Comultiplication on objects in an (abelian?) category

Looking for example at $$R$$-modules for some commutative $$R$$, we have the direct sum and the tensor product acting analogously to addition and multiplication. After studying a little bit about co-algebras and bi-algebras, I wondered if there is any way to define something that would be analogous to comultiplication in this setting.

What I might imagine is an additive functor $$C \to C \otimes C$$ when $$C$$ is an abelian category, under some suitable definition for the tensor (I looked u a little about it and I saw that there are a definitions for that, which I did not fully understand). Also, I can't really imagine what would be an analog for the counit except or maybe (the) functor $$C \to *$$.

Sure, we can define such things. Let's work in the Morita 2-category $$\text{Mor}(k)$$ over a commutative ring $$k$$, which has

• objects $$k$$-algebras $$A$$,
• morphisms $$k$$-bimodules (with composition given by tensor product), and
• 2-morphisms bimodule homomorphisms.

Equivalently, applying the forgetful functor $$\text{Hom}(k, -)$$, we can think of the Morita 2-category as having

• objects the cocomplete $$k$$-linear categories $$\text{Mod}(A)$$ of right modules over $$k$$-algebras,
• morphisms given by tensor product with a bimodule (equivalently, cocontinuous $$k$$-linear functors, by the Eilenberg-Watts theorem)
• 2-morphisms natural transformations.

The Morita 2-category is a categorified version of modules; specifically it can be thought of as a 2-category of module categories over $$\text{Mod}(k)$$, which itself can be thought of as a categorified commutative ring. It has a tensor-hom adjunction

$$\text{Hom}(A \otimes B, C) \cong \text{Hom}(A, [B, C])$$

where $$A \otimes B$$ is the ordinary tensor product over $$k$$ and $$[B, C] = B^{op} \otimes C$$ is the internal hom. This adjunction says that we can identify $$(A \otimes B, C)$$-bimodules naturally with $$(A, B^{op} \otimes C)$$-bimodules. It is moreover the case that $$\otimes$$ really deserves to be called the tensor product in this setting, in that $$\text{Mod}(A \otimes B)$$ is the universal recipient of a "bilinear" (cocontinuous and $$k$$-linear in each variable) functor out of $$\text{Mod}(A) \times \text{Mod}(B)$$.

The unit of the tensor product is $$\text{Mod}(k)$$, so we can define a "comonoidal object" in this setting to be equipped with a comultiplication $$\text{Mod}(A) \to \text{Mod}(A \otimes A)$$ (that is, an $$(A, A \otimes A)$$-bimodule) and a counit $$\text{Mod}(A) \to \text{Mod}(k)$$ (that is, a left $$A$$-module), plus an associator and stuff like that, satisfying some axioms that look like the axioms of a monoidal category. This is the dual of a sesquialgebra structure, in the very nice sense that a structure of this sort on $$A$$ is exactly a sesquialgebra structure on $$A^{op}$$.

• This seems a way less "canonical" or "natural" structure compared to direct sum and tensor product. It sure is interesting but it isn't what I had in mind. Is there some way to define a coproduct with, say, a universal property? Oct 4, 2019 at 17:47
• @Adi: the "most universal" construction I can think of is just the diagonal map $C \ni c \mapsto (c, c) \in C \times C$, which exists for any category. In any category with finite products (which here is $\text{Cat}$ itself) this is the unique comonoid structure on any object wrt the product. Oct 5, 2019 at 4:19