Coalgebra ($(C,\Delta ,\epsilon )$)

A comonoid in the monoidal category of modules over a commutative ring, with tensor ${\otimes }_k$ and unit $k$.

A linear subspace $I$ of a coalgebra satisfying the displayed conditions. These are exactly the conditions for the comultiplication and counit to descend to the quotient vector space $C/I$. An ideal that is also a coideal in a bialgebra gives a quotient bialgebra.

For unital algebras over a field $A,B$, a coalgebra $C$ measures $A$ into $B$ when a linear map $p:C\to {\mathrm{Hom}}_k(A,B)$ satisfies the displayed identity and $p(c)(1_A)=\epsilon (c)1_B$. Group-like elements therefore act by unital algebra homomorphisms over a field.

A terminal measuring coalgebra: every measuring map from $C$ factors uniquely through its universal measuring map by a coalgebra homomorphism $C\to P(A,B)$. Composition of universal measurings makes $P(A,A)$ a bialgebra, with the identity map providing its unit.

An element of a coalgebra satisfying the displayed identities. A group-like element in a Hopf algebra is invertible with inverse its antipode image.

A right comodule is an object $M$ with a coaction $\rho :M\to M\otimes C$ satisfying the coassociativity and counit laws. This definition extends from coalgebras to comonoids in a monoidal category.

A natural underlying transformation of tensor bifunctors on right comodules over a field has components ${\phi }_{M,N}(u\otimes v)=\sum u_{(0)}\otimes v_{(0)}\gamma (u_{(1)},v_{(1)})$. Recover $\gamma =(\epsilon \otimes \epsilon ){\phi }_{H,H}$; colinearity and monoidal axioms impose further equations.

For a transformation from the tensor built using $n$ to the tensor built using $m$, colinearity is $m\ast \gamma =\gamma \ast n$: $\sum m(a_{(1)},b_{(1)})\gamma (a_{(2)},b_{(2)})=\sum \gamma (a_{(1)},b_{(1)})n(a_{(2)},b_{(2)})$. Scalar maps enter this convolution product for coalgebra maps via the common algebra unit.

A comonoid morphism $f:H\to K$ induces a functor $f_{\ast }$ on right comodules, replacing $\rho$ by $(1\otimes f)\rho$ and preserving underlying objects and arrows. For bimonoids, it is strict monoidal exactly when $f$ is also a monoid morphism.

A natural transformation between corestriction functors over a field has components ${\beta }_M(u)=\sum u_{(0)}\alpha (u_{(1)})$, recovered by $\alpha =\epsilon {\beta }_H$. Monoidality forces $\alpha$ to be a unital algebra homomorphism over a field; for a Hopf algebra, $\alpha S$ supplies the inverse.

The structure map of a comodule; a right coaction satisfies $(\rho \otimes 1)\rho =(1\otimes \Delta )\rho$ with coherent parentheses, and $(1\otimes \epsilon )\rho$ is the inverse right unitor.

For a coalgebra $C$ and a unital associative algebra over a commutative ring $A$, maps $f,g:C\to A$ have convolution $f\ast g=m_A(f\otimes g){\Delta }_C$, with unit $j_A{\epsilon }_C$. The two-sided convolution inverse of $1_H$ is the antipode.

The notation $\Delta h=\sum h_{(1)}\otimes h_{(2)}$ suppresses the summation index of a comultiplication; iterated indices use coassociativity. For a comodule, write $\rho (u)=\sum u_{(0)}\otimes u_{(1)}$.