nLab comonoid



Monoid theory


Category theory



A comonoid (or comonoid object) in a monoidal category \mathcal{M} is a monoid object CC in the opposite category op\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 CC 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 CC is in addition equipped with a morphism η:IC\eta \colon I \rightarrow C

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



(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.)


(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 X𝒞X \,\in\, \mathcal{C} in a cartesian monoidal category (𝒞,*,×)(\mathcal{C}, \ast, \times) becomes (see also there) a (cocommutative) comonoid by taking the

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:


(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.

(co)monad nameunderlying endofunctor(co)monad structure induced by
reader monadW(-)W \to (\text{-}) on cartesian typesunique comonoid structure on WW
coreader comonadW×(-)W \times (\text{-}) on cartesian typesunique comonoid structure on WW
writer monadA(-)A \otimes (\text{-}) on monoidal typeschosen monoid structure on AA
cowriter comonadA(-) A(-)\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})} on monoidal typeschosen monoid structure on AA

chosen comonoid structure on AA
Frobenius (co)writerA(-) A(-)\array{A \to (\text{-}) \\ A \otimes (\text{-})} on monoidal typeschosen Frobenius monoid structure

Last revised on December 20, 2023 at 07:53:32. See the history of this page for a list of all contributions to it.