monoid theory in algebra:
symmetric monoidal (∞,1)-category of spectra
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:
(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 $X \,\in\, \mathcal{C}$ 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:
(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 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 |
Last revised on August 25, 2023 at 16:17:34. See the history of this page for a list of all contributions to it.