nLab comonoid

Definition

A comonoid (or comonoid object) in a monoidal category $\mathcal{M}$ is a monoid object $C$ in the opposite category $\mathcal{M}^{op}$ (which canonically becomes a monoidal category via the same tensor product operation as in $\mathcal{M}$).

With the usual definition of monoids as having a unit, this means that a comonoid $C$ is equipped with a counit, which in string diagram-notation for $\mathcal{M}$ is of this form:

One speaks of a unital comonoid (or unital comonoid object; internal to Vect also called an augmented coalgebra) if $C$ is in addition equipped with a morphism $\eta \colon I \rightarrow C$

which verifies the properties that the unit would verify if $C$ was a bimonoid, ie. in string diagrams:

Examples

Example

(coassociative coalgebras) A comonoid object in VectorSpaces (with its usual tensor product of vector spaces) is called a coalgebra.

(Beware though that the term “coalgebra” is overused in many ways, for instance in coalgebras for endofunctor?. To be more specific to the linear-algebraic context one can say coassociative coalgebra.)

Example

(cartesian comonoids)
Every set carries a unique structure of a comonoid in the category of Sets with respect to the usual cartesian product.

Generally, every object $X \,\in\, \mathcal{C}$ in a cartesian monoidal category $(\mathcal{C}, \ast, \times)$ becomes (see also there) a (cocommutative) comonoid by taking the

• counit to be the terminal morphism $X \xrightarrow{\exists !} \ast$

• coproduct to be the diagonal morphism $diag_X \,\colon\, X \to X \times X$.

The analogous statement remains true for cartesian monoidal (infinity,1)-categories (see there).

Obvious as Exp. may be, it plays a somewhat profound role in various contexts:

Example

(suspension coring spectra)
In the case of topological spaces or other models of classical homotopy types, and using that the suspension spectrum-construction is a strong monoidal functor, Exp. implies the remarkable fact that suspension spectra carry coring spectrum-structure via smash-monoidal diagonals.

Example

The frame of opens of any monoid internal to the category of locales is a comonoid with respect to the coproduct of frames, since the category of locales is the opposite category of the category of frames.

Example

In particular, the frame of opens of the locale of real numbers is a cocommutative comonoid with respect to the coproduct of frames in two different ways, since addition and multiplication on the Dedekind real numbers can both be extended to the locale of real numbers.

Related concepts

• cocategory

• coalgebra

• co-associativity

• co-unitality

• gs-monoidal category

(co)monad name	underlying endofunctor	(co)monad structure induced by
reader monad	$W \to (\text{-})$ on cartesian types	unique comonoid structure on $W$
coreader comonad	$W \times (\text{-})$ on cartesian types	unique comonoid structure on $W$
writer monad	$A \otimes (\text{-})$ on monoidal types	chosen monoid structure on $A$
cowriter comonad	$\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})}$ on monoidal types	chosen monoid structure on $A$ chosen comonoid structure on $A$
Frobenius (co)writer	$\array{A \to (\text{-}) \\ A \otimes (\text{-})}$ on monoidal types	chosen Frobenius monoid structure