcomodule of the tensor product $N \otimes_k M$. In particular, under the flatness assumption, if $\pi \colon D \rightarrow C$ is a surjection of coalgebras then $D$ is a left $D$- right $C$-bicomodule via $\Delta_D$ and $(\id \otimes \pi) \circ \Delta_D$ respectively, hence $\mathrm{Ind}^D_C \coloneqq D \Box^C -$ is a functor from left $C$- to left $D$-comodules called the induction functor for left comodules from $C$ to $D$.
Consider a commutative Hopf algebroid $\Gamma$ over $A$ (def.). Any left comodule $N$ over $\Gamma$ (def.) becomes a right comodule via the coaction
where the isomorphism in the middle the is braiding in $A Mod$ and where $c$ is the conjugation map of $\Gamma$.
Dually, a right comodule $N$ becoomes a left comodule with the coaction
Given a commutative Hopf algebroid $\Gamma$ over $A$, (def.), and given $N_1$ a right $\Gamma$-comodule and $N_2$ a left comodule (def.), then their cotensor product $N_1 \Box_\Gamma N_2$ is the kernel of the difference of the two coaction morphisms:
If both $N_1$ and $N_2$ are left comodules, then their cotensor product is the cotensor product of $N_2$ with $N_1$ regarded as a right comodule via prop. 1.
e.g. (Ravenel 86, def. A1.1.4).
Given a commutative Hopf algebroid $\Gamma$ over $A$, (def.), and given $N$ a left $\Gamma$-comodule (def.). Regard $A$ itself canonically as a right $\Gamma$-comodule Then the cotensor product
is called the primitive elements of $N$:
Given a commutative Hopf algebroid $\Gamma$ over $A$, and given $N_1, N_2$ two left $\Gamma$-comodules , then their cotensor product (def. 2) is commutative, in that there is an isomorphism
(e.g. Ravenel 86, prop. A1.1.5)
Given a commutative Hopf algebroid $\Gamma$ over $A$, and given $N_1, N_2$ two left $\Gamma$-comodules, such that $N_1$ is projective as an $A$-module, then
The morphism
gives $Hom_A(N_1,A)$ the structure of a right $\Gamma$-comodule;
The cotensor product (def. 2) with respect to this right comodule structure is isomorphic to the hom of $\Gamma$-comodules:
Hence in particular
(e.g. Ravenel 86, lemma A1.1.6)
In computing the second page of $E$-Adams spectral sequences, the second statement in lemma 1 is the key translation that makes the comodule Ext-groups on the page be equivalent to a Cotor-groups. The latter lend themselves to computation, for instance via Lambda-algebra or via the May spectral sequence.
Cotensor products in noncommutative geometry appear in the role of space of sections of 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.
A textbook account is in
An important problem is that the cotensor product of bicomodules is in general (even for $\mathcal{M}={}_k\mathrm{Mod}$) 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