symmetric monoidal (∞,1)-category of spectra
The notion of cocommutative coalgebra is the formal dual of commutative algebra.
Coommutative coalgebras form the category CocommCoalg.
For $(C,\otimes, I, \sigma)$ a symmetric monoidal category, a comonoid object in $C$ is an object $A$ equipped with a morphism
(“comultiplication?”) and a morphism
(“counit”) that satisfy the coassociativity? and couniticity? equations:
This is co-commutative if we have a commuting diagram
where $\sigma_{A,A} : A \otimes A \to A \otimes A$ is the braiding isomorphism of $C$.
In the case of the symmetric monoidal category of modules over a commutative ring $R$, such an object is called a cocommutative coalgebra over $R$. In the case of the symmetric monoidal category of chain complexes (or differential graded spaces), such an object is called a DG coalgebra.
Sometimes “cocommutative coalgebra” in a symmetric monoidal category is used as a synonym for cocommutative comonoid object. We will follow that convention below. If $M$ is symmetric monoidal, then $CocommCoalg(M)$ denotes the category of cocommutative algebras and coalgebra maps in $M$.
Cocommutative coalgebras form a bridge between the doctrines of symmetric monoidal categories and cartesian monoidal categories. Notice that every object in a cartesian monoidal category carries a unique cocommutative comonoid structure (the counit is the unique map to the terminal object, and the counicity equations then force the comultiplication map to be the diagonal), and every morphism in a cartesian monoidal category is thereby a coalgebra map.
If $C$ and $D$ are two cocommutative coalgebras, then $C \otimes D$ becomes a cocommutative coalgebra with comultiplication
and counit
$C \otimes D$ is the cartesian product of $C$ and $D$ in $CocommCoalg(M)$.
The forgetful 2-functor
from cartesian monoidal categories and product-preserving functors to symmetric monoidal categories and strong symmetric monoidal functors, is coreflective. That is to say, it has a right bi-adjoint which sends $M$ to $CocommCoalg(M)$, and the unit of the 2-adjunction is an equivalence.
The 2-category of cartesian monoidal categories is comonadic (in the bicategorical sense) over the 2-category of symmetric monoidal categories.