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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 $C \to *$.

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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 $A^{op}$.

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  • $\begingroup$ 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? $\endgroup$
    – Adi Ostrov
    Oct 4, 2019 at 17:47
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    $\begingroup$ @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. $\endgroup$ Oct 5, 2019 at 4:19

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